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Theorem omv 6099
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 4155 . . 3  |-  (/)  e.  On
2 omfnex 6093 . . . 4  |-  ( A  e.  On  ->  (
x  e.  _V  |->  ( x  +o  A ) )  Fn  _V )
3 rdgexggg 6026 . . . 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 1203 . . 3  |-  ( ( A  e.  On  /\  (/) 
e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )
51, 4mp3an2 1257 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )
6 oveq2 5551 . . . . . 6  |-  ( y  =  A  ->  (
x  +o  y )  =  ( x  +o  A ) )
76mpteq2dv 3877 . . . . 5  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  +o  y ) )  =  ( x  e.  _V  |->  ( x  +o  A ) ) )
8 rdgeq1 6020 . . . . 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 5211 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  y ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  z )
)
11 fveq2 5209 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  B )
)
12 df-omul 6070 . . 3  |-  .o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  ( x  +o  y ) ) ,  (/) ) `  z
) )
1310, 11, 12ovmpt2g 5666 . 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 1270 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 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   (/)c0 3258    |-> cmpt 3847   Oncon0 4126    Fn wfn 4927   ` cfv 4932  (class class class)co 5543   reccrdg 6018    +o coa 6062    .o comu 6063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-oadd 6069  df-omul 6070
This theorem is referenced by:  om0  6102  omcl  6105  omv2  6109
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