Theorem omv 7537

Description: Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)

Assertion

Ref	Expression
omv	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem omv

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

Step	Hyp	Ref	Expression
1		oveq2 6612	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 +𝑜 𝑦) = (𝑥 +𝑜 𝐴))
2	1	mpteq2dv 4705	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)))
3		rdgeq1 7452	. . . 4 ⊢ ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅) = rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅))
4	2, 3	syl 17	. . 3 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅) = rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅))
5	4	fveq1d 6150	. 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝑧))
6		fveq2 6148	. 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
7		df-omul 7510	. 2 ⊢ ·𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅)‘𝑧))
8		fvex 6158	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ V
9	5, 6, 7, 8	ovmpt2 6749	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ∅c0 3891   ↦ cmpt 4673  Oncon0 5682  ‘cfv 5847  (class class class)co 6604  reccrdg 7450   +_𝑜 coa 7502   ·_𝑜 comu 7503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-omul 7510

This theorem is referenced by:  om0  7542  omsuc  7551  onmsuc  7554  omlim  7558