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Theorem addpiord 8687
Description: Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addpiord  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  B
)  =  ( A  +o  B ) )

Proof of Theorem addpiord
StepHypRef Expression
1 opelxpi 4843 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 fvres 5678 . . 3  |-  ( <. A ,  B >.  e.  ( N.  X.  N. )  ->  ( (  +o  |`  ( N.  X.  N. ) ) `  <. A ,  B >. )  =  (  +o  `  <. A ,  B >. )
)
3 df-ov 6016 . . . 4  |-  ( A  +N  B )  =  (  +N  `  <. A ,  B >. )
4 df-pli 8676 . . . . 5  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
54fveq1i 5662 . . . 4  |-  (  +N 
`  <. A ,  B >. )  =  ( (  +o  |`  ( N.  X.  N. ) ) `  <. A ,  B >. )
63, 5eqtri 2400 . . 3  |-  ( A  +N  B )  =  ( (  +o  |`  ( N.  X.  N. ) ) `
 <. A ,  B >. )
7 df-ov 6016 . . 3  |-  ( A  +o  B )  =  (  +o  `  <. A ,  B >. )
82, 6, 73eqtr4g 2437 . 2  |-  ( <. A ,  B >.  e.  ( N.  X.  N. )  ->  ( A  +N  B )  =  ( A  +o  B ) )
91, 8syl 16 1  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  B
)  =  ( A  +o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3753    X. cxp 4809    |` cres 4813   ` cfv 5387  (class class class)co 6013    +o coa 6650   N.cnpi 8645    +N cpli 8646
This theorem is referenced by:  addclpi  8695  addcompi  8697  addasspi  8698  distrpi  8701  addcanpi  8702  addnidpi  8704  ltexpi  8705  ltapi  8706  1lt2pi  8708  indpi  8710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-res 4823  df-iota 5351  df-fv 5395  df-ov 6016  df-pli 8676
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