Theorem omcl3g 44121

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 4656	. . . . 5 ⊢ (𝐶 ∈ {∅, 1o, 2o} → (𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o))
2		df-3o 8461	. . . . . 6 ⊢ 3o = suc 2o
3		df2o3 8467	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	3	uneq1i 4118	. . . . . . 7 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
5		df-suc 6370	. . . . . . 7 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4596	. . . . . . 7 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
7	4, 5, 6	3eqtr4i 2798	. . . . . 6 ⊢ suc 2o = {∅, 1o, 2o}
8	2, 7	eqtri 2788	. . . . 5 ⊢ 3o = {∅, 1o, 2o}
9	1, 8	eleq2s 2883	. . . 4 ⊢ (𝐶 ∈ 3o → (𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o))
10		orc 881	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On)))
11		omcl2 44120	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
12	10, 11	sylan2 605	. . . . . 6 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐶 = ∅) → (𝐴 ·o 𝐵) ∈ 𝐶)
13	12	ex 418	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 = ∅ → (𝐴 ·o 𝐵) ∈ 𝐶))
14		el1o 8486	. . . . . . . . 9 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
15		el1o 8486	. . . . . . . . 9 ⊢ (𝐵 ∈ 1o ↔ 𝐵 = ∅)
16		oveq12 7428	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = (∅ ·o ∅))
17		0elon 6420	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
18		om0 8508	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅)
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ ·o ∅) = ∅
20		0lt1o 8495	. . . . . . . . . . 11 ⊢ ∅ ∈ 1o
21	19, 20	eqeltri 2861	. . . . . . . . . 10 ⊢ (∅ ·o ∅) ∈ 1o
22	16, 21	eqeltrdi 2873	. . . . . . . . 9 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 1o)
23	14, 15, 22	syl2anb 610	. . . . . . . 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 2854	. . . . . . . 8 ⊢ (𝐶 = 1o → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 1o))
26		eleq2 2854	. . . . . . . 8 ⊢ (𝐶 = 1o → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 1o))
27	25, 26	anbi12d 644	. . . . . . 7 ⊢ (𝐶 = 1o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ (𝐴 ∈ 1o ∧ 𝐵 ∈ 1o)))
28		eleq2 2854	. . . . . . 7 ⊢ (𝐶 = 1o → ((𝐴 ·o 𝐵) ∈ 𝐶 ↔ (𝐴 ·o 𝐵) ∈ 1o))
29	24, 27, 28	3imtr4d 297	. . . . . 6 ⊢ (𝐶 = 1o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ·o 𝐵) ∈ 𝐶))
30	29	com12 33	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 = 1o → (𝐴 ·o 𝐵) ∈ 𝐶))
31		elpri 4615	. . . . . . . . . 10 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
32	31, 3	eleq2s 2883	. . . . . . . . 9 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
33		elpri 4615	. . . . . . . . . 10 ⊢ (𝐵 ∈ {∅, 1o} → (𝐵 = ∅ ∨ 𝐵 = 1o))
34	33, 3	eleq2s 2883	. . . . . . . . 9 ⊢ (𝐵 ∈ 2o → (𝐵 = ∅ ∨ 𝐵 = 1o))
35		0ex 5272	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
36	35	prid1 4730	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1o}
37	36, 19, 3	3eltr4i 2878	. . . . . . . . . . 11 ⊢ (∅ ·o ∅) ∈ 2o
38	16, 37	eqeltrdi 2873	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
39		oveq12 7428	. . . . . . . . . . 11 ⊢ ((𝐴 = 1o ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = (1o ·o ∅))
40		1on 8472	. . . . . . . . . . . . 13 ⊢ 1o ∈ On
