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Theorem lcomf 26746
Description: A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Hypotheses
Ref Expression
lcomf.f  |-  F  =  (Scalar `  W )
lcomf.k  |-  K  =  ( Base `  F
)
lcomf.s  |-  .x.  =  ( .s `  W )
lcomf.b  |-  B  =  ( Base `  W
)
lcomf.w  |-  ( ph  ->  W  e.  LMod )
lcomf.g  |-  ( ph  ->  G : I --> K )
lcomf.h  |-  ( ph  ->  H : I --> B )
lcomf.i  |-  ( ph  ->  I  e.  V )
Assertion
Ref Expression
lcomf  |-  ( ph  ->  ( G  o F 
.x.  H ) : I --> B )

Proof of Theorem lcomf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lcomf.w . . 3  |-  ( ph  ->  W  e.  LMod )
2 lcomf.b . . . . 5  |-  B  =  ( Base `  W
)
3 lcomf.f . . . . 5  |-  F  =  (Scalar `  W )
4 lcomf.s . . . . 5  |-  .x.  =  ( .s `  W )
5 lcomf.k . . . . 5  |-  K  =  ( Base `  F
)
62, 3, 4, 5lmodvscl 15967 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  K  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B )
763expb 1154 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  K  /\  y  e.  B )
)  ->  ( x  .x.  y )  e.  B
)
81, 7sylan 458 . 2  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  B ) )  -> 
( x  .x.  y
)  e.  B )
9 lcomf.g . 2  |-  ( ph  ->  G : I --> K )
10 lcomf.h . 2  |-  ( ph  ->  H : I --> B )
11 lcomf.i . 2  |-  ( ph  ->  I  e.  V )
12 inidm 3550 . 2  |-  ( I  i^i  I )  =  I
138, 9, 10, 11, 11, 12off 6320 1  |-  ( ph  ->  ( G  o F 
.x.  H ) : I --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   LModclmod 15950
This theorem is referenced by:  lcomfsup  26747  frlmup2  27228  islindf4  27285
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-lmod 15952
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