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Theorem addpiord 8508
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 4721 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 fvres 5542 . . 3  |-  ( <. A ,  B >.  e.  ( N.  X.  N. )  ->  ( (  +o  |`  ( N.  X.  N. ) ) `  <. A ,  B >. )  =  (  +o  `  <. A ,  B >. )
)
3 df-ov 5861 . . . 4  |-  ( A  +N  B )  =  (  +N  `  <. A ,  B >. )
4 df-pli 8497 . . . . 5  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
54fveq1i 5526 . . . 4  |-  (  +N 
`  <. A ,  B >. )  =  ( (  +o  |`  ( N.  X.  N. ) ) `  <. A ,  B >. )
63, 5eqtri 2303 . . 3  |-  ( A  +N  B )  =  ( (  +o  |`  ( N.  X.  N. ) ) `
 <. A ,  B >. )
7 df-ov 5861 . . 3  |-  ( A  +o  B )  =  (  +o  `  <. A ,  B >. )
82, 6, 73eqtr4g 2340 . 2  |-  ( <. A ,  B >.  e.  ( N.  X.  N. )  ->  ( A  +N  B )  =  ( A  +o  B ) )
91, 8syl 15 1  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  B
)  =  ( A  +o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858    +o coa 6476   N.cnpi 8466    +N cpli 8467
This theorem is referenced by:  addclpi  8516  addcompi  8518  addasspi  8519  distrpi  8522  addcanpi  8523  addnidpi  8525  ltexpi  8526  ltapi  8527  1lt2pi  8529  indpi  8531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-res 4701  df-iota 5219  df-fv 5263  df-ov 5861  df-pli 8497
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