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Theorem lcomf 17091
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  oF  .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 17073 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  K  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B )
763expb 1189 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  K  /\  y  e.  B )
)  ->  ( x  .x.  y )  e.  B
)
81, 7sylan 471 . 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 3659 . 2  |-  ( I  i^i  I )  =  I
138, 9, 10, 11, 11, 12off 6436 1  |-  ( ph  ->  ( G  oF  .x.  H ) : I --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   -->wf 5514   ` cfv 5518  (class class class)co 6192    oFcof 6420   Basecbs 14278  Scalarcsca 14345   .scvsca 14346   LModclmod 17056
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 4503  ax-sep 4513  ax-nul 4521  ax-pr 4631
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 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-lmod 17058
This theorem is referenced by:  lcomfsupOLD  17092  lcomfsupp  17093  frlmup2  18338  islindf4  18378
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