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Theorem omsuc 7094
Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
omsuc  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )

Proof of Theorem omsuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rdgsuc 7008 . . 3  |-  ( B  e.  On  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B
)  =  ( ( x  e.  _V  |->  ( x  +o  A ) ) `  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) ) )
21adantl 464 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B )  =  ( ( x  e.  _V  |->  ( x  +o  A
) ) `  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) ) )
3 suceloni 6547 . . 3  |-  ( B  e.  On  ->  suc  B  e.  On )
4 omv 7080 . . 3  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  .o  suc  B )  =  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B ) )
53, 4sylan2 472 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B
) )
6 ovex 6224 . . . 4  |-  ( A  .o  B )  e. 
_V
7 oveq1 6203 . . . . 5  |-  ( x  =  ( A  .o  B )  ->  (
x  +o  A )  =  ( ( A  .o  B )  +o  A ) )
8 eqid 2382 . . . . 5  |-  ( x  e.  _V  |->  ( x  +o  A ) )  =  ( x  e. 
_V  |->  ( x  +o  A ) )
9 ovex 6224 . . . . 5  |-  ( ( A  .o  B )  +o  A )  e. 
_V
107, 8, 9fvmpt 5857 . . . 4  |-  ( ( A  .o  B )  e.  _V  ->  (
( x  e.  _V  |->  ( x  +o  A
) ) `  ( A  .o  B ) )  =  ( ( A  .o  B )  +o  A ) )
116, 10ax-mp 5 . . 3  |-  ( ( x  e.  _V  |->  ( x  +o  A ) ) `  ( A  .o  B ) )  =  ( ( A  .o  B )  +o  A )
12 omv 7080 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
1312fveq2d 5778 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  ( A  .o  B
) )  =  ( ( x  e.  _V  |->  ( x  +o  A
) ) `  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) ) )
1411, 13syl5eqr 2437 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  +o  A
)  =  ( ( x  e.  _V  |->  ( x  +o  A ) ) `  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) ) )
152, 5, 143eqtr4d 2433 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   (/)c0 3711    |-> cmpt 4425   Oncon0 4792   suc csuc 4794   ` cfv 5496  (class class class)co 6196   reccrdg 6993    +o coa 7045    .o comu 7046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-recs 6960  df-rdg 6994  df-omul 7053
This theorem is referenced by:  omcl  7104  om0r  7107  om1r  7110  omordi  7133  omwordri  7139  omlimcl  7145  odi  7146  omass  7147  oneo  7148  omeulem1  7149  omeulem2  7150  oeoelem  7165  oaabs2  7212  omxpenlem  7537  cantnflt  8004  cantnflem1d  8020  cantnfltOLD  8034  cantnflem1dOLD  8043  infxpenc  8308  infxpencOLD  8313
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