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Theorem onmsuc 7176
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 6698 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
2 nnon 6684 . . . . 5  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
31, 2syl 16 . . . 4  |-  ( B  e.  om  ->  suc  B  e.  On )
4 omv 7159 . . . 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 5878 . . . 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 2511 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
10 ovex 6307 . . . . 5  |-  ( A  .o  B )  e. 
_V
11 oveq1 6289 . . . . . 6  |-  ( x  =  ( A  .o  B )  ->  (
x  +o  A )  =  ( ( A  .o  B )  +o  A ) )
12 eqid 2467 . . . . . 6  |-  ( x  e.  _V  |->  ( x  +o  A ) )  =  ( x  e. 
_V  |->  ( x  +o  A ) )
13 ovex 6307 . . . . . 6  |-  ( ( A  .o  B )  +o  A )  e. 
_V
1411, 12, 13fvmpt 5948 . . . . 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 6684 . . . . . . 7  |-  ( B  e.  om  ->  B  e.  On )
17 omv 7159 . . . . . . 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 5878 . . . . . . 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 2511 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B ) )
2221fveq2d 5868 . . . 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 2522 . . 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 7099 . . . 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 2511 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( A  .o  B )  +o  A
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
279, 26eqtr4d 2511 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 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785    |-> cmpt 4505   Oncon0 4878   suc csuc 4880    |` cres 5001   ` cfv 5586  (class class class)co 6282   omcom 6678   reccrdg 7072    +o coa 7124    .o comu 7125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-omul 7132
This theorem is referenced by:  om1  7188  nnmsuc  7253
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