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Theorem scaffval 17401
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 5872 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
3 scaffval.f . . . . . . . 8  |-  F  =  (Scalar `  W )
42, 3syl6eqr 2526 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
54fveq2d 5876 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
6 scaffval.k . . . . . 6  |-  K  =  ( Base `  F
)
75, 6syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
8 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
9 scaffval.b . . . . . 6  |-  B  =  ( Base `  W
)
108, 9syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  B )
11 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
12 scaffval.s . . . . . . 7  |-  .x.  =  ( .s `  W )
1311, 12syl6eqr 2526 . . . . . 6  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
1413oveqd 6312 . . . . 5  |-  ( w  =  W  ->  (
x ( .s `  w ) y )  =  ( x  .x.  y ) )
157, 10, 14mpt2eq123dv 6354 . . . 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 17386 . . . 4  |-  .sf 
=  ( w  e. 
_V  |->  ( x  e.  ( Base `  (Scalar `  w ) ) ,  y  e.  ( Base `  w )  |->  ( x ( .s `  w
) y ) ) )
17 df-ov 6298 . . . . . . . 8  |-  ( x 
.x.  y )  =  (  .x.  `  <. x ,  y >. )
18 fvrn0 5894 . . . . . . . 8  |-  (  .x.  ` 
<. x ,  y >.
)  e.  ( ran 
.x.  u.  { (/) } )
1917, 18eqeltri 2551 . . . . . . 7  |-  ( x 
.x.  y )  e.  ( ran  .x.  u.  {
(/) } )
2019rgen2w 2829 . . . . . 6  |-  A. x  e.  K  A. y  e.  B  ( x  .x.  y )  e.  ( ran  .x.  u.  { (/) } )
21 eqid 2467 . . . . . . 7  |-  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) )  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) )
2221fmpt2 6862 . . . . . 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 5882 . . . . . . 7  |-  ( Base `  F )  e.  _V
256, 24eqeltri 2551 . . . . . 6  |-  K  e. 
_V
26 fvex 5882 . . . . . . 7  |-  ( Base `  W )  e.  _V
279, 26eqeltri 2551 . . . . . 6  |-  B  e. 
_V
2825, 27xpex 6599 . . . . 5  |-  ( K  X.  B )  e. 
_V
29 fvex 5882 . . . . . . . 8  |-  ( .s
`  W )  e. 
_V
3012, 29eqeltri 2551 . . . . . . 7  |-  .x.  e.  _V
3130rnex 6729 . . . . . 6  |-  ran  .x.  e.  _V
32 p0ex 4640 . . . . . 6  |-  { (/) }  e.  _V
3331, 32unex 6593 . . . . 5  |-  ( ran 
.x.  u.  { (/) } )  e.  _V
34 fex2 6750 . . . . 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 1324 . . . 4  |-  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) )  e.  _V
3615, 16, 35fvmpt 5957 . . 3  |-  ( W  e.  _V  ->  ( .sf `  W )  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) ) )
37 fvprc 5866 . . . . 5  |-  ( -.  W  e.  _V  ->  ( .sf `  W
)  =  (/) )
38 mpt20 6362 . . . . 5  |-  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) )  =  (/)
3937, 38syl6eqr 2526 . . . 4  |-  ( -.  W  e.  _V  ->  ( .sf `  W
)  =  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) ) )
40 fvprc 5866 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (Scalar `  W )  =  (/) )
413, 40syl5eq 2520 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  F  =  (/) )
4241fveq2d 5876 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  ( Base `  (/) ) )
436, 42syl5eq 2520 . . . . . 6  |-  ( -.  W  e.  _V  ->  K  =  ( Base `  (/) ) )
44 base0 14546 . . . . . 6  |-  (/)  =  (
Base `  (/) )
4543, 44syl6eqr 2526 . . . . 5  |-  ( -.  W  e.  _V  ->  K  =  (/) )
46 eqid 2467 . . . . 5  |-  B  =  B
47 mpt2eq12 6352 . . . . 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 2511 . . 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 2496 1  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    u. cun 3479   (/)c0 3790   {csn 4033   <.cop 4039    X. cxp 5003   ran crn 5006   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   Basecbs 14507  Scalarcsca 14575   .scvsca 14576   .sfcscaf 17384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-slot 14511  df-base 14512  df-scaf 17386
This theorem is referenced by:  scafval  17402  scafeq  17403  scaffn  17404  lmodscaf  17405  rlmscaf  17725
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