Theorem oaomoencom 42749

Description: Ordinal addition, multiplication, and exponentiation do not generally commute. Theorem 4.1 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)

Assertion

Ref	Expression
oaomoencom	⊢ (∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))

Distinct variable group:   𝑎,𝑏

Proof of Theorem oaomoencom

Step	Hyp	Ref	Expression
1		oancom 9680	. . . 4 ⊢ (1o +o ω) ≠ (ω +o 1o)
2	1	neii 2938	. . 3 ⊢ ¬ (1o +o ω) = (ω +o 1o)
3		1on 8503	. . . 4 ⊢ 1o ∈ On
4		omelon 9675	. . . . 5 ⊢ ω ∈ On
5		oveq2 7432	. . . . . . . 8 ⊢ (𝑏 = ω → (1o +o 𝑏) = (1o +o ω))
6		oveq1 7431	. . . . . . . 8 ⊢ (𝑏 = ω → (𝑏 +o 1o) = (ω +o 1o))
7	5, 6	eqeq12d 2743	. . . . . . 7 ⊢ (𝑏 = ω → ((1o +o 𝑏) = (𝑏 +o 1o) ↔ (1o +o ω) = (ω +o 1o)))
8	7	notbid 317	. . . . . 6 ⊢ (𝑏 = ω → (¬ (1o +o 𝑏) = (𝑏 +o 1o) ↔ ¬ (1o +o ω) = (ω +o 1o)))
9	8	rspcev 3609	. . . . 5 ⊢ ((ω ∈ On ∧ ¬ (1o +o ω) = (ω +o 1o)) → ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o))
10	4, 9	mpan 688	. . . 4 ⊢ (¬ (1o +o ω) = (ω +o 1o) → ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o))
11		oveq1 7431	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑎 +o 𝑏) = (1o +o 𝑏))
12		oveq2 7432	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑏 +o 𝑎) = (𝑏 +o 1o))
13	11, 12	eqeq12d 2743	. . . . . . 7 ⊢ (𝑎 = 1o → ((𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ (1o +o 𝑏) = (𝑏 +o 1o)))
14	13	notbid 317	. . . . . 6 ⊢ (𝑎 = 1o → (¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ¬ (1o +o 𝑏) = (𝑏 +o 1o)))
15	14	rexbidv 3174	. . . . 5 ⊢ (𝑎 = 1o → (∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o)))
16	15	rspcev 3609	. . . 4 ⊢ ((1o ∈ On ∧ ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎))
17	3, 10, 16	sylancr 585	. . 3 ⊢ (¬ (1o +o ω) = (ω +o 1o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎))
18	2, 17	ax-mp 5	. 2 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎)
19	4, 4	pm3.2i 469	. . . . . . 7 ⊢ (ω ∈ On ∧ ω ∈ On)
20		peano1 7898	. . . . . . 7 ⊢ ∅ ∈ ω
21	19, 20	pm3.2i 469	. . . . . 6 ⊢ ((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω)
22		oaord1 8576	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ ω ∈ (ω +o ω)))
23	22	biimpa 475	. . . . . 6 ⊢ (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → ω ∈ (ω +o ω))
24		elneq 9627	. . . . . 6 ⊢ (ω ∈ (ω +o ω) → ω ≠ (ω +o ω))
25	21, 23, 24	mp2b 10	. . . . 5 ⊢ ω ≠ (ω +o ω)
26		2omomeqom 42735	. . . . . 6 ⊢ (2o ·o ω) = ω
27		df-2o 8492	. . . . . . . 8 ⊢ 2o = suc 1o
28	27	oveq2i 7435	. . . . . . 7 ⊢ (ω ·o 2o) = (ω ·o suc 1o)
29		omsuc 8551	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω ·o suc 1o) = ((ω ·o 1o) +o ω))
30	4, 3, 29	mp2an 690	. . . . . . 7 ⊢ (ω ·o suc 1o) = ((ω ·o 1o) +o ω)
