Theorem omv 8437

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 7364	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 +o 𝑦) = (𝑥 +o 𝐴))
2	1	mpteq2dv 5166	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 +o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴)))
3		rdgeq1 8340	. . . 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 6829	. 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧))
6		fveq2 6827	. 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
7		df-omul 8400	. 2 ⊢ ·o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧))
8		fvex 6840	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V
9	5, 6, 7, 8	ovmpo 7516	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ∅c0 4261   ↦ cmpt 5153  Oncon0 6310  ‘cfv 6485  (class class class)co 7356  reccrdg 8338   +_o coa 8392   ·_o comu 8393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-omul 8400

This theorem is referenced by:  om0  8442  omsuc  8451  onmsuc  8454  omlim  8458