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Theorem nvsf 22098
Description: Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsf.1  |-  X  =  ( BaseSet `  U )
nvsf.4  |-  S  =  ( .s OLD `  U
)
Assertion
Ref Expression
nvsf  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  X ) --> X )

Proof of Theorem nvsf
StepHypRef Expression
1 eqid 2436 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 22094 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 eqid 2436 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 22082 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvsf.4 . . . 4  |-  S  =  ( .s OLD `  U
)
65smfval 22084 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvsf.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 22083 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vcsm 22028 . 2  |-  ( ( 1st `  U )  e.  CVec OLD  ->  S :
( CC  X.  X
) --> X )
102, 9syl 16 1  |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  X ) --> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    X. cxp 4876   -->wf 5450   ` cfv 5454   1stc1st 6347   CCcc 8988   CVec OLDcvc 22024   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .s
OLDcns 22066
This theorem is referenced by:  nvinvfval  22121  smcnlem  22193  ssps  22229  hlmulf  22406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-1st 6349  df-2nd 6350  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-nmcv 22079
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