Theorem ssfzoulel 13752

Description: If a half-open integer range is a subset of a half-open range of nonnegative integers, but its lower bound is greater than or equal to the upper bound of the containing range, or its upper bound is less than or equal to 0, then its upper bound is less than or equal to its lower bound (and therefore it is actually empty). (Contributed by Alexander van der Vekens, 24-May-2018.)

Assertion

Ref	Expression
ssfzoulel	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴)))

Proof of Theorem ssfzoulel

Step	Hyp	Ref	Expression
1		simpl2 1190	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℤ)
2		simpl3 1191	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℤ)
3		zre 12586	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4		zre 12586	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
5		ltnle 11317	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
6	3, 4, 5	syl2an 595	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
7	6	3adant1 1128	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
8	7	biimpar 477	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐴 < 𝐵)
9		ssfzo12 13751	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → (0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁)))
10	1, 2, 8, 9	syl3anc 1369	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → (0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁)))
11	4	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ)
12		0red 11241	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 ∈ ℝ)
13	3	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ)
14		letr 11332	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ 0 ∧ 0 ≤ 𝐴) → 𝐵 ≤ 𝐴))
15	11, 12, 13, 14	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 0 ∧ 0 ≤ 𝐴) → 𝐵 ≤ 𝐴))
16	15	expcomd 416	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (𝐵 ≤ 0 → 𝐵 ≤ 𝐴)))
17	16	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≤ 𝐴) → (𝐵 ≤ 0 → 𝐵 ≤ 𝐴))
18	17	con3d 152	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≤ 𝐴) → (¬ 𝐵 ≤ 𝐴 → ¬ 𝐵 ≤ 0))
19	18	ex 412	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (¬ 𝐵 ≤ 𝐴 → ¬ 𝐵 ≤ 0)))
20	19	3adant1 1128	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (¬ 𝐵 ≤ 𝐴 → ¬ 𝐵 ≤ 0)))
21	20	com23 86	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (¬ 𝐵 ≤ 𝐴 → (0 ≤ 𝐴 → ¬ 𝐵 ≤ 0)))
22	21	imp 406	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → (0 ≤ 𝐴 → ¬ 𝐵 ≤ 0))
23		nn0re 12505	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
24	4, 23, 3	3anim123i 1149	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
25	24	3coml 1125	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
26		letr 11332	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ 𝑁 ∧ 𝑁 ≤ 𝐴) → 𝐵 ≤ 𝐴))
27	25, 26	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 𝑁 ∧ 𝑁 ≤ 𝐴) → 𝐵 ≤ 𝐴))
28	27	expdimp 452	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐵 ≤ 𝑁) → (𝑁 ≤ 𝐴 → 𝐵 ≤ 𝐴))
29	28	con3d 152	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐵 ≤ 𝑁) → (¬ 𝐵 ≤ 𝐴 → ¬ 𝑁 ≤ 𝐴))
30	29	impancom 451	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → (𝐵 ≤ 𝑁 → ¬ 𝑁 ≤ 𝐴))
31	22, 30	anim12d 608	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁) → (¬ 𝐵 ≤ 0 ∧ ¬ 𝑁 ≤ 𝐴)))
32		ioran 982	. . . . . . . 8 ⊢ (¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) ↔ (¬ 𝑁 ≤ 𝐴 ∧ ¬ 𝐵 ≤ 0))
33	32	biancomi 462	. . . . . . 7 ⊢ (¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) ↔ (¬ 𝐵 ≤ 0 ∧ ¬ 𝑁 ≤ 𝐴))
34	31, 33	imbitrrdi 251	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁) → ¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0)))
35	10, 34	syld 47	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → ¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0)))
36	35	con2d 134	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ¬ (𝐴..^𝐵) ⊆ (0..^𝑁)))
37	36	impancom 451	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0)) → (¬ 𝐵 ≤ 𝐴 → ¬ (𝐴..^𝐵) ⊆ (0..^𝑁)))
38	37	con4d 115	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0)) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴))
39	38	ex 412	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   ∈ wcel 2099   ⊆ wss 3944   class class class wbr 5142  (class class class)co 7414  ℝcr 11131  0cc0 11132   < clt 11272   ≤ cle 11273  ℕ₀cn0 12496  ℤcz 12582  ..^cfzo 13653

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-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209

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-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-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-n0 12497  df-z 12583  df-uz 12847  df-fz 13511  df-fzo 13654

This theorem is referenced by:  swrdnd2  14631