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Theorem smfval 22037
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  =  ( .s OLD `  U
)
Assertion
Ref Expression
smfval  |-  S  =  ( 2nd `  ( 1st `  U ) )

Proof of Theorem smfval
StepHypRef Expression
1 smfval.4 . 2  |-  S  =  ( .s OLD `  U
)
2 df-sm 22029 . . . . 5  |-  .s OLD  =  ( 2nd  o.  1st )
32fveq1i 5688 . . . 4  |-  ( .s
OLD `  U )  =  ( ( 2nd 
o.  1st ) `  U
)
4 fo1st 6325 . . . . . 6  |-  1st : _V -onto-> _V
5 fof 5612 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
64, 5ax-mp 8 . . . . 5  |-  1st : _V
--> _V
7 fvco3 5759 . . . . 5  |-  ( ( 1st : _V --> _V  /\  U  e.  _V )  ->  ( ( 2nd  o.  1st ) `  U )  =  ( 2nd `  ( 1st `  U ) ) )
86, 7mpan 652 . . . 4  |-  ( U  e.  _V  ->  (
( 2nd  o.  1st ) `  U )  =  ( 2nd `  ( 1st `  U ) ) )
93, 8syl5eq 2448 . . 3  |-  ( U  e.  _V  ->  ( .s OLD `  U )  =  ( 2nd `  ( 1st `  U ) ) )
10 fvprc 5681 . . . 4  |-  ( -.  U  e.  _V  ->  ( .s OLD `  U
)  =  (/) )
11 fvprc 5681 . . . . . 6  |-  ( -.  U  e.  _V  ->  ( 1st `  U )  =  (/) )
1211fveq2d 5691 . . . . 5  |-  ( -.  U  e.  _V  ->  ( 2nd `  ( 1st `  U ) )  =  ( 2nd `  (/) ) )
13 2nd0 6313 . . . . 5  |-  ( 2nd `  (/) )  =  (/)
1412, 13syl6req 2453 . . . 4  |-  ( -.  U  e.  _V  ->  (/)  =  ( 2nd `  ( 1st `  U ) ) )
1510, 14eqtrd 2436 . . 3  |-  ( -.  U  e.  _V  ->  ( .s OLD `  U
)  =  ( 2nd `  ( 1st `  U
) ) )
169, 15pm2.61i 158 . 2  |-  ( .s
OLD `  U )  =  ( 2nd `  ( 1st `  U ) )
171, 16eqtri 2424 1  |-  S  =  ( 2nd `  ( 1st `  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588    o. ccom 4841   -->wf 5409   -onto->wfo 5411   ` cfv 5413   1stc1st 6306   2ndc2nd 6307   .s OLDcns 22019
This theorem is referenced by:  nvvop  22041  nvsf  22051  nvscl  22060  nvsid  22061  nvsass  22062  nvdi  22064  nvdir  22065  nv2  22066  nv0  22071  nvsz  22072  nvinv  22073  nvtri  22112  cnnvs  22125  phop  22272  phpar  22278  ipdirilem  22283  h2hsm  22431  hhsssm  22713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-1st 6308  df-2nd 6309  df-sm 22029
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