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Theorem lnof 22105
Description: A linear operator is a mapping. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnof.1  |-  X  =  ( BaseSet `  U )
lnof.2  |-  Y  =  ( BaseSet `  W )
lnof.7  |-  L  =  ( U  LnOp  W
)
Assertion
Ref Expression
lnof  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> Y )

Proof of Theorem lnof
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnof.1 . . . 4  |-  X  =  ( BaseSet `  U )
2 lnof.2 . . . 4  |-  Y  =  ( BaseSet `  W )
3 eqid 2388 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
4 eqid 2388 . . . 4  |-  ( +v
`  W )  =  ( +v `  W
)
5 eqid 2388 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
6 eqid 2388 . . . 4  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
7 lnof.7 . . . 4  |-  L  =  ( U  LnOp  W
)
81, 2, 3, 4, 5, 6, 7islno 22103 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  L  <->  ( T : X --> Y  /\  A. x  e.  CC  A. y  e.  X  A. z  e.  X  ( T `  ( ( x ( .s OLD `  U
) y ) ( +v `  U ) z ) )  =  ( ( x ( .s OLD `  W
) ( T `  y ) ) ( +v `  W ) ( T `  z
) ) ) ) )
98simprbda 607 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  T  e.  L )  ->  T : X --> Y )
1093impa 1148 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   -->wf 5391   ` cfv 5395  (class class class)co 6021   CCcc 8922   NrmCVeccnv 21912   +vcpv 21913   BaseSetcba 21914   .s
OLDcns 21915    LnOp clno 22090
This theorem is referenced by:  lno0  22106  lnocoi  22107  lnoadd  22108  lnosub  22109  lnomul  22110  isblo2  22133  blof  22135  nmlno0lem  22143  nmlnoubi  22146  nmlnogt0  22147  lnon0  22148  isblo3i  22151  blocnilem  22154  blocni  22155  htthlem  22269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-map 6957  df-lno 22094
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