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Theorem lmodscaf 15649
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 2283 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
4 scaffval.k . . . . 5  |-  K  =  ( Base `  F
)
51, 2, 3, 4lmodvscl 15644 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  K  /\  y  e.  B )  ->  (
x ( .s `  W ) y )  e.  B )
653expb 1152 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  K  /\  y  e.  B )
)  ->  ( x
( .s `  W
) y )  e.  B )
76ralrimivva 2635 . 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 15645 . . 3  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x ( .s
`  W ) y ) )
109fmpt2 6191 . 2  |-  ( A. x  e.  K  A. y  e.  B  (
x ( .s `  W ) y )  e.  B  <->  .xb  : ( K  X.  B ) --> B )
117, 10sylib 188 1  |-  ( W  e.  LMod  ->  .xb  : ( K  X.  B ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   LModclmod 15627   .s fcscaf 15628
This theorem is referenced by:  nlmvscn  18198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-slot 13152  df-base 13153  df-lmod 15629  df-scaf 15630
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