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Theorem lnof 22261
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 2438 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
4 eqid 2438 . . . 4  |-  ( +v
`  W )  =  ( +v `  W
)
5 eqid 2438 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
6 eqid 2438 . . . 4  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
7 lnof.7 . . . 4  |-  L  =  ( U  LnOp  W
)
81, 2, 3, 4, 5, 6, 7islno 22259 . . 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 608 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  T  e.  L )  ->  T : X --> Y )
1093impa 1149 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   -->wf 5453   ` cfv 5457  (class class class)co 6084   CCcc 8993   NrmCVeccnv 22068   +vcpv 22069   BaseSetcba 22070   .s
OLDcns 22071    LnOp clno 22246
This theorem is referenced by:  lno0  22262  lnocoi  22263  lnoadd  22264  lnosub  22265  lnomul  22266  isblo2  22289  blof  22291  nmlno0lem  22299  nmlnoubi  22302  nmlnogt0  22303  lnon0  22304  isblo3i  22307  blocnilem  22310  blocni  22311  htthlem  22425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-lno 22250
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