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Theorem smfval 21161
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 21153 . . . . 5  |-  .s OLD  =  ( 2nd  o.  1st )
32fveq1i 5526 . . . 4  |-  ( .s
OLD `  U )  =  ( ( 2nd 
o.  1st ) `  U
)
4 fo1st 6139 . . . . . 6  |-  1st : _V -onto-> _V
5 fof 5451 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
64, 5ax-mp 8 . . . . 5  |-  1st : _V
--> _V
7 fvco3 5596 . . . . 5  |-  ( ( 1st : _V --> _V  /\  U  e.  _V )  ->  ( ( 2nd  o.  1st ) `  U )  =  ( 2nd `  ( 1st `  U ) ) )
86, 7mpan 651 . . . 4  |-  ( U  e.  _V  ->  (
( 2nd  o.  1st ) `  U )  =  ( 2nd `  ( 1st `  U ) ) )
93, 8syl5eq 2327 . . 3  |-  ( U  e.  _V  ->  ( .s OLD `  U )  =  ( 2nd `  ( 1st `  U ) ) )
10 fvprc 5519 . . . 4  |-  ( -.  U  e.  _V  ->  ( .s OLD `  U
)  =  (/) )
11 fvprc 5519 . . . . . 6  |-  ( -.  U  e.  _V  ->  ( 1st `  U )  =  (/) )
1211fveq2d 5529 . . . . 5  |-  ( -.  U  e.  _V  ->  ( 2nd `  ( 1st `  U ) )  =  ( 2nd `  (/) ) )
13 2nd0 6127 . . . . 5  |-  ( 2nd `  (/) )  =  (/)
1412, 13syl6req 2332 . . . 4  |-  ( -.  U  e.  _V  ->  (/)  =  ( 2nd `  ( 1st `  U ) ) )
1510, 14eqtrd 2315 . . 3  |-  ( -.  U  e.  _V  ->  ( .s OLD `  U
)  =  ( 2nd `  ( 1st `  U
) ) )
169, 15pm2.61i 156 . 2  |-  ( .s
OLD `  U )  =  ( 2nd `  ( 1st `  U ) )
171, 16eqtri 2303 1  |-  S  =  ( 2nd `  ( 1st `  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   .s OLDcns 21143
This theorem is referenced by:  nvvop  21165  nvsf  21175  nvscl  21184  nvsid  21185  nvsass  21186  nvdi  21188  nvdir  21189  nv2  21190  nv0  21195  nvsz  21196  nvinv  21197  nvtri  21236  cnnvs  21249  phop  21396  phpar  21402  ipdirilem  21407  h2hsm  21555  hhsssm  21837
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-sm 21153
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