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Theorem addpiord 10881
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 5713 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 6910 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +o ‘⟨𝐴, 𝐵⟩))
3 df-ov 7414 . . . 4 (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4 df-pli 10870 . . . . 5 +N = ( +o ↾ (N × N))
54fveq1i 6892 . . . 4 ( +N ‘⟨𝐴, 𝐵⟩) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2760 . . 3 (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 7414 . . 3 (𝐴 +o 𝐵) = ( +o ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2797 . 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 396   = wceq 1541  wcel 2106  cop 4634   × cxp 5674  cres 5678  cfv 6543  (class class class)co 7411   +o coa 8465  Ncnpi 10841   +N cpli 10842
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-res 5688  df-iota 6495  df-fv 6551  df-ov 7414  df-pli 10870
This theorem is referenced by:  addclpi  10889  addcompi  10891  addasspi  10892  distrpi  10895  addcanpi  10896  addnidpi  10898  ltexpi  10899  ltapi  10900  1lt2pi  10902  indpi  10904
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