Theorem addpiord 10801

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 5662	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		fvres 6854	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +o ‘⟨𝐴, 𝐵⟩))
3		df-ov 7364	. . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4		df-pli 10790	. . . . 5 ⊢ +N = ( +o ↾ (N × N))
5	4	fveq1i 6836	. . . 4 ⊢ ( +N ‘⟨𝐴, 𝐵⟩) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2760	. . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7		df-ov 7364	. . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘⟨𝐴, 𝐵⟩)
8	2, 6, 7	3eqtr4g 2797	. 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 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4574   × cxp 5623   ↾ cres 5627  ‘cfv 6493  (class class class)co 7361   +_o coa 8396  Ncnpi 10761   +_N cpli 10762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-res 5637  df-iota 6449  df-fv 6501  df-ov 7364  df-pli 10790

This theorem is referenced by:  addclpi  10809  addcompi  10811  addasspi  10812  distrpi  10815  addcanpi  10816  addnidpi  10818  ltexpi  10819  ltapi  10820  1lt2pi  10822  indpi  10824