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Theorem addpiord 6472
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 4404 . 2 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 fvres 5226 . . 3 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +𝑜 ‘⟨𝐴, 𝐵⟩))
3 df-ov 5543 . . . 4 (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩)
4 df-pli 6461 . . . . 5 +N = ( +𝑜 ↾ (N × N))
54fveq1i 5207 . . . 4 ( +N ‘⟨𝐴, 𝐵⟩) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
63, 5eqtri 2076 . . 3 (𝐴 +N 𝐵) = (( +𝑜 ↾ (N × N))‘⟨𝐴, 𝐵⟩)
7 df-ov 5543 . . 3 (𝐴 +𝑜 𝐵) = ( +𝑜 ‘⟨𝐴, 𝐵⟩)
82, 6, 73eqtr4g 2113 . 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 101   = wceq 1259  wcel 1409  cop 3406   × cxp 4371  cres 4375  cfv 4930  (class class class)co 5540   +𝑜 coa 6029  Ncnpi 6428   +N cpli 6429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-res 4385  df-iota 4895  df-fv 4938  df-ov 5543  df-pli 6461
This theorem is referenced by:  addclpi  6483  addcompig  6485  addasspig  6486  distrpig  6489  addcanpig  6490  addnidpig  6492  ltexpi  6493  ltapig  6494  1lt2pi  6496  indpi  6498  archnqq  6573  prarloclemarch2  6575  nqnq0a  6610
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