31		om1 8567	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ·o 1o) = ω)
32	4, 31	ax-mp 5	. . . . . . . 8 ⊢ (ω ·o 1o) = ω
33	32	oveq1i 7434	. . . . . . 7 ⊢ ((ω ·o 1o) +o ω) = (ω +o ω)
34	28, 30, 33	3eqtri 2759	. . . . . 6 ⊢ (ω ·o 2o) = (ω +o ω)
35	26, 34	neeq12i 3003	. . . . 5 ⊢ ((2o ·o ω) ≠ (ω ·o 2o) ↔ ω ≠ (ω +o ω))
36	25, 35	mpbir 230	. . . 4 ⊢ (2o ·o ω) ≠ (ω ·o 2o)
37	36	neii 2938	. . 3 ⊢ ¬ (2o ·o ω) = (ω ·o 2o)
38		2on 8505	. . . 4 ⊢ 2o ∈ On
39		oveq2 7432	. . . . . . . 8 ⊢ (𝑏 = ω → (2o ·o 𝑏) = (2o ·o ω))
40		oveq1 7431	. . . . . . . 8 ⊢ (𝑏 = ω → (𝑏 ·o 2o) = (ω ·o 2o))
41	39, 40	eqeq12d 2743	. . . . . . 7 ⊢ (𝑏 = ω → ((2o ·o 𝑏) = (𝑏 ·o 2o) ↔ (2o ·o ω) = (ω ·o 2o)))
42	41	notbid 317	. . . . . 6 ⊢ (𝑏 = ω → (¬ (2o ·o 𝑏) = (𝑏 ·o 2o) ↔ ¬ (2o ·o ω) = (ω ·o 2o)))
43	42	rspcev 3609	. . . . 5 ⊢ ((ω ∈ On ∧ ¬ (2o ·o ω) = (ω ·o 2o)) → ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o))
44	4, 43	mpan 688	. . . 4 ⊢ (¬ (2o ·o ω) = (ω ·o 2o) → ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o))
45		oveq1 7431	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ·o 𝑏) = (2o ·o 𝑏))
46		oveq2 7432	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑏 ·o 𝑎) = (𝑏 ·o 2o))
47	45, 46	eqeq12d 2743	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ (2o ·o 𝑏) = (𝑏 ·o 2o)))
48	47	notbid 317	. . . . . 6 ⊢ (𝑎 = 2o → (¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)))
49	48	rexbidv 3174	. . . . 5 ⊢ (𝑎 = 2o → (∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)))
50	49	rspcev 3609	. . . 4 ⊢ ((2o ∈ On ∧ ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎))
51	38, 44, 50	sylancr 585	. . 3 ⊢ (¬ (2o ·o ω) = (ω ·o 2o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎))
52	37, 51	ax-mp 5	. 2 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎)
53		1onn 8665	. . . . . . 7 ⊢ 1o ∈ ω
54	21, 53	pm3.2i 469	. . . . . 6 ⊢ (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω)
55	4, 31	mp1i 13	. . . . . . 7 ⊢ ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → (ω ·o 1o) = ω)
56		omordi 8591	. . . . . . . 8 ⊢ (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → (1o ∈ ω → (ω ·o 1o) ∈ (ω ·o ω)))
57	56	imp 405	. . . . . . 7 ⊢ ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → (ω ·o 1o) ∈ (ω ·o ω))
58	55, 57	eqeltrrd 2829	. . . . . 6 ⊢ ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → ω ∈ (ω ·o ω))
59		elneq 9627	. . . . . 6 ⊢ (ω ∈ (ω ·o ω) → ω ≠ (ω ·o ω))
60	54, 58, 59	mp2b 10	. . . . 5 ⊢ ω ≠ (ω ·o ω)
61		2onn 8667	. . . . . . 7 ⊢ 2o ∈ ω
62		1oex 8501	. . . . . . . . 9 ⊢ 1o ∈ V
63	62	prid2 4770	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
64		df2o3 8499	. . . . . . . 8 ⊢ 2o = {∅, 1o}
65	63, 64	eleqtrri 2827	. . . . . . 7 ⊢ 1o ∈ 2o
