Theorem omcl 8460

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 7363	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
2	1	eleq1d 2818	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o ∅) ∈ On))
3		oveq2 7363	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
4	3	eleq1d 2818	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝑦) ∈ On))
5		oveq2 7363	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
6	5	eleq1d 2818	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o suc 𝑦) ∈ On))
7		oveq2 7363	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
8	7	eleq1d 2818	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝐵) ∈ On))
9		om0 8441	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
10		0elon 6369	. . . 4 ⊢ ∅ ∈ On
11	9, 10	eqeltrdi 2841	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) ∈ On)
12		oacl 8459	. . . . . . 7 ⊢ (((𝐴 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On)
13	12	expcom 413	. . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ·o 𝑦) ∈ On → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
14	13	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝑦) ∈ On → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
15		omsuc 8450	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
16	15	eleq1d 2818	. . . . 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 413	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ·o 𝑦) ∈ On → (𝐴 ·o suc 𝑦) ∈ On)))
19		vex 3441	. . . . . 6 ⊢ 𝑥 ∈ V
20		iunon 8268	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On)
21	19, 20	mpan 690	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On)
22		omlim 8457	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
23	19, 22	mpanr1 703	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦))
24	23	eleq1d 2818	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 ·o 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On))
25	21, 24	imbitrrid 246	. . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On → (𝐴 ·o 𝑥) ∈ On))
26	25	expcom 413	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 ·o 𝑦) ∈ On → (𝐴 ·o 𝑥) ∈ On)))
27	2, 4, 6, 8, 11, 18, 26	tfinds3 7804	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 ·o 𝐵) ∈ On))
28	27	impcom 407	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437  ∅c0 4282  ∪ ciun 4943  Oncon0 6314  Lim wlim 6315  suc csuc 6316  (class class class)co 7355   +_o coa 8391   ·_o comu 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-oadd 8398  df-omul 8399

This theorem is referenced by:  oecl  8461  omordi  8490  omord2  8491  omcan  8493  omword  8494  omwordri  8496  om00  8499  om00el  8500  omlimcl  8502  odi  8503  omass  8504  oneo  8505  omeulem1  8506  omeulem2  8507  omopth2  8508  oeoelem  8522  oeoe  8523  oeeui  8526  oaabs2  8573  omxpenlem  9002  omxpen  9003  cantnfle  9572  cantnflt  9573  cantnflem1d  9589  cantnflem1  9590  cantnflem3  9592  cantnflem4  9593  cnfcomlem  9600  xpnum  9855  infxpenc  9920  dfac12lem2  10047  onexomgt  43398  omlimcl2  43399  onexlimgt  43400  onexoegt  43401  oaomoecl  43435  oaabsb  43451  dflim5  43486  omabs2  43489  naddwordnexlem0  43553  naddwordnexlem1  43554  naddwordnexlem3  43556  oawordex3  43557  naddwordnexlem4  43558