Theorem addpiord 9706

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 5148	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		fvres 6207	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +𝑜 ‘⟨𝐴, 𝐵⟩))
3		df-ov 6653	. . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4		df-pli 9695	. . . . 5 ⊢ +N = ( +𝑜 ↾ (N × N))
5	4	fveq1i 6192	. . . 4 ⊢ ( +N ‘⟨𝐴, 𝐵⟩) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2644	. . 3 ⊢ (𝐴 +N 𝐵) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7		df-ov 6653	. . 3 ⊢ (𝐴 +𝑜 𝐵) = ( +𝑜 ‘⟨𝐴, 𝐵⟩)
8	2, 6, 7	3eqtr4g 2681	. 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 384   = wceq 1483   ∈ wcel 1990  ⟨cop 4183   × cxp 5112   ↾ cres 5116  ‘cfv 5888  (class class class)co 6650   +_𝑜 coa 7557  Ncnpi 9666   +_N cpli 9667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896  df-ov 6653  df-pli 9695

This theorem is referenced by:  addclpi  9714  addcompi  9716  addasspi  9717  distrpi  9720  addcanpi  9721  addnidpi  9723  ltexpi  9724  ltapi  9725  1lt2pi  9727  indpi  9729