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Theorem omv 6527
Description: Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
omv  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem omv
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5882 . . . . 5  |-  ( y  =  A  ->  (
x  +o  y )  =  ( x  +o  A ) )
21mpteq2dv 4123 . . . 4  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  +o  y ) )  =  ( x  e.  _V  |->  ( x  +o  A ) ) )
3 rdgeq1 6440 . . . 4  |-  ( ( x  e.  _V  |->  ( x  +o  y ) )  =  ( x  e.  _V  |->  ( x  +o  A ) )  ->  rec ( ( x  e.  _V  |->  ( x  +o  y ) ) ,  (/) )  =  rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) )
42, 3syl 15 . . 3  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  ( x  +o  y ) ) ,  (/) )  =  rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) )
54fveq1d 5543 . 2  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  y ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  z )
)
6 fveq2 5541 . 2  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  B )
)
7 df-omul 6500 . 2  |-  .o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  ( x  +o  y ) ) ,  (/) ) `  z
) )
8 fvex 5555 . 2  |-  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V
95, 6, 7, 8ovmpt2 5999 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
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   ` cfv 5271  (class class class)co 5874   reccrdg 6438    +o coa 6492    .o comu 6493
This theorem is referenced by:  om0  6532  omsuc  6541  onmsuc  6544  omlim  6548
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  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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