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Theorem smfval 24155
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 24147 . . . . 5  |-  .sOLD  =  ( 2nd  o.  1st )
32fveq1i 5803 . . . 4  |-  ( .sOLD `  U )  =  ( ( 2nd 
o.  1st ) `  U
)
4 fo1st 6709 . . . . . 6  |-  1st : _V -onto-> _V
5 fof 5731 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
64, 5ax-mp 5 . . . . 5  |-  1st : _V
--> _V
7 fvco3 5880 . . . . 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 2507 . . 3  |-  ( U  e.  _V  ->  ( .sOLD `  U )  =  ( 2nd `  ( 1st `  U ) ) )
10 fvprc 5796 . . . 4  |-  ( -.  U  e.  _V  ->  ( .sOLD `  U
)  =  (/) )
11 fvprc 5796 . . . . . 6  |-  ( -.  U  e.  _V  ->  ( 1st `  U )  =  (/) )
1211fveq2d 5806 . . . . 5  |-  ( -.  U  e.  _V  ->  ( 2nd `  ( 1st `  U ) )  =  ( 2nd `  (/) ) )
13 2nd0 6697 . . . . 5  |-  ( 2nd `  (/) )  =  (/)
1412, 13syl6req 2512 . . . 4  |-  ( -.  U  e.  _V  ->  (/)  =  ( 2nd `  ( 1st `  U ) ) )
1510, 14eqtrd 2495 . . 3  |-  ( -.  U  e.  _V  ->  ( .sOLD `  U
)  =  ( 2nd `  ( 1st `  U
) ) )
169, 15pm2.61i 164 . 2  |-  ( .sOLD `  U )  =  ( 2nd `  ( 1st `  U ) )
171, 16eqtri 2483 1  |-  S  =  ( 2nd `  ( 1st `  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3748    o. ccom 4955   -->wf 5525   -onto->wfo 5527   ` cfv 5529   1stc1st 6688   2ndc2nd 6689   .sOLDcns 24137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-1st 6690  df-2nd 6691  df-sm 24147
This theorem is referenced by:  nvvop  24159  nvsf  24169  nvscl  24178  nvsid  24179  nvsass  24180  nvdi  24182  nvdir  24183  nv2  24184  nv0  24189  nvsz  24190  nvinv  24191  nvtri  24230  cnnvs  24243  phop  24390  phpar  24396  ipdirilem  24401  h2hsm  24549  hhsssm  24833
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