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Theorem mulone 25685
Description: Multiplication of a vector by 1. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)
Hypothesis
Ref Expression
mulone.1  |-  . t  =  ( . cv `  N )
Assertion
Ref Expression
mulone  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( 1 . t U )  =  U )

Proof of Theorem mulone
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . . . 6  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  ->  U  e.  ( CC  ^m  ( 1 ... N
) ) )
2 cnex 8818 . . . . . . 7  |-  CC  e.  _V
3 ovex 5883 . . . . . . 7  |-  ( 1 ... N )  e. 
_V
42, 3elmap 6796 . . . . . 6  |-  ( U  e.  ( CC  ^m  ( 1 ... N
) )  <->  U :
( 1 ... N
) --> CC )
51, 4sylib 188 . . . . 5  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  ->  U : ( 1 ... N ) --> CC )
6 ffvelrn 5663 . . . . 5  |-  ( ( U : ( 1 ... N ) --> CC 
/\  x  e.  ( 1 ... N ) )  ->  ( U `  x )  e.  CC )
75, 6sylan 457 . . . 4  |-  ( ( ( N  e.  NN  /\  U  e.  ( CC 
^m  ( 1 ... N ) ) )  /\  x  e.  ( 1 ... N ) )  ->  ( U `  x )  e.  CC )
87mulid2d 8853 . . 3  |-  ( ( ( N  e.  NN  /\  U  e.  ( CC 
^m  ( 1 ... N ) ) )  /\  x  e.  ( 1 ... N ) )  ->  ( 1  x.  ( U `  x ) )  =  ( U `  x
) )
98mpteq2dva 4106 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( x  e.  ( 1 ... N ) 
|->  ( 1  x.  ( U `  x )
) )  =  ( x  e.  ( 1 ... N )  |->  ( U `  x ) ) )
10 ax-1cn 8795 . . 3  |-  1  e.  CC
11 mulone.1 . . . 4  |-  . t  =  ( . cv `  N )
1211ismulcv 25681 . . 3  |-  ( ( N  e.  NN  /\  1  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( 1 . t U )  =  ( x  e.  ( 1 ... N
)  |->  ( 1  x.  ( U `  x
) ) ) )
1310, 12mp3an2 1265 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( 1 . t U )  =  ( x  e.  ( 1 ... N )  |->  ( 1  x.  ( U `
 x ) ) ) )
145feqmptd 5575 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  ->  U  =  ( x  e.  ( 1 ... N
)  |->  ( U `  x ) ) )
159, 13, 143eqtr4d 2325 1  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( 1 . t U )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   1c1 8738    x. cmul 8742   NNcn 9746   ...cfz 10782   . cvcsmcv 25679
This theorem is referenced by:  vecscmonto  25686  tcnvec  25690
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-mulcom 8801  ax-mulass 8803  ax-distr 8804  ax-1rid 8807  ax-cnre 8810
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-reu 2550  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-mulcv 25680
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