Theorem addpiord 10300

Description: Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
addpiord	⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))

Proof of Theorem addpiord

Step	Hyp	Ref	Expression
1		opelxpi 5586	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		fvres 6683	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +o ‘⟨𝐴, 𝐵⟩))
3		df-ov 7153	. . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4		df-pli 10289	. . . . 5 ⊢ +N = ( +o ↾ (N × N))
5	4	fveq1i 6665	. . . 4 ⊢ ( +N ‘⟨𝐴, 𝐵⟩) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2844	. . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7		df-ov 7153	. . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘⟨𝐴, 𝐵⟩)
8	2, 6, 7	3eqtr4g 2881	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
9	1, 8	syl 17	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ⟨cop 4566   × cxp 5547   ↾ cres 5551  ‘cfv 6349  (class class class)co 7150   +_o coa 8093  Ncnpi 10260   +_N cpli 10261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-res 5561  df-iota 6308  df-fv 6357  df-ov 7153  df-pli 10289

This theorem is referenced by:  addclpi  10308  addcompi  10310  addasspi  10311  distrpi  10314  addcanpi  10315  addnidpi  10317  ltexpi  10318  ltapi  10319  1lt2pi  10321  indpi  10323