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Theorem onmsuc 7066
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 6593 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
2 nnon 6579 . . . . 5  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
31, 2syl 16 . . . 4  |-  ( B  e.  om  ->  suc  B  e.  On )
4 omv 7049 . . . 4  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  .o  suc  B )  =  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B ) )
53, 4sylan2 474 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B
) )
61adantl 466 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  suc  B  e.  om )
7 fvres 5800 . . . 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 2494 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
10 ovex 6212 . . . . 5  |-  ( A  .o  B )  e. 
_V
11 oveq1 6194 . . . . . 6  |-  ( x  =  ( A  .o  B )  ->  (
x  +o  A )  =  ( ( A  .o  B )  +o  A ) )
12 eqid 2451 . . . . . 6  |-  ( x  e.  _V  |->  ( x  +o  A ) )  =  ( x  e. 
_V  |->  ( x  +o  A ) )
13 ovex 6212 . . . . . 6  |-  ( ( A  .o  B )  +o  A )  e. 
_V
1411, 12, 13fvmpt 5870 . . . . 5  |-  ( ( A  .o  B )  e.  _V  ->  (
( x  e.  _V  |->  ( x  +o  A
) ) `  ( A  .o  B ) )  =  ( ( A  .o  B )  +o  A ) )
1510, 14ax-mp 5 . . . 4  |-  ( ( x  e.  _V  |->  ( x  +o  A ) ) `  ( A  .o  B ) )  =  ( ( A  .o  B )  +o  A )
16 nnon 6579 . . . . . . 7  |-  ( B  e.  om  ->  B  e.  On )
17 omv 7049 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
1816, 17sylan2 474 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
19 fvres 5800 . . . . . . 7  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  B )
)
2019adantl 466 . . . . . 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 2494 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B ) )
2221fveq2d 5790 . . . 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 2505 . . 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 6989 . . . 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 466 . . 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 2494 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( A  .o  B )  +o  A
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
279, 26eqtr4d 2494 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( ( A  .o  B )  +o  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3065   (/)c0 3732    |-> cmpt 4445   Oncon0 4814   suc csuc 4816    |` cres 4937   ` cfv 5513  (class class class)co 6187   omcom 6573   reccrdg 6962    +o coa 7014    .o comu 7015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-recs 6929  df-rdg 6963  df-omul 7022
This theorem is referenced by:  om1  7078  nnmsuc  7143
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