Theorem omv 8511

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 7416	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 +o 𝑦) = (𝑥 +o 𝐴))
2	1	mpteq2dv 5250	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 +o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴)))
3		rdgeq1 8410	. . . 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 6893	. 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧))
6		fveq2 6891	. 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
7		df-omul 8470	. 2 ⊢ ·o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧))
8		fvex 6904	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V
9	5, 6, 7, 8	ovmpo 7567	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∅c0 4322   ↦ cmpt 5231  Oncon0 6364  ‘cfv 6543  (class class class)co 7408  reccrdg 8408   +_o coa 8462   ·_o comu 8463

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-omul 8470

This theorem is referenced by:  om0  8516  omsuc  8525  onmsuc  8528  omlim  8532