Theorem omcl3g 43586

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 4645	. . . . 5 ⊢ (𝐶 ∈ {∅, 1o, 2o} → (𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o))
2		df-3o 8399	. . . . . 6 ⊢ 3o = suc 2o
3		df2o3 8405	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	3	uneq1i 4116	. . . . . . 7 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
5		df-suc 6323	. . . . . . 7 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4585	. . . . . . 7 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
7	4, 5, 6	3eqtr4i 2769	. . . . . 6 ⊢ suc 2o = {∅, 1o, 2o}
8	2, 7	eqtri 2759	. . . . 5 ⊢ 3o = {∅, 1o, 2o}
9	1, 8	eleq2s 2854	. . . 4 ⊢ (𝐶 ∈ 3o → (𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o))
10		orc 867	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On)))
11		omcl2 43585	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
12	10, 11	sylan2 593	. . . . . 6 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐶 = ∅) → (𝐴 ·o 𝐵) ∈ 𝐶)
13	12	ex 412	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 = ∅ → (𝐴 ·o 𝐵) ∈ 𝐶))
14		el1o 8422	. . . . . . . . 9 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
15		el1o 8422	. . . . . . . . 9 ⊢ (𝐵 ∈ 1o ↔ 𝐵 = ∅)
16		oveq12 7367	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = (∅ ·o ∅))
17		0elon 6372	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
18		om0 8444	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅)
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ ·o ∅) = ∅
20		0lt1o 8431	. . . . . . . . . . 11 ⊢ ∅ ∈ 1o
21	19, 20	eqeltri 2832	. . . . . . . . . 10 ⊢ (∅ ·o ∅) ∈ 1o
22	16, 21	eqeltrdi 2844	. . . . . . . . 9 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 1o)
23	14, 15, 22	syl2anb 598	. . . . . . . 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 2825	. . . . . . . 8 ⊢ (𝐶 = 1o → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 1o))
26		eleq2 2825	. . . . . . . 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 2825	. . . . . . 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 4604	. . . . . . . . . 10 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
32	31, 3	eleq2s 2854	. . . . . . . . 9 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
33		elpri 4604	. . . . . . . . . 10 ⊢ (𝐵 ∈ {∅, 1o} → (𝐵 = ∅ ∨ 𝐵 = 1o))
34	33, 3	eleq2s 2854	. . . . . . . . 9 ⊢ (𝐵 ∈ 2o → (𝐵 = ∅ ∨ 𝐵 = 1o))
35		0ex 5252	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
36	35	prid1 4719	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1o}
37	36, 19, 3	3eltr4i 2849	. . . . . . . . . . 11 ⊢ (∅ ·o ∅) ∈ 2o
38	16, 37	eqeltrdi 2844	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
39		oveq12 7367	. . . . . . . . . . 11 ⊢ ((𝐴 = 1o ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = (1o ·o ∅))
40		1on 8409	. . . . . . . . . . . . 13 ⊢ 1o ∈ On
41		om0 8444	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (1o ·o ∅) = ∅)
42	40, 41	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ·o ∅) = ∅
43	36, 42, 3	3eltr4i 2849	. . . . . . . . . . 11 ⊢ (1o ·o ∅) ∈ 2o
44	39, 43	eqeltrdi 2844	. . . . . . . . . 10 ⊢ ((𝐴 = 1o ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
45		oveq12 7367	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) = (∅ ·o 1o))
46		om0r 8466	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (∅ ·o 1o) = ∅)
47	40, 46	ax-mp 5	. . . . . . . . . . . 12 ⊢ (∅ ·o 1o) = ∅
48	36, 47, 3	3eltr4i 2849	. . . . . . . . . . 11 ⊢ (∅ ·o 1o) ∈ 2o
49	45, 48	eqeltrdi 2844	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) ∈ 2o)
50		oveq12 7367	. . . . . . . . . . 11 ⊢ ((𝐴 = 1o ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) = (1o ·o 1o))
51		1oex 8407	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
52	51	prid2 4720	. . . . . . . . . . . 12 ⊢ 1o ∈ {∅, 1o}
53		om1 8469	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
54	40, 53	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ·o 1o) = 1o
55	52, 54, 3	3eltr4i 2849	. . . . . . . . . . 11 ⊢ (1o ·o 1o) ∈ 2o
56	50, 55	eqeltrdi 2844	. . . . . . . . . 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 596	. . . . . . . 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 2825	. . . . . . . 8 ⊢ (𝐶 = 2o → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 2o))
61		eleq2 2825	. . . . . . . 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 2825	. . . . . . 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 1431	. . . 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 868	. . . . 5 ⊢ ((𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On) → (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)))
69		omcl2 43585	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
70	68, 69	sylan2 593	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)) → (𝐴 ·o 𝐵) ∈ 𝐶)
71	70	ex 412	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐵) ∈ 𝐶))
72	67, 71	jaod 859	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)) → (𝐴 ·o 𝐵) ∈ 𝐶))
73	72	imp 406	1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1541   ∈ wcel 2113   ∪ cun 3899  ∅c0 4285  {csn 4580  {cpr 4582  {ctp 4584  Oncon0 6317  suc csuc 6319  (class class class)co 7358  ωcom 7808  1_oc1o 8390  2_oc2o 8391  3_oc3o 8392   ·_o comu 8395   ↑_o coe 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680  ax-reg 9497  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-3o 8399  df-oadd 8401  df-omul 8402  df-oexp 8403

This theorem is referenced by: (None)