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Theorem smfval 22089
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 22081 . . . . 5  |-  .s OLD  =  ( 2nd  o.  1st )
32fveq1i 5732 . . . 4  |-  ( .s
OLD `  U )  =  ( ( 2nd 
o.  1st ) `  U
)
4 fo1st 6369 . . . . . 6  |-  1st : _V -onto-> _V
5 fof 5656 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
64, 5ax-mp 5 . . . . 5  |-  1st : _V
--> _V
7 fvco3 5803 . . . . 5  |-  ( ( 1st : _V --> _V  /\  U  e.  _V )  ->  ( ( 2nd  o.  1st ) `  U )  =  ( 2nd `  ( 1st `  U ) ) )
86, 7mpan 653 . . . 4  |-  ( U  e.  _V  ->  (
( 2nd  o.  1st ) `  U )  =  ( 2nd `  ( 1st `  U ) ) )
93, 8syl5eq 2482 . . 3  |-  ( U  e.  _V  ->  ( .s OLD `  U )  =  ( 2nd `  ( 1st `  U ) ) )
10 fvprc 5725 . . . 4  |-  ( -.  U  e.  _V  ->  ( .s OLD `  U
)  =  (/) )
11 fvprc 5725 . . . . . 6  |-  ( -.  U  e.  _V  ->  ( 1st `  U )  =  (/) )
1211fveq2d 5735 . . . . 5  |-  ( -.  U  e.  _V  ->  ( 2nd `  ( 1st `  U ) )  =  ( 2nd `  (/) ) )
13 2nd0 6357 . . . . 5  |-  ( 2nd `  (/) )  =  (/)
1412, 13syl6req 2487 . . . 4  |-  ( -.  U  e.  _V  ->  (/)  =  ( 2nd `  ( 1st `  U ) ) )
1510, 14eqtrd 2470 . . 3  |-  ( -.  U  e.  _V  ->  ( .s OLD `  U
)  =  ( 2nd `  ( 1st `  U
) ) )
169, 15pm2.61i 159 . 2  |-  ( .s
OLD `  U )  =  ( 2nd `  ( 1st `  U ) )
171, 16eqtri 2458 1  |-  S  =  ( 2nd `  ( 1st `  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630    o. ccom 4885   -->wf 5453   -onto->wfo 5455   ` cfv 5457   1stc1st 6350   2ndc2nd 6351   .s OLDcns 22071
This theorem is referenced by:  nvvop  22093  nvsf  22103  nvscl  22112  nvsid  22113  nvsass  22114  nvdi  22116  nvdir  22117  nv2  22118  nv0  22123  nvsz  22124  nvinv  22125  nvtri  22164  cnnvs  22177  phop  22324  phpar  22330  ipdirilem  22335  h2hsm  22483  hhsssm  22765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-1st 6352  df-2nd 6353  df-sm 22081
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