Theorem omv 8503

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 7427	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 +o 𝑦) = (𝑥 +o 𝐴))
2	1	mpteq2dv 5207	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 +o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴)))
3		rdgeq1 8404	. . . 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 18	. . 3 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅) = rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅))
5	4	fveq1d 6887	. 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧))
6		fveq2 6885	. 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
7		df-omul 8464	. 2 ⊢ ·o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝑦)), ∅)‘𝑧))
8		fvex 6898	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ V
9	5, 6, 7, 8	ovmpo 7579	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ∅c0 4286   ↦ cmpt 5194  Oncon0 6364  ‘cfv 6540  (class class class)co 7419  reccrdg 8402   +_o coa 8456   ·_o comu 8457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-omul 8464

This theorem is used by:  om0  8508  omsuc  8517  onmsuc  8520  omlim  8524