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Theorem addpiord 10886
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 5700 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 6904 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +o ‘⟨𝐴, 𝐵⟩))
3 df-ov 7422 . . . 4 (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4 df-pli 10875 . . . . 5 +N = ( +o ↾ (N × N))
54fveq1i 6886 . . . 4 ( +N ‘⟨𝐴, 𝐵⟩) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2788 . . 3 (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 7422 . . 3 (𝐴 +o 𝐵) = ( +o ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2825 . 2 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
91, 8syl 18 1 ((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wa 401   = wceq 1570  wcel 2146  cop 4597   × cxp 5661  cres 5665  cfv 6540  (class class class)co 7419   +o coa 8456  Ncnpi 10846   +N cpli 10847
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-res 5675  df-iota 6496  df-fv 6548  df-ov 7422  df-pli 10875
This theorem is used by:  addclpi  10894  addcompi  10896  addasspi  10897  distrpi  10900  addcanpi  10901  addnidpi  10903  ltexpi  10904  ltapi  10905  1lt2pi  10907  indpi  10909
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