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Theorem ofcc 26560
Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcc.1  |-  ( ph  ->  A  e.  V )
ofcc.2  |-  ( ph  ->  B  e.  W )
ofcc.3  |-  ( ph  ->  C  e.  X )
Assertion
Ref Expression
ofcc  |-  ( ph  ->  ( ( A  X.  { B } )𝑓/𝑐 R C )  =  ( A  X.  { ( B R C ) } ) )

Proof of Theorem ofcc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ofcc.2 . . . 4  |-  ( ph  ->  B  e.  W )
2 fnconstg 5610 . . . 4  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
31, 2syl 16 . . 3  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
4 ofcc.1 . . 3  |-  ( ph  ->  A  e.  V )
5 ofcc.3 . . 3  |-  ( ph  ->  C  e.  X )
6 fvconst2g 5943 . . . 4  |-  ( ( B  e.  W  /\  x  e.  A )  ->  ( ( A  X.  { B } ) `  x )  =  B )
71, 6sylan 471 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( A  X.  { B } ) `  x
)  =  B )
83, 4, 5, 7ofcfval 26552 . 2  |-  ( ph  ->  ( ( A  X.  { B } )𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
9 fconstmpt 4894 . 2  |-  ( A  X.  { ( B R C ) } )  =  ( x  e.  A  |->  ( B R C ) )
108, 9syl6eqr 2493 1  |-  ( ph  ->  ( ( A  X.  { B } )𝑓/𝑐 R C )  =  ( A  X.  { ( B R C ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {csn 3889    e. cmpt 4362    X. cxp 4850    Fn wfn 5425   ` cfv 5430  (class class class)co 6103  ∘𝑓/𝑐cofc 26549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-ofc 26550
This theorem is referenced by: (None)
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