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Theorem onmsuc 6702
Description: Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Assertion
Ref Expression
onmsuc  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )

Proof of Theorem onmsuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 peano2 4798 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
2 nnon 4784 . . . . 5  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
31, 2syl 16 . . . 4  |-  ( B  e.  om  ->  suc  B  e.  On )
4 omv 6685 . . . 4  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  .o  suc  B )  =  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B ) )
53, 4sylan2 461 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B
) )
61adantl 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  suc  B  e.  om )
7 fvres 5678 . . . 4  |-  ( suc 
B  e.  om  ->  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B
) )
86, 7syl 16 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) )  |`  om ) `  suc  B )  =  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B ) )
95, 8eqtr4d 2415 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
10 ovex 6038 . . . . 5  |-  ( A  .o  B )  e. 
_V
11 oveq1 6020 . . . . . 6  |-  ( x  =  ( A  .o  B )  ->  (
x  +o  A )  =  ( ( A  .o  B )  +o  A ) )
12 eqid 2380 . . . . . 6  |-  ( x  e.  _V  |->  ( x  +o  A ) )  =  ( x  e. 
_V  |->  ( x  +o  A ) )
13 ovex 6038 . . . . . 6  |-  ( ( A  .o  B )  +o  A )  e. 
_V
1411, 12, 13fvmpt 5738 . . . . 5  |-  ( ( A  .o  B )  e.  _V  ->  (
( x  e.  _V  |->  ( x  +o  A
) ) `  ( A  .o  B ) )  =  ( ( A  .o  B )  +o  A ) )
1510, 14ax-mp 8 . . . 4  |-  ( ( x  e.  _V  |->  ( x  +o  A ) ) `  ( A  .o  B ) )  =  ( ( A  .o  B )  +o  A )
16 nnon 4784 . . . . . . 7  |-  ( B  e.  om  ->  B  e.  On )
17 omv 6685 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
1816, 17sylan2 461 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
19 fvres 5678 . . . . . . 7  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  B )
)
2019adantl 453 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) )  |`  om ) `  B )  =  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  B
) )
2118, 20eqtr4d 2415 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B ) )
2221fveq2d 5665 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  ( A  .o  B
) )  =  ( ( x  e.  _V  |->  ( x  +o  A
) ) `  (
( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B ) ) )
2315, 22syl5eqr 2426 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( A  .o  B )  +o  A
)  =  ( ( x  e.  _V  |->  ( x  +o  A ) ) `  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B ) ) )
24 frsuc 6623 . . . 4  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B
)  =  ( ( x  e.  _V  |->  ( x  +o  A ) ) `  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B ) ) )
2524adantl 453 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) )  |`  om ) `  suc  B )  =  ( ( x  e.  _V  |->  ( x  +o  A
) ) `  (
( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B ) ) )
2623, 25eqtr4d 2415 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( A  .o  B )  +o  A
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
279, 26eqtr4d 2415 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892   (/)c0 3564    e. cmpt 4200   Oncon0 4515   suc csuc 4517   omcom 4778    |` cres 4813   ` cfv 5387  (class class class)co 6013   reccrdg 6596    +o coa 6650    .o comu 6651
This theorem is referenced by:  om1  6714  nnmsuc  6779
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-13 1719  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-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-recs 6562  df-rdg 6597  df-omul 6658
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