Theorem omcl3g 43311

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 4640	. . . . 5 ⊢ (𝐶 ∈ {∅, 1o, 2o} → (𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o))
2		df-3o 8390	. . . . . 6 ⊢ 3o = suc 2o
3		df2o3 8396	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	3	uneq1i 4115	. . . . . . 7 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
5		df-suc 6313	. . . . . . 7 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4582	. . . . . . 7 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
7	4, 5, 6	3eqtr4i 2762	. . . . . 6 ⊢ suc 2o = {∅, 1o, 2o}
8	2, 7	eqtri 2752	. . . . 5 ⊢ 3o = {∅, 1o, 2o}
9	1, 8	eleq2s 2846	. . . 4 ⊢ (𝐶 ∈ 3o → (𝐶 = ∅ ∨ 𝐶 = 1o ∨ 𝐶 = 2o))
10		orc 867	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On)))
11		omcl2 43310	. . . . . . 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 8413	. . . . . . . . 9 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
15		el1o 8413	. . . . . . . . 9 ⊢ (𝐵 ∈ 1o ↔ 𝐵 = ∅)
16		oveq12 7358	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = (∅ ·o ∅))
17		0elon 6362	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
18		om0 8435	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅)
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ ·o ∅) = ∅
20		0lt1o 8422	. . . . . . . . . . 11 ⊢ ∅ ∈ 1o
21	19, 20	eqeltri 2824	. . . . . . . . . 10 ⊢ (∅ ·o ∅) ∈ 1o
22	16, 21	eqeltrdi 2836	. . . . . . . . 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 2817	. . . . . . . 8 ⊢ (𝐶 = 1o → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 1o))
26		eleq2 2817	. . . . . . . 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 2817	. . . . . . 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 4601	. . . . . . . . . 10 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
32	31, 3	eleq2s 2846	. . . . . . . . 9 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
33		elpri 4601	. . . . . . . . . 10 ⊢ (𝐵 ∈ {∅, 1o} → (𝐵 = ∅ ∨ 𝐵 = 1o))
34	33, 3	eleq2s 2846	. . . . . . . . 9 ⊢ (𝐵 ∈ 2o → (𝐵 = ∅ ∨ 𝐵 = 1o))
35		0ex 5246	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
36	35	prid1 4714	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1o}
37	36, 19, 3	3eltr4i 2841	. . . . . . . . . . 11 ⊢ (∅ ·o ∅) ∈ 2o
38	16, 37	eqeltrdi 2836	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
39		oveq12 7358	. . . . . . . . . . 11 ⊢ ((𝐴 = 1o ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = (1o ·o ∅))
40		1on 8400	. . . . . . . . . . . . 13 ⊢ 1o ∈ On
41		om0 8435	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (1o ·o ∅) = ∅)
42	40, 41	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ·o ∅) = ∅
43	36, 42, 3	3eltr4i 2841	. . . . . . . . . . 11 ⊢ (1o ·o ∅) ∈ 2o
44	39, 43	eqeltrdi 2836	. . . . . . . . . 10 ⊢ ((𝐴 = 1o ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
45		oveq12 7358	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) = (∅ ·o 1o))
46		om0r 8457	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (∅ ·o 1o) = ∅)
47	40, 46	ax-mp 5	. . . . . . . . . . . 12 ⊢ (∅ ·o 1o) = ∅
48	36, 47, 3	3eltr4i 2841	. . . . . . . . . . 11 ⊢ (∅ ·o 1o) ∈ 2o
49	45, 48	eqeltrdi 2836	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) ∈ 2o)
50		oveq12 7358	. . . . . . . . . . 11 ⊢ ((𝐴 = 1o ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) = (1o ·o 1o))
51		1oex 8398	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
52	51	prid2 4715	. . . . . . . . . . . 12 ⊢ 1o ∈ {∅, 1o}
53		om1 8460	. . . . . . . . . . . . 13 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
54	40, 53	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1o ·o 1o) = 1o
55	52, 54, 3	3eltr4i 2841	. . . . . . . . . . 11 ⊢ (1o ·o 1o) ∈ 2o
56	50, 55	eqeltrdi 2836	. . . . . . . . . 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 2817	. . . . . . . 8 ⊢ (𝐶 = 2o → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 2o))
61		eleq2 2817	. . . . . . . 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 2817	. . . . . . 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 43310	. . . . 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 1540   ∈ wcel 2109   ∪ cun 3901  ∅c0 4284  {csn 4577  {cpr 4579  {ctp 4581  Oncon0 6307  suc csuc 6309  (class class class)co 7349  ωcom 7799  1_oc1o 8381  2_oc2o 8382  3_oc3o 8383   ·_o comu 8386   ↑_o coe 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671  ax-reg 9484  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-3o 8390  df-oadd 8392  df-omul 8393  df-oexp 8394

This theorem is referenced by: (None)