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Theorem ofc1 6266
Description: Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
ofc1.1  |-  ( ph  ->  A  e.  V )
ofc1.2  |-  ( ph  ->  B  e.  W )
ofc1.3  |-  ( ph  ->  F  Fn  A )
ofc1.4  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
Assertion
Ref Expression
ofc1  |-  ( (
ph  /\  X  e.  A )  ->  (
( ( A  X.  { B } )  o F R F ) `
 X )  =  ( B R C ) )

Proof of Theorem ofc1
StepHypRef Expression
1 ofc1.2 . . 3  |-  ( ph  ->  B  e.  W )
2 fnconstg 5571 . . 3  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
31, 2syl 16 . 2  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
4 ofc1.3 . 2  |-  ( ph  ->  F  Fn  A )
5 ofc1.1 . 2  |-  ( ph  ->  A  e.  V )
6 inidm 3493 . 2  |-  ( A  i^i  A )  =  A
7 fvconst2g 5884 . . 3  |-  ( ( B  e.  W  /\  X  e.  A )  ->  ( ( A  X.  { B } ) `  X )  =  B )
81, 7sylan 458 . 2  |-  ( (
ph  /\  X  e.  A )  ->  (
( A  X.  { B } ) `  X
)  =  B )
9 ofc1.4 . 2  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
103, 4, 5, 5, 6, 8, 9ofval 6253 1  |-  ( (
ph  /\  X  e.  A )  ->  (
( ( A  X.  { B } )  o F R F ) `
 X )  =  ( B R C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3757    X. cxp 4816    Fn wfn 5389   ` cfv 5394  (class class class)co 6020    o Fcof 6242
This theorem is referenced by:  ofnegsub  9930  pwsvscaval  13644  lmhmvsca  16048  psrvscaval  16383  mplvscaval  16438  coe1sclmulfv  16602  mbfmulc2lem  19406  i1fmulclem  19461  itg1mulc  19463  itg2monolem1  19509  uc1pmon1p  19941  coemulc  20040  basellem9  20738  mamuvs1  27132  mamuvs2  27133  ofdivrec  27212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244
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