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Theorem addpiord 9072
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 4890 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( N.  X.  N. ) )
2 fvres 5723 . . 3  |-  ( <. A ,  B >.  e.  ( N.  X.  N. )  ->  ( (  +o  |`  ( N.  X.  N. ) ) `  <. A ,  B >. )  =  (  +o  `  <. A ,  B >. )
)
3 df-ov 6113 . . . 4  |-  ( A  +N  B )  =  (  +N  `  <. A ,  B >. )
4 df-pli 9061 . . . . 5  |-  +N  =  (  +o  |`  ( N.  X.  N. ) )
54fveq1i 5711 . . . 4  |-  (  +N 
`  <. A ,  B >. )  =  ( (  +o  |`  ( N.  X.  N. ) ) `  <. A ,  B >. )
63, 5eqtri 2463 . . 3  |-  ( A  +N  B )  =  ( (  +o  |`  ( N.  X.  N. ) ) `
 <. A ,  B >. )
7 df-ov 6113 . . 3  |-  ( A  +o  B )  =  (  +o  `  <. A ,  B >. )
82, 6, 73eqtr4g 2500 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3902    X. cxp 4857    |` cres 4861   ` cfv 5437  (class class class)co 6110    +o coa 6936   N.cnpi 9030    +N cpli 9031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-xp 4865  df-res 4871  df-iota 5400  df-fv 5445  df-ov 6113  df-pli 9061
This theorem is referenced by:  addclpi  9080  addcompi  9082  addasspi  9083  distrpi  9086  addcanpi  9087  addnidpi  9089  ltexpi  9090  ltapi  9091  1lt2pi  9093  indpi  9095
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