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Theorem ofc1 6319
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 5623 . . 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 3542 . 2  |-  ( A  i^i  A )  =  A
7 fvconst2g 5937 . . 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 6306 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 1652    e. wcel 1725   {csn 3806    X. cxp 4868    Fn wfn 5441   ` cfv 5446  (class class class)co 6073    o Fcof 6295
This theorem is referenced by:  ofnegsub  9990  pwsvscaval  13709  lmhmvsca  16113  psrvscaval  16448  mplvscaval  16503  coe1sclmulfv  16667  mbfmulc2lem  19531  i1fmulclem  19586  itg1mulc  19588  itg2monolem1  19634  uc1pmon1p  20066  coemulc  20165  basellem9  20863  mamuvs1  27431  mamuvs2  27432  ofdivrec  27511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297
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