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Theorem omv 6449
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 0elon 4388 . . 3  |-  (/)  e.  On
2 omfnex 6443 . . . 4  |-  ( A  e.  On  ->  (
x  e.  _V  |->  ( x  +o  A ) )  Fn  _V )
3 rdgexggg 6371 . . . 4  |-  ( ( ( x  e.  _V  |->  ( x  +o  A
) )  Fn  _V  /\  (/)  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )
42, 3syl3an1 1271 . . 3  |-  ( ( A  e.  On  /\  (/) 
e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )
51, 4mp3an2 1325 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )
6 oveq2 5876 . . . . . 6  |-  ( y  =  A  ->  (
x  +o  y )  =  ( x  +o  A ) )
76mpteq2dv 4091 . . . . 5  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  +o  y ) )  =  ( x  e.  _V  |->  ( x  +o  A ) ) )
8 rdgeq1 6365 . . . . 5  |-  ( ( 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 ) ) ,  (/) ) )
97, 8syl 14 . . . 4  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  ( x  +o  y ) ) ,  (/) )  =  rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) )
109fveq1d 5512 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  y ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  z )
)
11 fveq2 5510 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  B )
)
12 df-omul 6415 . . 3  |-  .o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  ( x  +o  y ) ) ,  (/) ) `  z
) )
1310, 11, 12ovmpog 6002 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )  -> 
( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
145, 13mpd3an3 1338 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 104    = wceq 1353    e. wcel 2148   _Vcvv 2737   (/)c0 3422    |-> cmpt 4061   Oncon0 4359    Fn wfn 5206   ` cfv 5211  (class class class)co 5868   reccrdg 6363    +o coa 6407    .o comu 6408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-oadd 6414  df-omul 6415
This theorem is referenced by:  om0  6452  omcl  6455  omv2  6459
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