MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onmsuc Structured version   Unicode version

Theorem onmsuc 6773
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 4865 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
2 nnon 4851 . . . . 5  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
31, 2syl 16 . . . 4  |-  ( B  e.  om  ->  suc  B  e.  On )
4 omv 6756 . . . 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 5745 . . . 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 2471 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
10 ovex 6106 . . . . 5  |-  ( A  .o  B )  e. 
_V
11 oveq1 6088 . . . . . 6  |-  ( x  =  ( A  .o  B )  ->  (
x  +o  A )  =  ( ( A  .o  B )  +o  A ) )
12 eqid 2436 . . . . . 6  |-  ( x  e.  _V  |->  ( x  +o  A ) )  =  ( x  e. 
_V  |->  ( x  +o  A ) )
13 ovex 6106 . . . . . 6  |-  ( ( A  .o  B )  +o  A )  e. 
_V
1411, 12, 13fvmpt 5806 . . . . 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 4851 . . . . . . 7  |-  ( B  e.  om  ->  B  e.  On )
17 omv 6756 . . . . . . 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 5745 . . . . . . 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 2471 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B ) )
2221fveq2d 5732 . . . 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 2482 . . 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 6694 . . . 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 2471 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( A  .o  B )  +o  A
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
279, 26eqtr4d 2471 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 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628    e. cmpt 4266   Oncon0 4581   suc csuc 4583   omcom 4845    |` cres 4880   ` cfv 5454  (class class class)co 6081   reccrdg 6667    +o coa 6721    .o comu 6722
This theorem is referenced by:  om1  6785  nnmsuc  6850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-omul 6729
  Copyright terms: Public domain W3C validator