Theorem oav 8494

Description: Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oav

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

Step	Hyp	Ref	Expression
1		rdgeq2 8397	. . 3 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦) = rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴))
2	1	fveq1d 6883	. 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧))
3		fveq2 6881	. 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
4		df-oadd 8455	. 2 ⊢ +o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧))
5		fvex 6894	. 2 ⊢ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵) ∈ V
6	2, 3, 4, 5	ovmpo 7572	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ↦ cmpt 5191  Oncon0 6360  suc csuc 6362  ‘cfv 6536  (class class class)co 7412  reccrdg 8394   +_o coa 8448

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-oadd 8455

This theorem is used by:  oa0  8499  oasuc  8507  onasuc  8511  oalim  8515