Theorem omcl 8442

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 7350	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
2	1	eleq1d 2822	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o ∅) ∈ On))
3		oveq2 7350	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
4	3	eleq1d 2822	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝑦) ∈ On))
5		oveq2 7350	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
6	5	eleq1d 2822	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o suc 𝑦) ∈ On))
7		oveq2 7350	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
8	7	eleq1d 2822	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝐵) ∈ On))
9		om0 8423	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
10		0elon 6360	. . . 4 ⊢ ∅ ∈ On
11	9, 10	eqeltrdi 2846	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ∈ On)
12		oacl 8441	. . . . . . 7 ⊢ (((𝐴 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On)
13	12	expcom 415	. . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ·o 𝑦) ∈ On → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
14	13	adantr 482	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝑦) ∈ On → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
15		omsuc 8432	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
16	15	eleq1d 2822	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o suc 𝑦) ∈ On ↔ ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
17	14, 16	sylibrd 259	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝑦) ∈ On → (𝐴 ·o suc 𝑦) ∈ On))
18	17	expcom 415	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ·o 𝑦) ∈ On → (𝐴 ·o suc 𝑦) ∈ On)))
19		vex 3446	. . . . . 6 ⊢ 𝑥 ∈ V
20		iunon 8245	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On)
21	19, 20	mpan 688	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On)
22		omlim 8439	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
23	19, 22	mpanr1 701	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
24	23	eleq1d 2822	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 ·o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On))
25	21, 24	syl5ibr 246	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On → (𝐴 ·o 𝑥) ∈ On))
26	25	expcom 415	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On → (𝐴 ·o 𝑥) ∈ On)))
27	2, 4, 6, 8, 11, 18, 26	tfinds3 7784	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 ·o 𝐵) ∈ On))
28	27	impcom 409	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1541   ∈ wcel 2106  ∀wral 3062  Vcvv 3442  ∅c0 4274  ∪ ciun 4946  Oncon0 6307  Lim wlim 6308  suc csuc 6309  (class class class)co 7342   +_o coa 8369   ·_o comu 8370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-oadd 8376  df-omul 8377

This theorem is referenced by:  oecl  8443  omordi  8473  omord2  8474  omcan  8476  omword  8477  omwordri  8479  om00  8482  om00el  8483  omlimcl  8485  odi  8486  omass  8487  oneo  8488  omeulem1  8489  omeulem2  8490  omopth2  8491  oeoelem  8505  oeoe  8506  oeeui  8509  oaabs2  8555  omxpenlem  8943  omxpen  8944  cantnfle  9533  cantnflt  9534  cantnflem1d  9550  cantnflem1  9551  cantnflem3  9553  cantnflem4  9554  cnfcomlem  9561  xpnum  9813  infxpenc  9880  dfac12lem2  10006  omlimcl2  41361  dflim5  41365  omabs2  41367