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Theorem addpiord 9918
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 5305 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 6369 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +𝑜 ‘⟨𝐴, 𝐵⟩))
3 df-ov 6817 . . . 4 (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4 df-pli 9907 . . . . 5 +N = ( +𝑜 ↾ (N × N))
54fveq1i 6354 . . . 4 ( +N ‘⟨𝐴, 𝐵⟩) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2782 . . 3 (𝐴 +N 𝐵) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 6817 . . 3 (𝐴 +𝑜 𝐵) = ( +𝑜 ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2819 . 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 383   = wceq 1632  wcel 2139  cop 4327   × cxp 5264  cres 5268  cfv 6049  (class class class)co 6814   +𝑜 coa 7727  Ncnpi 9878   +N cpli 9879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-res 5278  df-iota 6012  df-fv 6057  df-ov 6817  df-pli 9907
This theorem is referenced by:  addclpi  9926  addcompi  9928  addasspi  9929  distrpi  9932  addcanpi  9933  addnidpi  9935  ltexpi  9936  ltapi  9937  1lt2pi  9939  indpi  9941
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