Theorem txsconn 34787

Description: The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)

Assertion

Ref	Expression
txsconn	⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)

Proof of Theorem txsconn

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sconnpconn 34773	. . 3 ⊢ (𝑅 ∈ SConn → 𝑅 ∈ PConn)
2		sconnpconn 34773	. . 3 ⊢ (𝑆 ∈ SConn → 𝑆 ∈ PConn)
3		txpconn 34778	. . 3 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
4	1, 2, 3	syl2an 595	. 2 ⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ PConn)
5		simpll 766	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ SConn)
6		simprl 770	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝑅 ×t 𝑆)))
7		sconntop 34774	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ SConn → 𝑅 ∈ Top)
8	7	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ Top)
9		eqid 2727	. . . . . . . . . . . . 13 ⊢ ∪ 𝑅 = ∪ 𝑅
10	9	toptopon 22806	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
11	8, 10	sylib 217	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ (TopOn‘∪ 𝑅))
12		sconntop 34774	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ SConn → 𝑆 ∈ Top)
13	12	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ Top)
14		eqid 2727	. . . . . . . . . . . . 13 ⊢ ∪ 𝑆 = ∪ 𝑆
15	14	toptopon 22806	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
16	13, 15	sylib 217	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ (TopOn‘∪ 𝑆))
17		tx1cn 23500	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
18	11, 16, 17	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
19		cnco 23157	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅))
20	6, 18, 19	syl2anc 583	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅))
21		simprr 772	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) = (𝑓‘1))
22	21	fveq2d 6895	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
23		iitopon 24786	. . . . . . . . . . . . 13 ⊢ II ∈ (TopOn‘(0[,]1))
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → II ∈ (TopOn‘(0[,]1)))
25		txtopon 23482	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
26	11, 16, 25	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
27		cnf2 23140	. . . . . . . . . . . 12 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)) ∧ 𝑓 ∈ (II Cn (𝑅 ×t 𝑆))) → 𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆))
28	24, 26, 6, 27	syl3anc 1369	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆))
29		0elunit 13470	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,]1)
30		fvco3 6991	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
31	28, 29, 30	sylancl 585	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
32		1elunit 13471	. . . . . . . . . . 11 ⊢ 1 ∈ (0[,]1)
33		fvco3 6991	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
34	28, 32, 33	sylancl 585	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
35	22, 31, 34	3eqtr4d 2777	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1))
36		sconnpht 34775	. . . . . . . . 9 ⊢ ((𝑅 ∈ SConn ∧ ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅) ∧ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1)) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))
37	5, 20, 35, 36	syl3anc 1369	. . . . . . . 8 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))
38		isphtpc 24907	. . . . . . . 8 ⊢ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ↔ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅) ∧ ((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑅) ∧ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
39	37, 38	sylib 217	. . . . . . 7 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅) ∧ ((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑅) ∧ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
40	39	simp3d 1142	. . . . . 6 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅)
41		n0 4342	. . . . . 6 ⊢ ((((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
42	40, 41	sylib 217	. . . . 5 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ∃𝑔 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
43		simplr 768	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ SConn)
44		tx2cn 23501	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
45	11, 16, 44	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
46		cnco 23157	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆))
47	6, 45, 46	syl2anc 583	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆))
48	21	fveq2d 6895	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
49		fvco3 6991	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
50	28, 29, 49	sylancl 585	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
51		fvco3 6991	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
52	28, 32, 51	sylancl 585	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
53	48, 50, 52	3eqtr4d 2777	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1))
54		sconnpht 34775	. . . . . . . . 9 ⊢ ((𝑆 ∈ SConn ∧ ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆) ∧ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1)) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))
55	43, 47, 53, 54	syl3anc 1369	. . . . . . . 8 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))
56		isphtpc 24907	. . . . . . . 8 ⊢ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ↔ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆) ∧ ((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑆) ∧ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
57	55, 56	sylib 217	. . . . . . 7 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆) ∧ ((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑆) ∧ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
58	57	simp3d 1142	. . . . . 6 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅)
59		n0 4342	. . . . . 6 ⊢ ((((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅ ↔ ∃ℎ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
60	58, 59	sylib 217	. . . . 5 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ∃ℎ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
61		exdistrv 1952	. . . . . 6 ⊢ (∃𝑔∃ℎ(𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))) ↔ (∃𝑔 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ∃ℎ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))))
62	8	adantr 480	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑅 ∈ Top)
63	13	adantr 480	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑆 ∈ Top)
64	6	adantr 480	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑓 ∈ (II Cn (𝑅 ×t 𝑆)))
65		eqid 2727	. . . . . . . . 9 ⊢ ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)
66		eqid 2727	. . . . . . . . 9 ⊢ ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)
67		simprl 770	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
68		simprr 772	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
69	62, 63, 64, 65, 66, 67, 68	txsconnlem 34786	. . . . . . . 8 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)}))
70	69	ex 412	. . . . . . 7 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
71	70	exlimdvv 1930	. . . . . 6 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (∃𝑔∃ℎ(𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
72	61, 71	biimtrrid 242	. . . . 5 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((∃𝑔 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ∃ℎ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
73	42, 60, 72	mp2and 698	. . . 4 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)}))
74	73	expr 456	. . 3 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ 𝑓 ∈ (II Cn (𝑅 ×t 𝑆))) → ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
75	74	ralrimiva 3141	. 2 ⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → ∀𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
76		issconn 34772	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ SConn ↔ ((𝑅 ×t 𝑆) ∈ PConn ∧ ∀𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)}))))
77	4, 75, 76	sylanbrc 582	1 ⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2935  ∀wral 3056  ∅c0 4318  {csn 4624  ∪ cuni 4903   class class class wbr 5142   × cxp 5670   ↾ cres 5674   ∘ ccom 5676  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986  0cc0 11130  1c1 11131  [,]cicc 13351  Topctop 22782  TopOnctopon 22799   Cn ccn 23115   ×_t ctx 23451  IIcii 24782  PHtpycphtpy 24881   ≃_phcphtpc 24882  PConncpconn 34765  SConncsconn 34766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9457  df-inf 9458  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-q 12955  df-rp 12999  df-xneg 13116  df-xadd 13117  df-xmul 13118  df-icc 13355  df-seq 13991  df-exp 14051  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-topgen 17416  df-psmet 21258  df-xmet 21259  df-met 21260  df-bl 21261  df-mopn 21262  df-top 22783  df-topon 22800  df-bases 22836  df-cn 23118  df-cnp 23119  df-tx 23453  df-ii 24784  df-htpy 24883  df-phtpy 24884  df-phtpc 24905  df-pconn 34767  df-sconn 34768

This theorem is referenced by: (None)