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Theorem ofc1 6548
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 5763 . . 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 3692 . 2  |-  ( A  i^i  A )  =  A
7 fvconst2g 6109 . . 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 6534 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 1383    e. wcel 1804   {csn 4014    X. cxp 4987    Fn wfn 5573   ` cfv 5578  (class class class)co 6281    oFcof 6523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525
This theorem is referenced by:  ofnegsub  10540  pwsvscaval  14769  lmhmvsca  17565  psrvscaval  17919  mplvscaval  17984  coe1sclmulfv  18198  mamuvs1  18780  mamuvs2  18781  matvscacell  18811  mdetrsca  18978  mbfmulc2lem  21927  i1fmulclem  21982  itg1mulc  21984  itg2monolem1  22030  uc1pmon1p  22425  coemulc  22524  basellem9  23234  ofdivrec  31207
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