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Theorem addpiord 6162
Description: Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.)
Assertion
Ref Expression
addpiord ((A N B N) → (A +N B) = (A +𝑜 B))

Proof of Theorem addpiord
StepHypRef Expression
1 opelxpi 4291 . 2 ((A N B N) → ⟨A, B (N × N))
2 fvres 5111 . . 3 (⟨A, B (N × N) → (( +𝑜 ↾ (N × N))‘⟨A, B⟩) = ( +𝑜 ‘⟨A, B⟩))
3 df-ov 5427 . . . 4 (A +N B) = ( +N ‘⟨A, B⟩)
4 df-pli 6151 . . . . 5 +N = ( +𝑜 ↾ (N × N))
54fveq1i 5092 . . . 4 ( +N ‘⟨A, B⟩) = (( +𝑜 ↾ (N × N))‘⟨A, B⟩)
63, 5eqtri 2033 . . 3 (A +N B) = (( +𝑜 ↾ (N × N))‘⟨A, B⟩)
7 df-ov 5427 . . 3 (A +𝑜 B) = ( +𝑜 ‘⟨A, B⟩)
82, 6, 73eqtr4g 2070 . 2 (⟨A, B (N × N) → (A +N B) = (A +𝑜 B))
91, 8syl 14 1 ((A N B N) → (A +N B) = (A +𝑜 B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1223   wcel 1366  cop 3342   × cxp 4258  cres 4262  cfv 4817  (class class class)co 5424   +𝑜 coa 5901  Ncnpi 6118   +N cpli 6119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-opab 3782  df-xp 4266  df-res 4272  df-iota 4782  df-fv 4825  df-ov 5427  df-pli 6151
This theorem is referenced by:  addclpi  6173  addcompig  6175  addasspig  6176  distrpig  6179  addcanpig  6180  addnidpig  6182  ltexpi  6183  ltapig  6184  1lt2pi  6186  archnqq  6260  prarloclemarch2  6262  nqnq0a  6295
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