Theorem addpiord 7673

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

Assertion

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

Proof of Theorem addpiord

Step	Hyp	Ref	Expression
1		opelxpi 4801	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		fvres 5714	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +o ‘⟨𝐴, 𝐵⟩))
3		df-ov 6078	. . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4		df-pli 7662	. . . . 5 ⊢ +N = ( +o ↾ (N × N))
5	4	fveq1i 5691	. . . 4 ⊢ ( +N ‘⟨𝐴, 𝐵⟩) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2259	. . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7		df-ov 6078	. . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘⟨𝐴, 𝐵⟩)
8	2, 6, 7	3eqtr4g 2296	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
9	1, 8	syl 14	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ⟨cop 3708   × cxp 4767   ↾ cres 4771  ‘cfv 5372  (class class class)co 6075   +_o coa 6674  Ncnpi 7629   +_N cpli 7630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-res 4781  df-iota 5332  df-fv 5380  df-ov 6078  df-pli 7662

This theorem is referenced by:  addclpi  7684  addcompig  7686  addasspig  7687  distrpig  7690  addcanpig  7691  addnidpig  7693  ltexpi  7694  ltapig  7695  1lt2pi  7697  indpi  7699  archnqq  7774  prarloclemarch2  7776  nqnq0a  7811