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Theorem smfval 25625
Description: Value of the function for the scalar multiplication operation on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
smfval.4  |-  S  =  ( .sOLD `  U )
Assertion
Ref Expression
smfval  |-  S  =  ( 2nd `  ( 1st `  U ) )

Proof of Theorem smfval
StepHypRef Expression
1 smfval.4 . 2  |-  S  =  ( .sOLD `  U )
2 df-sm 25617 . . . . 5  |-  .sOLD  =  ( 2nd  o.  1st )
32fveq1i 5873 . . . 4  |-  ( .sOLD `  U )  =  ( ( 2nd 
o.  1st ) `  U
)
4 fo1st 6819 . . . . . 6  |-  1st : _V -onto-> _V
5 fof 5801 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
64, 5ax-mp 5 . . . . 5  |-  1st : _V
--> _V
7 fvco3 5950 . . . . 5  |-  ( ( 1st : _V --> _V  /\  U  e.  _V )  ->  ( ( 2nd  o.  1st ) `  U )  =  ( 2nd `  ( 1st `  U ) ) )
86, 7mpan 670 . . . 4  |-  ( U  e.  _V  ->  (
( 2nd  o.  1st ) `  U )  =  ( 2nd `  ( 1st `  U ) ) )
93, 8syl5eq 2510 . . 3  |-  ( U  e.  _V  ->  ( .sOLD `  U )  =  ( 2nd `  ( 1st `  U ) ) )
10 fvprc 5866 . . . 4  |-  ( -.  U  e.  _V  ->  ( .sOLD `  U
)  =  (/) )
11 fvprc 5866 . . . . . 6  |-  ( -.  U  e.  _V  ->  ( 1st `  U )  =  (/) )
1211fveq2d 5876 . . . . 5  |-  ( -.  U  e.  _V  ->  ( 2nd `  ( 1st `  U ) )  =  ( 2nd `  (/) ) )
13 2nd0 6806 . . . . 5  |-  ( 2nd `  (/) )  =  (/)
1412, 13syl6req 2515 . . . 4  |-  ( -.  U  e.  _V  ->  (/)  =  ( 2nd `  ( 1st `  U ) ) )
1510, 14eqtrd 2498 . . 3  |-  ( -.  U  e.  _V  ->  ( .sOLD `  U
)  =  ( 2nd `  ( 1st `  U
) ) )
169, 15pm2.61i 164 . 2  |-  ( .sOLD `  U )  =  ( 2nd `  ( 1st `  U ) )
171, 16eqtri 2486 1  |-  S  =  ( 2nd `  ( 1st `  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793    o. ccom 5012   -->wf 5590   -onto->wfo 5592   ` cfv 5594   1stc1st 6797   2ndc2nd 6798   .sOLDcns 25607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-1st 6799  df-2nd 6800  df-sm 25617
This theorem is referenced by:  nvvop  25629  nvsf  25639  nvscl  25648  nvsid  25649  nvsass  25650  nvdi  25652  nvdir  25653  nv2  25654  nv0  25659  nvsz  25660  nvinv  25661  nvtri  25700  cnnvs  25713  phop  25860  phpar  25866  ipdirilem  25871  h2hsm  26019  hhsssm  26303
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