Theorem oav 8567

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 8468	. . 3 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦) = rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴))
2	1	fveq1d 6922	. 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧))
3		fveq2 6920	. 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
4		df-oadd 8526	. 2 ⊢ +o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧))
5		fvex 6933	. 2 ⊢ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵) ∈ V
6	2, 3, 4, 5	ovmpo 7610	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ↦ cmpt 5249  Oncon0 6395  suc csuc 6397  ‘cfv 6573  (class class class)co 7448  reccrdg 8465   +_o coa 8519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526

This theorem is referenced by:  oa0  8572  oasuc  8580  onasuc  8584  oalim  8588