MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lcomf Structured version   Unicode version

Theorem lcomf 17661
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 17642 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  K  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B )
763expb 1195 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  K  /\  y  e.  B )
)  ->  ( x  .x.  y )  e.  B
)
81, 7sylan 469 . 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 3621 . 2  |-  ( I  i^i  I )  =  I
138, 9, 10, 11, 11, 12off 6453 1  |-  ( ph  ->  ( G  oF  .x.  H ) : I --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   -->wf 5492   ` cfv 5496  (class class class)co 6196    oFcof 6437   Basecbs 14634  Scalarcsca 14705   .scvsca 14706   LModclmod 17625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-lmod 17627
This theorem is referenced by:  lcomfsupOLD  17662  lcomfsupp  17663  frlmup2  18919  islindf4  18958
  Copyright terms: Public domain W3C validator