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Theorem addpiord 6300
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 4319 . 2 ((A N B N) → ⟨A, B (N × N))
2 fvres 5141 . . 3 (⟨A, B (N × N) → (( +𝑜 ↾ (N × N))‘⟨A, B⟩) = ( +𝑜 ‘⟨A, B⟩))
3 df-ov 5458 . . . 4 (A +N B) = ( +N ‘⟨A, B⟩)
4 df-pli 6289 . . . . 5 +N = ( +𝑜 ↾ (N × N))
54fveq1i 5122 . . . 4 ( +N ‘⟨A, B⟩) = (( +𝑜 ↾ (N × N))‘⟨A, B⟩)
63, 5eqtri 2057 . . 3 (A +N B) = (( +𝑜 ↾ (N × N))‘⟨A, B⟩)
7 df-ov 5458 . . 3 (A +𝑜 B) = ( +𝑜 ‘⟨A, B⟩)
82, 6, 73eqtr4g 2094 . 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 1242   wcel 1390  cop 3370   × cxp 4286  cres 4290  cfv 4845  (class class class)co 5455   +𝑜 coa 5937  Ncnpi 6256   +N cpli 6257
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-res 4300  df-iota 4810  df-fv 4853  df-ov 5458  df-pli 6289
This theorem is referenced by:  addclpi  6311  addcompig  6313  addasspig  6314  distrpig  6317  addcanpig  6318  addnidpig  6320  ltexpi  6321  ltapig  6322  1lt2pi  6324  indpi  6326  archnqq  6400  prarloclemarch2  6402  nqnq0a  6437
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