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Theorem smfval 21474
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 21466 . . . . 5  |-  .s OLD  =  ( 2nd  o.  1st )
32fveq1i 5633 . . . 4  |-  ( .s
OLD `  U )  =  ( ( 2nd 
o.  1st ) `  U
)
4 fo1st 6266 . . . . . 6  |-  1st : _V -onto-> _V
5 fof 5557 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
64, 5ax-mp 8 . . . . 5  |-  1st : _V
--> _V
7 fvco3 5703 . . . . 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 2410 . . 3  |-  ( U  e.  _V  ->  ( .s OLD `  U )  =  ( 2nd `  ( 1st `  U ) ) )
10 fvprc 5626 . . . 4  |-  ( -.  U  e.  _V  ->  ( .s OLD `  U
)  =  (/) )
11 fvprc 5626 . . . . . 6  |-  ( -.  U  e.  _V  ->  ( 1st `  U )  =  (/) )
1211fveq2d 5636 . . . . 5  |-  ( -.  U  e.  _V  ->  ( 2nd `  ( 1st `  U ) )  =  ( 2nd `  (/) ) )
13 2nd0 6254 . . . . 5  |-  ( 2nd `  (/) )  =  (/)
1412, 13syl6req 2415 . . . 4  |-  ( -.  U  e.  _V  ->  (/)  =  ( 2nd `  ( 1st `  U ) ) )
1510, 14eqtrd 2398 . . 3  |-  ( -.  U  e.  _V  ->  ( .s OLD `  U
)  =  ( 2nd `  ( 1st `  U
) ) )
169, 15pm2.61i 156 . 2  |-  ( .s
OLD `  U )  =  ( 2nd `  ( 1st `  U ) )
171, 16eqtri 2386 1  |-  S  =  ( 2nd `  ( 1st `  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1647    e. wcel 1715   _Vcvv 2873   (/)c0 3543    o. ccom 4796   -->wf 5354   -onto->wfo 5356   ` cfv 5358   1stc1st 6247   2ndc2nd 6248   .s OLDcns 21456
This theorem is referenced by:  nvvop  21478  nvsf  21488  nvscl  21497  nvsid  21498  nvsass  21499  nvdi  21501  nvdir  21502  nv2  21503  nv0  21508  nvsz  21509  nvinv  21510  nvtri  21549  cnnvs  21562  phop  21709  phpar  21715  ipdirilem  21720  h2hsm  21868  hhsssm  22150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366  df-1st 6249  df-2nd 6250  df-sm 21466
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