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Theorem omv 6319
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 4284 . . 3  |-  (/)  e.  On
2 omfnex 6313 . . . 4  |-  ( A  e.  On  ->  (
x  e.  _V  |->  ( x  +o  A ) )  Fn  _V )
3 rdgexggg 6242 . . . 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 1234 . . 3  |-  ( ( A  e.  On  /\  (/) 
e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )
51, 4mp3an2 1288 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )
6 oveq2 5750 . . . . . 6  |-  ( y  =  A  ->  (
x  +o  y )  =  ( x  +o  A ) )
76mpteq2dv 3989 . . . . 5  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  +o  y ) )  =  ( x  e.  _V  |->  ( x  +o  A ) ) )
8 rdgeq1 6236 . . . . 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 5391 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  y ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  z )
)
11 fveq2 5389 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  B )
)
12 df-omul 6286 . . 3  |-  .o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  ( x  +o  y ) ) ,  (/) ) `  z
) )
1310, 11, 12ovmpog 5873 . 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 1301 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 103    = wceq 1316    e. wcel 1465   _Vcvv 2660   (/)c0 3333    |-> cmpt 3959   Oncon0 4255    Fn wfn 5088   ` cfv 5093  (class class class)co 5742   reccrdg 6234    +o coa 6278    .o comu 6279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285  df-omul 6286
This theorem is referenced by:  om0  6322  omcl  6325  omv2  6329
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