Theorem addpiord 9585

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 5072	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		fvres 6117	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +𝑜 ‘⟨𝐴, 𝐵⟩))
3		df-ov 6552	. . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4		df-pli 9574	. . . . 5 ⊢ +N = ( +𝑜 ↾ (N × N))
5	4	fveq1i 6104	. . . 4 ⊢ ( +N ‘⟨𝐴, 𝐵⟩) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2632	. . 3 ⊢ (𝐴 +N 𝐵) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7		df-ov 6552	. . 3 ⊢ (𝐴 +𝑜 𝐵) = ( +𝑜 ‘⟨𝐴, 𝐵⟩)
8	2, 6, 7	3eqtr4g 2669	. 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 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549   +_𝑜 coa 7444  Ncnpi 9545   +_N cpli 9546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812  df-ov 6552  df-pli 9574

This theorem is referenced by:  addclpi  9593  addcompi  9595  addasspi  9596  distrpi  9599  addcanpi  9600  addnidpi  9602  ltexpi  9603  ltapi  9604  1lt2pi  9606  indpi  9608