Theorem omcl3g 42084

Description: Closure law for ordinal multiplication. (Contributed by RP, 14-Jan-2025.)

Assertion

Ref	Expression
omcl3g	⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)

Proof of Theorem omcl3g

Step	Hyp	Ref	Expression
1		eltpi 4692	. . . . 5 ⊢ (𝐶 ∈ {∅, 1o, 2o} → (𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o))
2		df-3o 8468	. . . . . 6 ⊢ 3o = suc 2o
3		df2o3 8474	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	3	uneq1i 4160	. . . . . . 7 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
5		df-suc 6371	. . . . . . 7 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4634	. . . . . . 7 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
7	4, 5, 6	3eqtr4i 2771	. . . . . 6 ⊢ suc 2o = {∅, 1o, 2o}
8	2, 7	eqtri 2761	. . . . 5 ⊢ 3o = {∅, 1o, 2o}
9	1, 8	eleq2s 2852	. . . 4 ⊢ (𝐶 ∈ 3o → (𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o))
10		orc 866	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On)))
11		omcl2 42083	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
12	10, 11	sylan2 594	. . . . . 6 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐶 = ∅) → (𝐴 ·o 𝐵) ∈ 𝐶)
13	12	ex 414	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 = ∅ → (𝐴 ·o 𝐵) ∈ 𝐶))
14		el1o 8495	. . . . . . . . 9 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
15		el1o 8495	. . . . . . . . 9 ⊢ (𝐵 ∈ 1o ↔ 𝐵 = ∅)
16		oveq12 7418	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = (∅ ·o ∅))
17		0elon 6419	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
18		om0 8517	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅)
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ ·o ∅) = ∅
20		0lt1o 8504	. . . . . . . . . . 11 ⊢ ∅ ∈ 1o
21	19, 20	eqeltri 2830	. . . . . . . . . 10 ⊢ (∅ ·o ∅) ∈ 1o
22	16, 21	eqeltrdi 2842	. . . . . . . . 9 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 1o)
23	14, 15, 22	syl2anb 599	. . . . . . . 8 ⊢ ((𝐴 ∈ 1o ∧ 𝐵 ∈ 1o) → (𝐴 ·o 𝐵) ∈ 1o)
24	23	a1i 11	. . . . . . 7 ⊢ (𝐶 = 1o → ((𝐴 ∈ 1o ∧ 𝐵 ∈ 1o) → (𝐴 ·o 𝐵) ∈ 1o))
25		eleq2 2823	. . . . . . . 8 ⊢ (𝐶 = 1o → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 1o))
26		eleq2 2823	. . . . . . . 8 ⊢ (𝐶 = 1o → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 1o))
27	25, 26	anbi12d 632	. . . . . . 7 ⊢ (𝐶 = 1o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ (𝐴 ∈ 1o ∧ 𝐵 ∈ 1o)))
28		eleq2 2823	. . . . . . 7 ⊢ (𝐶 = 1o → ((𝐴 ·o 𝐵) ∈ 𝐶 ↔ (𝐴 ·o 𝐵) ∈ 1o))
29	24, 27, 28	3imtr4d 294	. . . . . 6 ⊢ (𝐶 = 1o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ·o 𝐵) ∈ 𝐶))
30	29	com12 32	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 = 1o → (𝐴 ·o 𝐵) ∈ 𝐶))
31		elpri 4651	. . . . . . . . . 10 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
32	31, 3	eleq2s 2852	. . . . . . . . 9 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
33		elpri 4651	. . . . . . . . . 10 ⊢ (𝐵 ∈ {∅, 1o} → (𝐵 = ∅ ∨ 𝐵 = 1o))
34	33, 3	eleq2s 2852	. . . . . . . . 9 ⊢ (𝐵 ∈ 2o → (𝐵 = ∅ ∨ 𝐵 = 1o))
35		0ex 5308	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
36	35	prid1 4767	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1o}
37	36, 19, 3	3eltr4i 2847	. . . . . . . . . . 11 ⊢ (∅ ·o ∅) ∈ 2o
38	16, 37	eqeltrdi 2842	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
39		oveq12 7418	. . . . . . . . . . 11 ⊢ ((𝐴 = 1o ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = (1o ·o ∅))
