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Theorem lcomf 26870
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 15660 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  K  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B )
763expb 1152 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  K  /\  y  e.  B )
)  ->  ( x  .x.  y )  e.  B
)
81, 7sylan 457 . 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 3391 . 2  |-  ( I  i^i  I )  =  I
138, 9, 10, 11, 11, 12off 6109 1  |-  ( ph  ->  ( G  o F 
.x.  H ) : I --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   LModclmod 15643
This theorem is referenced by:  lcomfsup  26871  frlmup2  27354  islindf4  27411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-lmod 15645
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