Theorem omv 7479

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 6557	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 +𝑜 𝑦) = (𝑥 +𝑜 𝐴))
2	1	mpteq2dv 4673	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)))
3		rdgeq1 7394	. . . 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 6105	. 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝑧))
6		fveq2 6103	. 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
7		df-omul 7452	. 2 ⊢ ·𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅)‘𝑧))
8		fvex 6113	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ V
9	5, 6, 7, 8	ovmpt2 6694	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   ↦ cmpt 4643  Oncon0 5640  ‘cfv 5804  (class class class)co 6549  reccrdg 7392   +_𝑜 coa 7444   ·_𝑜 comu 7445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-omul 7452

This theorem is referenced by:  om0  7484  omsuc  7493  onmsuc  7496  omlim  7500