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Theorem onmsuc 6544
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 4692 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
2 nnon 4678 . . . . 5  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
31, 2syl 15 . . . 4  |-  ( B  e.  om  ->  suc  B  e.  On )
4 omv 6527 . . . 4  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  .o  suc  B )  =  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B ) )
53, 4sylan2 460 . . 3  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  suc  B
) )
61adantl 452 . . . 4  |-  ( ( A  e.  On  /\  B  e.  om )  ->  suc  B  e.  om )
7 fvres 5558 . . . 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 15 . . 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 2331 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc  B )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
10 ovex 5899 . . . . 5  |-  ( A  .o  B )  e. 
_V
11 oveq1 5881 . . . . . 6  |-  ( x  =  ( A  .o  B )  ->  (
x  +o  A )  =  ( ( A  .o  B )  +o  A ) )
12 eqid 2296 . . . . . 6  |-  ( x  e.  _V  |->  ( x  +o  A ) )  =  ( x  e. 
_V  |->  ( x  +o  A ) )
13 ovex 5899 . . . . . 6  |-  ( ( A  .o  B )  +o  A )  e. 
_V
1411, 12, 13fvmpt 5618 . . . . 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 4678 . . . . . . 7  |-  ( B  e.  om  ->  B  e.  On )
17 omv 6527 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
1816, 17sylan2 460 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
19 fvres 5558 . . . . . . 7  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  B )
)
2019adantl 452 . . . . . 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 2331 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  B ) )
2221fveq2d 5545 . . . 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 2342 . . 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 6465 . . . 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 452 . . 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 2331 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( ( A  .o  B )  +o  A
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) )  |`  om ) `  suc  B ) )
279, 26eqtr4d 2331 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468    e. cmpt 4093   Oncon0 4408   suc csuc 4410   omcom 4672    |` cres 4707   ` cfv 5271  (class class class)co 5874   reccrdg 6438    +o coa 6492    .o comu 6493
This theorem is referenced by:  om1  6556  nnmsuc  6621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-omul 6500
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