Theorem omcl 7771

Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (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 6802	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
2	1	eleq1d 2835	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 ∅) ∈ On))
3		oveq2 6802	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
4	3	eleq1d 2835	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 𝑦) ∈ On))
5		oveq2 6802	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
6	5	eleq1d 2835	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 suc 𝑦) ∈ On))
7		oveq2 6802	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
8	7	eleq1d 2835	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 𝐵) ∈ On))
9		om0 7752	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
10		0elon 5922	. . . 4 ⊢ ∅ ∈ On
11	9, 10	syl6eqel 2858	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) ∈ On)
12		oacl 7770	. . . . . . 7 ⊢ (((𝐴 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On)
13	12	expcom 398	. . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ·𝑜 𝑦) ∈ On → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On))
14	13	adantr 466	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ∈ On → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On))
15		omsuc 7761	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
16	15	eleq1d 2835	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 suc 𝑦) ∈ On ↔ ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On))
17	14, 16	sylibrd 249	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 suc 𝑦) ∈ On))
18	17	expcom 398	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 suc 𝑦) ∈ On)))
19		vex 3354	. . . . . 6 ⊢ 𝑥 ∈ V
20		iunon 7590	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On)
21	19, 20	mpan 664	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On)
22		omlim 7768	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦))
23	19, 22	mpanr1 677	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦))
24	23	eleq1d 2835	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On))
25	21, 24	syl5ibr 236	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 𝑥) ∈ On))
26	25	expcom 398	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 𝑥) ∈ On)))
27	2, 4, 6, 8, 11, 18, 26	tfinds3 7212	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 ·𝑜 𝐵) ∈ On))
28	27	impcom 394	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351  ∅c0 4064  ∪ ciun 4655  Oncon0 5867  Lim wlim 5868  suc csuc 5869  (class class class)co 6794   +_𝑜 coa 7711   ·_𝑜 comu 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-oadd 7718  df-omul 7719

This theorem is referenced by:  oecl  7772  omordi  7801  omord2  7802  omcan  7804  omword  7805  omwordri  7807  om00  7810  om00el  7811  omlimcl  7813  odi  7814  omass  7815  oneo  7816  omeulem1  7817  omeulem2  7818  omopth2  7819  oeoelem  7833  oeoe  7834  oeeui  7837  oaabs2  7880  omxpenlem  8218  omxpen  8219  cantnfle  8733  cantnflt  8734  cantnflem1d  8750  cantnflem1  8751  cantnflem3  8753  cantnflem4  8754  cnfcomlem  8761  xpnum  8978  infxpenc  9042  dfac12lem2  9169