Theorem omcl 8523

Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. Remark 2.8 of [Schloeder] p. 5. (Contributed by NM, 3-Aug-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
omcl	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)

Proof of Theorem omcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7421	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
2	1	eleq1d 2845	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o ∅) ∈ On))
3		oveq2 7421	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
4	3	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝑦) ∈ On))
5		oveq2 7421	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
6	5	eleq1d 2845	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o suc 𝑦) ∈ On))
7		oveq2 7421	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
8	7	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝐵) ∈ On))
9		om0 8504	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
10		0elon 6413	. . . 4 ⊢ ∅ ∈ On
11	9, 10	eqeltrdi 2868	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ∈ On)
12		oacl 8522	. . . . . . 7 ⊢ (((𝐴 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On)
13	12	expcom 419	. . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ·o 𝑦) ∈ On → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
14	13	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝑦) ∈ On → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
15		omsuc 8513	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
16	15	eleq1d 2845	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o suc 𝑦) ∈ On ↔ ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
17	14, 16	sylibrd 262	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝑦) ∈ On → (𝐴 ·o suc 𝑦) ∈ On))
18	17	expcom 419	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ·o 𝑦) ∈ On → (𝐴 ·o suc 𝑦) ∈ On)))
19		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
20		iunon 8328	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On)
21	19, 20	mpan 703	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On)
22		omlim 8520	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
23	19, 22	mpanr1 716	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
24	23	eleq1d 2845	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 ·o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On))
25	21, 24	imbitrrid 249	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On → (𝐴 ·o 𝑥) ∈ On))
26	25	expcom 419	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On → (𝐴 ·o 𝑥) ∈ On)))
27	2, 4, 6, 8, 11, 18, 26	tfinds3 7861	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 ·o 𝐵) ∈ On))
28	27	impcom 413	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  ∅c0 4279  ∪ ciun 4951  Oncon0 6357  Lim wlim 6358  suc csuc 6359  (class class class)co 7413   +_o coa 8452   ·_o comu 8453

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-oadd 8459  df-omul 8460

This theorem is used by:  oecl  8524  omordi  8553  omord2  8554  omcan  8556  omword  8557  omwordri  8559  om00  8562  om00el  8563  omlimcl  8565  odi  8566  omass  8567  oneo  8568  omeulem1  8569  omeulem2  8570  omopth2  8571  oeoelem  8586  oeoe  8587  oeeui  8590  oaabs2  8637  omxpenlem  9076  omxpen  9077  cantnfle  9650  cantnflt  9651  cantnflem1d  9667  cantnflem1  9668  cantnflem3  9670  cantnflem4  9671  cnfcomlem  9678  xpnum  9956  infxpenc  10021  dfac12lem2  10147  onexomgt  44082  omlimcl2  44083  onexlimgt  44084  onexoegt  44085  oaomoecl  44119  oaabsb  44135  dflim5  44170  omabs2  44173  naddwordnexlem0  44237  naddwordnexlem1  44238  naddwordnexlem3  44240  oawordex3  44241  naddwordnexlem4  44242