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Theorem ofc1 6440
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 } )  oF R F ) `
 X )  =  ( B R C ) )

Proof of Theorem ofc1
StepHypRef Expression
1 ofc1.2 . . 3  |-  ( ph  ->  B  e.  W )
2 fnconstg 5693 . . 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 3654 . 2  |-  ( A  i^i  A )  =  A
7 fvconst2g 6027 . . 3  |-  ( ( B  e.  W  /\  X  e.  A )  ->  ( ( A  X.  { B } ) `  X )  =  B )
81, 7sylan 471 . 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 6426 1  |-  ( (
ph  /\  X  e.  A )  ->  (
( ( A  X.  { B } )  oF R F ) `
 X )  =  ( B R C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {csn 3972    X. cxp 4933    Fn wfn 5508   ` cfv 5513  (class class class)co 6187    oFcof 6415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-of 6417
This theorem is referenced by:  ofnegsub  10418  pwsvscaval  14532  lmhmvsca  17229  psrvscaval  17566  mplvscaval  17631  coe1sclmulfv  17841  mamuvs1  18415  mamuvs2  18416  mdetrsca  18522  mbfmulc2lem  21238  i1fmulclem  21293  itg1mulc  21295  itg2monolem1  21341  uc1pmon1p  21736  coemulc  21835  basellem9  22539  ofdivrec  29735  matvscacell  31006
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