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Theorem addpiord 9650
Description: Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addpiord ((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵))

Proof of Theorem addpiord
StepHypRef Expression
1 opelxpi 5108 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 6164 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +𝑜 ‘⟨𝐴, 𝐵⟩))
3 df-ov 6607 . . . 4 (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4 df-pli 9639 . . . . 5 +N = ( +𝑜 ↾ (N × N))
54fveq1i 6149 . . . 4 ( +N ‘⟨𝐴, 𝐵⟩) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2643 . . 3 (𝐴 +N 𝐵) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 6607 . . 3 (𝐴 +𝑜 𝐵) = ( +𝑜 ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2680 . 2 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵))
91, 8syl 17 1 ((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cop 4154   × cxp 5072  cres 5076  cfv 5847  (class class class)co 6604   +𝑜 coa 7502  Ncnpi 9610   +N cpli 9611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-res 5086  df-iota 5810  df-fv 5855  df-ov 6607  df-pli 9639
This theorem is referenced by:  addclpi  9658  addcompi  9660  addasspi  9661  distrpi  9664  addcanpi  9665  addnidpi  9667  ltexpi  9668  ltapi  9669  1lt2pi  9671  indpi  9673
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