40		1on 8478	. . . . . . . . . . . . 13 ⊢ 1o ∈ On
41		om0 8517	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (1o ·o ∅) = ∅)
42	40, 41	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ·o ∅) = ∅
43	36, 42, 3	3eltr4i 2847	. . . . . . . . . . 11 ⊢ (1o ·o ∅) ∈ 2o
44	39, 43	eqeltrdi 2842	. . . . . . . . . 10 ⊢ ((𝐴 = 1o ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
45		oveq12 7418	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) = (∅ ·o 1o))
46		om0r 8539	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (∅ ·o 1o) = ∅)
47	40, 46	ax-mp 5	. . . . . . . . . . . 12 ⊢ (∅ ·o 1o) = ∅
48	36, 47, 3	3eltr4i 2847	. . . . . . . . . . 11 ⊢ (∅ ·o 1o) ∈ 2o
49	45, 48	eqeltrdi 2842	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) ∈ 2o)
50		oveq12 7418	. . . . . . . . . . 11 ⊢ ((𝐴 = 1o ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) = (1o ·o 1o))
51		1oex 8476	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
52	51	prid2 4768	. . . . . . . . . . . 12 ⊢ 1o ∈ {∅, 1o}
53		om1 8542	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
54	40, 53	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ·o 1o) = 1o
55	52, 54, 3	3eltr4i 2847	. . . . . . . . . . 11 ⊢ (1o ·o 1o) ∈ 2o
56	50, 55	eqeltrdi 2842	. . . . . . . . . 10 ⊢ ((𝐴 = 1o ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) ∈ 2o)
57	38, 44, 49, 56	ccase 1037	. . . . . . . . 9 ⊢ (((𝐴 = ∅ ∨ 𝐴 = 1o) ∧ (𝐵 = ∅ ∨ 𝐵 = 1o)) → (𝐴 ·o 𝐵) ∈ 2o)
58	32, 34, 57	syl2an 597	. . . . . . . 8 ⊢ ((𝐴 ∈ 2o ∧ 𝐵 ∈ 2o) → (𝐴 ·o 𝐵) ∈ 2o)
59	58	a1i 11	. . . . . . 7 ⊢ (𝐶 = 2o → ((𝐴 ∈ 2o ∧ 𝐵 ∈ 2o) → (𝐴 ·o 𝐵) ∈ 2o))
60		eleq2 2823	. . . . . . . 8 ⊢ (𝐶 = 2o → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 2o))
61		eleq2 2823	. . . . . . . 8 ⊢ (𝐶 = 2o → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 2o))
62	60, 61	anbi12d 632	. . . . . . 7 ⊢ (𝐶 = 2o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ (𝐴 ∈ 2o ∧ 𝐵 ∈ 2o)))
63		eleq2 2823	. . . . . . 7 ⊢ (𝐶 = 2o → ((𝐴 ·o 𝐵) ∈ 𝐶 ↔ (𝐴 ·o 𝐵) ∈ 2o))
64	59, 62, 63	3imtr4d 294	. . . . . 6 ⊢ (𝐶 = 2o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ·o 𝐵) ∈ 𝐶))
65	64	com12 32	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 = 2o → (𝐴 ·o 𝐵) ∈ 𝐶))
66	13, 30, 65	3jaod 1429	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o) → (𝐴 ·o 𝐵) ∈ 𝐶))
67	9, 66	syl5 34	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 ∈ 3o → (𝐴 ·o 𝐵) ∈ 𝐶))
68		olc 867	. . . . 5 ⊢ ((𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On) → (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)))
69		omcl2 42083	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
70	68, 69	sylan2 594	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)) → (𝐴 ·o 𝐵) ∈ 𝐶)
71	70	ex 414	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐵) ∈ 𝐶))
72	67, 71	jaod 858	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)) → (𝐴 ·o 𝐵) ∈ 𝐶))
73	72	imp 408	1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 846   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107   ∪ cun 3947  ∅c0 4323  {csn 4629  {cpr 4631  {ctp 4633  Oncon0 6365  suc csuc 6367  (class class class)co 7409  ωcom 7855  1_oc1o 8459  2_oc2o 8460  3_oc3o 8461   ·_o comu 8464   ↑_o coe 8465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-3o 8468  df-oadd 8470  df-omul 8471  df-oexp 8472

This theorem is referenced by: (None)