41		om0 8508	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (1o ·o ∅) = ∅)
42	40, 41	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ·o ∅) = ∅
43	36, 42, 3	3eltr4i 2878	. . . . . . . . . . 11 ⊢ (1o ·o ∅) ∈ 2o
44	39, 43	eqeltrdi 2873	. . . . . . . . . 10 ⊢ ((𝐴 = 1o ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
45		oveq12 7428	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) = (∅ ·o 1o))
46		om0r 8530	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (∅ ·o 1o) = ∅)
47	40, 46	ax-mp 5	. . . . . . . . . . . 12 ⊢ (∅ ·o 1o) = ∅
48	36, 47, 3	3eltr4i 2878	. . . . . . . . . . 11 ⊢ (∅ ·o 1o) ∈ 2o
49	45, 48	eqeltrdi 2873	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) ∈ 2o)
50		oveq12 7428	. . . . . . . . . . 11 ⊢ ((𝐴 = 1o ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) = (1o ·o 1o))
51		1oelpr 8470	. . . . . . . . . . . 12 ⊢ 1o ∈ {∅, 1o}
52		om1 8533	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
53	40, 52	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ·o 1o) = 1o
54	51, 53, 3	3eltr4i 2878	. . . . . . . . . . 11 ⊢ (1o ·o 1o) ∈ 2o
55	50, 54	eqeltrdi 2873	. . . . . . . . . 10 ⊢ ((𝐴 = 1o ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) ∈ 2o)
56	38, 44, 49, 55	ccase 1053	. . . . . . . . 9 ⊢ (((𝐴 = ∅ ∨ 𝐴 = 1o) ∧ (𝐵 = ∅ ∨ 𝐵 = 1o)) → (𝐴 ·o 𝐵) ∈ 2o)
57	32, 34, 56	syl2an 608	. . . . . . . 8 ⊢ ((𝐴 ∈ 2o ∧ 𝐵 ∈ 2o) → (𝐴 ·o 𝐵) ∈ 2o)
58	57	a1i 11	. . . . . . 7 ⊢ (𝐶 = 2o → ((𝐴 ∈ 2o ∧ 𝐵 ∈ 2o) → (𝐴 ·o 𝐵) ∈ 2o))
59		eleq2 2854	. . . . . . . 8 ⊢ (𝐶 = 2o → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 2o))
60		eleq2 2854	. . . . . . . 8 ⊢ (𝐶 = 2o → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 2o))
61	59, 60	anbi12d 644	. . . . . . 7 ⊢ (𝐶 = 2o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ (𝐴 ∈ 2o ∧ 𝐵 ∈ 2o)))
62		eleq2 2854	. . . . . . 7 ⊢ (𝐶 = 2o → ((𝐴 ·o 𝐵) ∈ 𝐶 ↔ (𝐴 ·o 𝐵) ∈ 2o))
63	58, 61, 62	3imtr4d 297	. . . . . 6 ⊢ (𝐶 = 2o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ·o 𝐵) ∈ 𝐶))
64	63	com12 33	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 = 2o → (𝐴 ·o 𝐵) ∈ 𝐶))
65	13, 30, 64	3jaod 1456	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o) → (𝐴 ·o 𝐵) ∈ 𝐶))
66	9, 65	syl5 35	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 ∈ 3o → (𝐴 ·o 𝐵) ∈ 𝐶))
67		olc 882	. . . . 5 ⊢ ((𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On) → (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)))
68		omcl2 44120	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
69	67, 68	sylan2 605	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)) → (𝐴 ·o 𝐵) ∈ 𝐶)
70	69	ex 418	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐵) ∈ 𝐶))
71	66, 70	jaod 873	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)) → (𝐴 ·o 𝐵) ∈ 𝐶))
72	71	imp 412	1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2146   ∪ cun 3904  ∅c0 4286  {csn 4591  {cpr 4593  {ctp 4595  Oncon0 6364  suc csuc 6366  (class class class)co 7419  ωcom 7868  1_oc1o 8452  2_oc2o 8453  3_oc3o 8454   ·_o comu 8457   ↑_o coe 8458

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742  ax-inf2 9617

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-3o 8461  df-oadd 8463  df-omul 8464  df-oexp 8465

This theorem is used by: (None)