66		nnoeomeqom 42744	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
67	61, 65, 66	mp2an 690	. . . . . 6 ⊢ (2o ↑o ω) = ω
68	27	oveq2i 7435	. . . . . . 7 ⊢ (ω ↑o 2o) = (ω ↑o suc 1o)
69		oesuc 8552	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω))
70	4, 3, 69	mp2an 690	. . . . . . 7 ⊢ (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω)
71		oe1 8569	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ↑o 1o) = ω)
72	4, 71	ax-mp 5	. . . . . . . 8 ⊢ (ω ↑o 1o) = ω
73	72	oveq1i 7434	. . . . . . 7 ⊢ ((ω ↑o 1o) ·o ω) = (ω ·o ω)
74	68, 70, 73	3eqtri 2759	. . . . . 6 ⊢ (ω ↑o 2o) = (ω ·o ω)
75	67, 74	neeq12i 3003	. . . . 5 ⊢ ((2o ↑o ω) ≠ (ω ↑o 2o) ↔ ω ≠ (ω ·o ω))
76	60, 75	mpbir 230	. . . 4 ⊢ (2o ↑o ω) ≠ (ω ↑o 2o)
77	76	neii 2938	. . 3 ⊢ ¬ (2o ↑o ω) = (ω ↑o 2o)
78		oveq2 7432	. . . . . . . 8 ⊢ (𝑏 = ω → (2o ↑o 𝑏) = (2o ↑o ω))
79		oveq1 7431	. . . . . . . 8 ⊢ (𝑏 = ω → (𝑏 ↑o 2o) = (ω ↑o 2o))
80	78, 79	eqeq12d 2743	. . . . . . 7 ⊢ (𝑏 = ω → ((2o ↑o 𝑏) = (𝑏 ↑o 2o) ↔ (2o ↑o ω) = (ω ↑o 2o)))
81	80	notbid 317	. . . . . 6 ⊢ (𝑏 = ω → (¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o) ↔ ¬ (2o ↑o ω) = (ω ↑o 2o)))
82	81	rspcev 3609	. . . . 5 ⊢ ((ω ∈ On ∧ ¬ (2o ↑o ω) = (ω ↑o 2o)) → ∃𝑏 ∈ On ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o))
83	4, 82	mpan 688	. . . 4 ⊢ (¬ (2o ↑o ω) = (ω ↑o 2o) → ∃𝑏 ∈ On ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o))
84		oveq1 7431	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o ↑o 𝑏))
85		oveq2 7432	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑏 ↑o 𝑎) = (𝑏 ↑o 2o))
86	84, 85	eqeq12d 2743	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ (2o ↑o 𝑏) = (𝑏 ↑o 2o)))
87	86	notbid 317	. . . . . 6 ⊢ (𝑎 = 2o → (¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o)))
88	87	rexbidv 3174	. . . . 5 ⊢ (𝑎 = 2o → (∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o)))
89	88	rspcev 3609	. . . 4 ⊢ ((2o ∈ On ∧ ∃𝑏 ∈ On ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))
90	38, 83, 89	sylancr 585	. . 3 ⊢ (¬ (2o ↑o ω) = (ω ↑o 2o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))
91	77, 90	ax-mp 5	. 2 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎)
92	18, 52, 91	3pm3.2i 1336	1 ⊢ (∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))

Syntax hints:  ¬ wn 3   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2936  ∃wrex 3066  ∅c0 4324  {cpr 4632  Oncon0 6372  suc csuc 6374  (class class class)co 7424  ωcom 7874  1_oc1o 8484  2_oc2o 8485   +_o coa 8488   ·_o comu 8489   ↑_o coe 8490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744  ax-reg 9621  ax-inf2 9670

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-2o 8492  df-oadd 8495  df-omul 8496  df-oexp 8497

This theorem is referenced by: (None)