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Theorem addpiord 8745
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 4896 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 fvres 5731 . . 3  |-  ( <. A ,  B >.  e.  ( N.  X.  N. )  ->  ( (  +o  |`  ( N.  X.  N. ) ) `  <. A ,  B >. )  =  (  +o  `  <. A ,  B >. )
)
3 df-ov 6070 . . . 4  |-  ( A  +N  B )  =  (  +N  `  <. A ,  B >. )
4 df-pli 8734 . . . . 5  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
54fveq1i 5715 . . . 4  |-  (  +N 
`  <. A ,  B >. )  =  ( (  +o  |`  ( N.  X.  N. ) ) `  <. A ,  B >. )
63, 5eqtri 2450 . . 3  |-  ( A  +N  B )  =  ( (  +o  |`  ( N.  X.  N. ) ) `
 <. A ,  B >. )
7 df-ov 6070 . . 3  |-  ( A  +o  B )  =  (  +o  `  <. A ,  B >. )
82, 6, 73eqtr4g 2487 . 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 1652    e. wcel 1725   <.cop 3804    X. cxp 4862    |` cres 4866   ` cfv 5440  (class class class)co 6067    +o coa 6707   N.cnpi 8703    +N cpli 8704
This theorem is referenced by:  addclpi  8753  addcompi  8755  addasspi  8756  distrpi  8759  addcanpi  8760  addnidpi  8762  ltexpi  8763  ltapi  8764  1lt2pi  8766  indpi  8768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-xp 4870  df-res 4876  df-iota 5404  df-fv 5448  df-ov 6070  df-pli 8734
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