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Theorem scaffval 16988
Description: The scalar multiplication operation as 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  =  ( .sf `  W
)
scaffval.s  |-  .x.  =  ( .s `  W )
Assertion
Ref Expression
scaffval  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) )
Distinct variable groups:    x, y, B    x, K, y    x,  .x. , y    x, W, y
Allowed substitution hints:    .xb ( x, y)    F( x, y)

Proof of Theorem scaffval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 scaffval.a . 2  |-  .xb  =  ( .sf `  W
)
2 fveq2 5712 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
3 scaffval.f . . . . . . . 8  |-  F  =  (Scalar `  W )
42, 3syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
54fveq2d 5716 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
6 scaffval.k . . . . . 6  |-  K  =  ( Base `  F
)
75, 6syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
8 fveq2 5712 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
9 scaffval.b . . . . . 6  |-  B  =  ( Base `  W
)
108, 9syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  B )
11 fveq2 5712 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
12 scaffval.s . . . . . . 7  |-  .x.  =  ( .s `  W )
1311, 12syl6eqr 2493 . . . . . 6  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
1413oveqd 6129 . . . . 5  |-  ( w  =  W  ->  (
x ( .s `  w ) y )  =  ( x  .x.  y ) )
157, 10, 14mpt2eq123dv 6169 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  (Scalar `  w )
) ,  y  e.  ( Base `  w
)  |->  ( x ( .s `  w ) y ) )  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) ) )
16 df-scaf 16973 . . . 4  |-  .sf 
=  ( w  e. 
_V  |->  ( x  e.  ( Base `  (Scalar `  w ) ) ,  y  e.  ( Base `  w )  |->  ( x ( .s `  w
) y ) ) )
17 df-ov 6115 . . . . . . . 8  |-  ( x 
.x.  y )  =  (  .x.  `  <. x ,  y >. )
18 fvrn0 5733 . . . . . . . 8  |-  (  .x.  ` 
<. x ,  y >.
)  e.  ( ran 
.x.  u.  { (/) } )
1917, 18eqeltri 2513 . . . . . . 7  |-  ( x 
.x.  y )  e.  ( ran  .x.  u.  {
(/) } )
2019rgen2w 2805 . . . . . 6  |-  A. x  e.  K  A. y  e.  B  ( x  .x.  y )  e.  ( ran  .x.  u.  { (/) } )
21 eqid 2443 . . . . . . 7  |-  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) )  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) )
2221fmpt2 6662 . . . . . 6  |-  ( A. x  e.  K  A. y  e.  B  (
x  .x.  y )  e.  ( ran  .x.  u.  {
(/) } )  <->  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) ) : ( K  X.  B
) --> ( ran  .x.  u.  { (/) } ) )
2320, 22mpbi 208 . . . . 5  |-  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) ) : ( K  X.  B ) --> ( ran 
.x.  u.  { (/) } )
24 fvex 5722 . . . . . . 7  |-  ( Base `  F )  e.  _V
256, 24eqeltri 2513 . . . . . 6  |-  K  e. 
_V
26 fvex 5722 . . . . . . 7  |-  ( Base `  W )  e.  _V
279, 26eqeltri 2513 . . . . . 6  |-  B  e. 
_V
2825, 27xpex 6529 . . . . 5  |-  ( K  X.  B )  e. 
_V
29 fvex 5722 . . . . . . . 8  |-  ( .s
`  W )  e. 
_V
3012, 29eqeltri 2513 . . . . . . 7  |-  .x.  e.  _V
3130rnex 6533 . . . . . 6  |-  ran  .x.  e.  _V
32 p0ex 4500 . . . . . 6  |-  { (/) }  e.  _V
3331, 32unex 6399 . . . . 5  |-  ( ran 
.x.  u.  { (/) } )  e.  _V
34 fex2 6553 . . . . 5  |-  ( ( ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) ) : ( K  X.  B ) --> ( ran  .x.  u.  {
(/) } )  /\  ( K  X.  B )  e. 
_V  /\  ( ran  .x. 
u.  { (/) } )  e.  _V )  -> 
( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) )  e.  _V )
3523, 28, 33, 34mp3an 1314 . . . 4  |-  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) )  e.  _V
3615, 16, 35fvmpt 5795 . . 3  |-  ( W  e.  _V  ->  ( .sf `  W )  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) ) )
37 fvprc 5706 . . . . 5  |-  ( -.  W  e.  _V  ->  ( .sf `  W
)  =  (/) )
38 mpt20 6177 . . . . 5  |-  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) )  =  (/)
3937, 38syl6eqr 2493 . . . 4  |-  ( -.  W  e.  _V  ->  ( .sf `  W
)  =  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) ) )
40 fvprc 5706 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (Scalar `  W )  =  (/) )
413, 40syl5eq 2487 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  F  =  (/) )
4241fveq2d 5716 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  ( Base `  (/) ) )
436, 42syl5eq 2487 . . . . . 6  |-  ( -.  W  e.  _V  ->  K  =  ( Base `  (/) ) )
44 base0 14234 . . . . . 6  |-  (/)  =  (
Base `  (/) )
4543, 44syl6eqr 2493 . . . . 5  |-  ( -.  W  e.  _V  ->  K  =  (/) )
46 eqid 2443 . . . . 5  |-  B  =  B
47 mpt2eq12 6167 . . . . 5  |-  ( ( K  =  (/)  /\  B  =  B )  ->  (
x  e.  K , 
y  e.  B  |->  ( x  .x.  y ) )  =  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) ) )
4845, 46, 47sylancl 662 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  K , 
y  e.  B  |->  ( x  .x.  y ) )  =  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) ) )
4939, 48eqtr4d 2478 . . 3  |-  ( -.  W  e.  _V  ->  ( .sf `  W
)  =  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) ) )
5036, 49pm2.61i 164 . 2  |-  ( .sf `  W )  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) )
511, 50eqtri 2463 1  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   A.wral 2736   _Vcvv 2993    u. cun 3347   (/)c0 3658   {csn 3898   <.cop 3904    X. cxp 4859   ran crn 4862   -->wf 5435   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   Basecbs 14195  Scalarcsca 14262   .scvsca 14263   .sfcscaf 16971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-slot 14199  df-base 14200  df-scaf 16973
This theorem is referenced by:  scafval  16989  scafeq  16990  scaffn  16991  lmodscaf  16992  rlmscaf  17311
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