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Theorem clsmulrv 25786
Description: Closure of scalar multiplication. (Contributed by FL, 29-May-2014.)
Hypothesis
Ref Expression
ismulcv.1  |-  . t  =  ( . cv `  N )
Assertion
Ref Expression
clsmulrv  |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( S . t U )  e.  ( RR  ^m  (
1 ... N ) ) )

Proof of Theorem clsmulrv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 19 . . 3  |-  ( N  e.  NN  ->  N  e.  NN )
2 recn 8843 . . 3  |-  ( S  e.  RR  ->  S  e.  CC )
3 ax-resscn 8810 . . . . 5  |-  RR  C_  CC
4 fss 5413 . . . . 5  |-  ( ( U : ( 1 ... N ) --> RR 
/\  RR  C_  CC )  ->  U : ( 1 ... N ) --> CC )
53, 4mpan2 652 . . . 4  |-  ( U : ( 1 ... N ) --> RR  ->  U : ( 1 ... N ) --> CC )
6 reex 8844 . . . . 5  |-  RR  e.  _V
7 ovex 5899 . . . . 5  |-  ( 1 ... N )  e. 
_V
86, 7elmap 6812 . . . 4  |-  ( U  e.  ( RR  ^m  ( 1 ... N
) )  <->  U :
( 1 ... N
) --> RR )
9 cnex 8834 . . . . 5  |-  CC  e.  _V
109, 7elmap 6812 . . . 4  |-  ( U  e.  ( CC  ^m  ( 1 ... N
) )  <->  U :
( 1 ... N
) --> CC )
115, 8, 103imtr4i 257 . . 3  |-  ( U  e.  ( RR  ^m  ( 1 ... N
) )  ->  U  e.  ( CC  ^m  (
1 ... N ) ) )
12 ismulcv.1 . . . 4  |-  . t  =  ( . cv `  N )
1312ismulcv 25784 . . 3  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( S . t U )  =  ( x  e.  ( 1 ... N ) 
|->  ( S  x.  ( U `  x )
) ) )
141, 2, 11, 13syl3an 1224 . 2  |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( S . t U )  =  ( x  e.  ( 1 ... N ) 
|->  ( S  x.  ( U `  x )
) ) )
15 simp1 955 . . . . . . . . 9  |-  ( ( S  e.  RR  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  x  e.  ( 1 ... N
) )  ->  S  e.  RR )
16 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( U : ( 1 ... N ) --> RR 
/\  x  e.  ( 1 ... N ) )  ->  ( U `  x )  e.  RR )
178, 16sylanb 458 . . . . . . . . . 10  |-  ( ( U  e.  ( RR 
^m  ( 1 ... N ) )  /\  x  e.  ( 1 ... N ) )  ->  ( U `  x )  e.  RR )
18173adant1 973 . . . . . . . . 9  |-  ( ( S  e.  RR  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  x  e.  ( 1 ... N
) )  ->  ( U `  x )  e.  RR )
1915, 18jca 518 . . . . . . . 8  |-  ( ( S  e.  RR  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  x  e.  ( 1 ... N
) )  ->  ( S  e.  RR  /\  ( U `  x )  e.  RR ) )
20193expia 1153 . . . . . . 7  |-  ( ( S  e.  RR  /\  U  e.  ( RR  ^m  ( 1 ... N
) ) )  -> 
( x  e.  ( 1 ... N )  ->  ( S  e.  RR  /\  ( U `
 x )  e.  RR ) ) )
21203adant1 973 . . . . . 6  |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( x  e.  ( 1 ... N
)  ->  ( S  e.  RR  /\  ( U `
 x )  e.  RR ) ) )
22 remulcl 8838 . . . . . 6  |-  ( ( S  e.  RR  /\  ( U `  x )  e.  RR )  -> 
( S  x.  ( U `  x )
)  e.  RR )
2321, 22syl6 29 . . . . 5  |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( x  e.  ( 1 ... N
)  ->  ( S  x.  ( U `  x
) )  e.  RR ) )
2423ralrimiv 2638 . . . 4  |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  (
1 ... N ) ) )  ->  A. x  e.  ( 1 ... N
) ( S  x.  ( U `  x ) )  e.  RR )
25 eqid 2296 . . . . 5  |-  ( x  e.  ( 1 ... N )  |->  ( S  x.  ( U `  x ) ) )  =  ( x  e.  ( 1 ... N
)  |->  ( S  x.  ( U `  x ) ) )
2625fmpt 5697 . . . 4  |-  ( A. x  e.  ( 1 ... N ) ( S  x.  ( U `
 x ) )  e.  RR  <->  ( x  e.  ( 1 ... N
)  |->  ( S  x.  ( U `  x ) ) ) : ( 1 ... N ) --> RR )
2724, 26sylib 188 . . 3  |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( x  e.  ( 1 ... N
)  |->  ( S  x.  ( U `  x ) ) ) : ( 1 ... N ) --> RR )
286, 7pm3.2i 441 . . . 4  |-  ( RR  e.  _V  /\  (
1 ... N )  e. 
_V )
29 elmapg 6801 . . . 4  |-  ( ( RR  e.  _V  /\  ( 1 ... N
)  e.  _V )  ->  ( ( x  e.  ( 1 ... N
)  |->  ( S  x.  ( U `  x ) ) )  e.  ( RR  ^m  ( 1 ... N ) )  <-> 
( x  e.  ( 1 ... N ) 
|->  ( S  x.  ( U `  x )
) ) : ( 1 ... N ) --> RR ) )
3028, 29mp1i 11 . . 3  |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( (
x  e.  ( 1 ... N )  |->  ( S  x.  ( U `
 x ) ) )  e.  ( RR 
^m  ( 1 ... N ) )  <->  ( x  e.  ( 1 ... N
)  |->  ( S  x.  ( U `  x ) ) ) : ( 1 ... N ) --> RR ) )
3127, 30mpbird 223 . 2  |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( x  e.  ( 1 ... N
)  |->  ( S  x.  ( U `  x ) ) )  e.  ( RR  ^m  ( 1 ... N ) ) )
3214, 31eqeltrd 2370 1  |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( S . t U )  e.  ( RR  ^m  (
1 ... N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   RRcr 8752   1c1 8754    x. cmul 8758   NNcn 9762   ...cfz 10798   . cvcsmcv 25782
This theorem is referenced by:  divclrvd  25798
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-mulrcl 8816
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-reu 2563  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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-mulcv 25783
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