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Theorem smfval 25271
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 25263 . . . . 5  |-  .sOLD  =  ( 2nd  o.  1st )
32fveq1i 5867 . . . 4  |-  ( .sOLD `  U )  =  ( ( 2nd 
o.  1st ) `  U
)
4 fo1st 6805 . . . . . 6  |-  1st : _V -onto-> _V
5 fof 5795 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
64, 5ax-mp 5 . . . . 5  |-  1st : _V
--> _V
7 fvco3 5945 . . . . 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 2520 . . 3  |-  ( U  e.  _V  ->  ( .sOLD `  U )  =  ( 2nd `  ( 1st `  U ) ) )
10 fvprc 5860 . . . 4  |-  ( -.  U  e.  _V  ->  ( .sOLD `  U
)  =  (/) )
11 fvprc 5860 . . . . . 6  |-  ( -.  U  e.  _V  ->  ( 1st `  U )  =  (/) )
1211fveq2d 5870 . . . . 5  |-  ( -.  U  e.  _V  ->  ( 2nd `  ( 1st `  U ) )  =  ( 2nd `  (/) ) )
13 2nd0 6792 . . . . 5  |-  ( 2nd `  (/) )  =  (/)
1412, 13syl6req 2525 . . . 4  |-  ( -.  U  e.  _V  ->  (/)  =  ( 2nd `  ( 1st `  U ) ) )
1510, 14eqtrd 2508 . . 3  |-  ( -.  U  e.  _V  ->  ( .sOLD `  U
)  =  ( 2nd `  ( 1st `  U
) ) )
169, 15pm2.61i 164 . 2  |-  ( .sOLD `  U )  =  ( 2nd `  ( 1st `  U ) )
171, 16eqtri 2496 1  |-  S  =  ( 2nd `  ( 1st `  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785    o. ccom 5003   -->wf 5584   -onto->wfo 5586   ` cfv 5588   1stc1st 6783   2ndc2nd 6784   .sOLDcns 25253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-1st 6785  df-2nd 6786  df-sm 25263
This theorem is referenced by:  nvvop  25275  nvsf  25285  nvscl  25294  nvsid  25295  nvsass  25296  nvdi  25298  nvdir  25299  nv2  25300  nv0  25305  nvsz  25306  nvinv  25307  nvtri  25346  cnnvs  25359  phop  25506  phpar  25512  ipdirilem  25517  h2hsm  25665  hhsssm  25949
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