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Theorem addpiord 7345
Description: Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.)
Assertion
Ref Expression
addpiord ((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))

Proof of Theorem addpiord
StepHypRef Expression
1 opelxpi 4676 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 5558 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +o ‘⟨𝐴, 𝐵⟩))
3 df-ov 5899 . . . 4 (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4 df-pli 7334 . . . . 5 +N = ( +o ↾ (N × N))
54fveq1i 5535 . . . 4 ( +N ‘⟨𝐴, 𝐵⟩) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2210 . . 3 (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 5899 . . 3 (𝐴 +o 𝐵) = ( +o ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2247 . 2 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
91, 8syl 14 1 ((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  cop 3610   × cxp 4642  cres 4646  cfv 5235  (class class class)co 5896   +o coa 6438  Ncnpi 7301   +N cpli 7302
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-res 4656  df-iota 5196  df-fv 5243  df-ov 5899  df-pli 7334
This theorem is referenced by:  addclpi  7356  addcompig  7358  addasspig  7359  distrpig  7362  addcanpig  7363  addnidpig  7365  ltexpi  7366  ltapig  7367  1lt2pi  7369  indpi  7371  archnqq  7446  prarloclemarch2  7448  nqnq0a  7483
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