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Theorem omv 6170
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 4193 . . 3  |-  (/)  e.  On
2 omfnex 6164 . . . 4  |-  ( A  e.  On  ->  (
x  e.  _V  |->  ( x  +o  A ) )  Fn  _V )
3 rdgexggg 6096 . . . 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 1205 . . 3  |-  ( ( A  e.  On  /\  (/) 
e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )
51, 4mp3an2 1259 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  _V )
6 oveq2 5621 . . . . . 6  |-  ( y  =  A  ->  (
x  +o  y )  =  ( x  +o  A ) )
76mpteq2dv 3904 . . . . 5  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  +o  y ) )  =  ( x  e.  _V  |->  ( x  +o  A ) ) )
8 rdgeq1 6090 . . . . 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 5270 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  y ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  z )
)
11 fveq2 5268 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  z )  =  ( rec (
( x  e.  _V  |->  ( x  +o  A
) ) ,  (/) ) `  B )
)
12 df-omul 6140 . . 3  |-  .o  =  ( y  e.  On ,  z  e.  On  |->  ( rec ( ( x  e.  _V  |->  ( x  +o  y ) ) ,  (/) ) `  z
) )
1310, 11, 12ovmpt2g 5736 . 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 1272 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 1287    e. wcel 1436   _Vcvv 2615   (/)c0 3275    |-> cmpt 3874   Oncon0 4164    Fn wfn 4976   ` cfv 4981  (class class class)co 5613   reccrdg 6088    +o coa 6132    .o comu 6133
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-oadd 6139  df-omul 6140
This theorem is referenced by:  om0  6173  omcl  6176  omv2  6180
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