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 7424	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
2	1	eleq1d 2850	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o ∅) ∈ On))
3		oveq2 7424	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
4	3	eleq1d 2850	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝑦) ∈ On))
5		oveq2 7424	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
6	5	eleq1d 2850	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o suc 𝑦) ∈ On))
7		oveq2 7424	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
8	7	eleq1d 2850	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝐵) ∈ On))
9		om0 8504	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
10		0elon 6420	. . . 4 ⊢ ∅ ∈ On
11	9, 10	eqeltrdi 2873	. . 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 2850	. . . . 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 3461	. . . . . 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 2850	. . . . 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 7863	. 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 2146  ∀wral 3081  Vcvv 3457  ∅c0 4286  ∪ ciun 4958  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7416   +_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 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 7738

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-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-op 4598  df-uni 4875  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 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-2nd 7989  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  9069  omxpen  9070  cantnfle  9643  cantnflt  9644  cantnflem1d  9660  cantnflem1  9661  cantnflem3  9663  cantnflem4  9664  cnfcomlem  9671  xpnum  9949  infxpenc  10014  dfac12lem2  10140  onexomgt  44001  omlimcl2  44002  onexlimgt  44003  onexoegt  44004  oaomoecl  44038  oaabsb  44054  dflim5  44089  omabs2  44092  naddwordnexlem0  44156  naddwordnexlem1  44157  naddwordnexlem3  44159  oawordex3  44160  naddwordnexlem4  44161