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Theorem lnof 22246
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 2435 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
4 eqid 2435 . . . 4  |-  ( +v
`  W )  =  ( +v `  W
)
5 eqid 2435 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
6 eqid 2435 . . . 4  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
7 lnof.7 . . . 4  |-  L  =  ( U  LnOp  W
)
81, 2, 3, 4, 5, 6, 7islno 22244 . . 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 1652    e. wcel 1725   A.wral 2697   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8978   NrmCVeccnv 22053   +vcpv 22054   BaseSetcba 22055   .s
OLDcns 22056    LnOp clno 22231
This theorem is referenced by:  lno0  22247  lnocoi  22248  lnoadd  22249  lnosub  22250  lnomul  22251  isblo2  22274  blof  22276  nmlno0lem  22284  nmlnoubi  22287  nmlnogt0  22288  lnon0  22289  isblo3i  22292  blocnilem  22295  blocni  22296  htthlem  22410
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-lno 22235
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