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

Theorem lmodscaf 15973
Description: The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
scaffval.b  |-  B  =  ( Base `  W
)
scaffval.f  |-  F  =  (Scalar `  W )
scaffval.k  |-  K  =  ( Base `  F
)
scaffval.a  |-  .xb  =  ( .s f `  W
)
Assertion
Ref Expression
lmodscaf  |-  ( W  e.  LMod  ->  .xb  : ( K  X.  B ) --> B )

Proof of Theorem lmodscaf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scaffval.b . . . . 5  |-  B  =  ( Base `  W
)
2 scaffval.f . . . . 5  |-  F  =  (Scalar `  W )
3 eqid 2437 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
4 scaffval.k . . . . 5  |-  K  =  ( Base `  F
)
51, 2, 3, 4lmodvscl 15968 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  K  /\  y  e.  B )  ->  (
x ( .s `  W ) y )  e.  B )
653expb 1155 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  K  /\  y  e.  B )
)  ->  ( x
( .s `  W
) y )  e.  B )
76ralrimivva 2799 . 2  |-  ( W  e.  LMod  ->  A. x  e.  K  A. y  e.  B  ( x
( .s `  W
) y )  e.  B )
8 scaffval.a . . . 4  |-  .xb  =  ( .s f `  W
)
91, 2, 4, 8, 3scaffval 15969 . . 3  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x ( .s
`  W ) y ) )
109fmpt2 6419 . 2  |-  ( A. x  e.  K  A. y  e.  B  (
x ( .s `  W ) y )  e.  B  <->  .xb  : ( K  X.  B ) --> B )
117, 10sylib 190 1  |-  ( W  e.  LMod  ->  .xb  : ( K  X.  B ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2706    X. cxp 4877   -->wf 5451   ` cfv 5455  (class class class)co 6082   Basecbs 13470  Scalarcsca 13533   .scvsca 13534   LModclmod 15951   .s fcscaf 15952
This theorem is referenced by:  nlmvscn  18724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-slot 13474  df-base 13475  df-lmod 15953  df-scaf 15954
  Copyright terms: Public domain W3C validator