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Theorem scaffval 15661
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  =  ( .s f `  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  =  ( .s f `  W
)
2 fveq2 5541 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
3 scaffval.f . . . . . . . 8  |-  F  =  (Scalar `  W )
42, 3syl6eqr 2346 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
54fveq2d 5545 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
6 scaffval.k . . . . . 6  |-  K  =  ( Base `  F
)
75, 6syl6eqr 2346 . . . . 5  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
8 fveq2 5541 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
9 scaffval.b . . . . . 6  |-  B  =  ( Base `  W
)
108, 9syl6eqr 2346 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  B )
11 fveq2 5541 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
12 scaffval.s . . . . . . 7  |-  .x.  =  ( .s `  W )
1311, 12syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
1413oveqd 5891 . . . . 5  |-  ( w  =  W  ->  (
x ( .s `  w ) y )  =  ( x  .x.  y ) )
157, 10, 14mpt2eq123dv 5926 . . . 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 15646 . . . 4  |-  .s f  =  ( w  e. 
_V  |->  ( x  e.  ( Base `  (Scalar `  w ) ) ,  y  e.  ( Base `  w )  |->  ( x ( .s `  w
) y ) ) )
17 df-ov 5877 . . . . . . . 8  |-  ( x 
.x.  y )  =  (  .x.  `  <. x ,  y >. )
18 fvrn0 5566 . . . . . . . 8  |-  (  .x.  ` 
<. x ,  y >.
)  e.  ( ran 
.x.  u.  { (/) } )
1917, 18eqeltri 2366 . . . . . . 7  |-  ( x 
.x.  y )  e.  ( ran  .x.  u.  {
(/) } )
2019rgen2w 2624 . . . . . 6  |-  A. x  e.  K  A. y  e.  B  ( x  .x.  y )  e.  ( ran  .x.  u.  { (/) } )
21 eqid 2296 . . . . . . 7  |-  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) )  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) )
2221fmpt2 6207 . . . . . 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 199 . . . . 5  |-  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) ) : ( K  X.  B ) --> ( ran 
.x.  u.  { (/) } )
24 fvex 5555 . . . . . . 7  |-  ( Base `  F )  e.  _V
256, 24eqeltri 2366 . . . . . 6  |-  K  e. 
_V
26 fvex 5555 . . . . . . 7  |-  ( Base `  W )  e.  _V
279, 26eqeltri 2366 . . . . . 6  |-  B  e. 
_V
2825, 27xpex 4817 . . . . 5  |-  ( K  X.  B )  e. 
_V
29 fvex 5555 . . . . . . . 8  |-  ( .s
`  W )  e. 
_V
3012, 29eqeltri 2366 . . . . . . 7  |-  .x.  e.  _V
3130rnex 4958 . . . . . 6  |-  ran  .x.  e.  _V
32 p0ex 4213 . . . . . 6  |-  { (/) }  e.  _V
3331, 32unex 4534 . . . . 5  |-  ( ran 
.x.  u.  { (/) } )  e.  _V
34 fex2 5417 . . . . 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 1277 . . . 4  |-  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) )  e.  _V
3615, 16, 35fvmpt 5618 . . 3  |-  ( W  e.  _V  ->  ( .s f `  W )  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) ) )
37 fvprc 5535 . . . . 5  |-  ( -.  W  e.  _V  ->  ( .s f `  W
)  =  (/) )
38 mpt20 6215 . . . . 5  |-  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) )  =  (/)
3937, 38syl6eqr 2346 . . . 4  |-  ( -.  W  e.  _V  ->  ( .s f `  W
)  =  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) ) )
40 fvprc 5535 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  (Scalar `  W )  =  (/) )
413, 40syl5eq 2340 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  F  =  (/) )
4241fveq2d 5545 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  F )  =  ( Base `  (/) ) )
436, 42syl5eq 2340 . . . . . 6  |-  ( -.  W  e.  _V  ->  K  =  ( Base `  (/) ) )
44 base0 13201 . . . . . 6  |-  (/)  =  (
Base `  (/) )
4543, 44syl6eqr 2346 . . . . 5  |-  ( -.  W  e.  _V  ->  K  =  (/) )
46 eqid 2296 . . . . 5  |-  B  =  B
47 mpt2eq12 5924 . . . . 5  |-  ( ( K  =  (/)  /\  B  =  B )  ->  (
x  e.  K , 
y  e.  B  |->  ( x  .x.  y ) )  =  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) ) )
4845, 46, 47sylancl 643 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  K , 
y  e.  B  |->  ( x  .x.  y ) )  =  ( x  e.  (/) ,  y  e.  B  |->  ( x  .x.  y ) ) )
4939, 48eqtr4d 2331 . . 3  |-  ( -.  W  e.  _V  ->  ( .s f `  W
)  =  ( x  e.  K ,  y  e.  B  |->  ( x 
.x.  y ) ) )
5036, 49pm2.61i 156 . 2  |-  ( .s f `  W )  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) )
511, 50eqtri 2316 1  |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    u. cun 3163   (/)c0 3468   {csn 3653   <.cop 3656    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   .s fcscaf 15644
This theorem is referenced by:  scafval  15662  scafeq  15663  scaffn  15664  lmodscaf  15665  rlmscaf  15976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-slot 13168  df-base 13169  df-scaf 15646
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