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Theorem lno0 22106
Description: The value of a linear operator at zero is zero. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lno0.1  |-  X  =  ( BaseSet `  U )
lno0.2  |-  Y  =  ( BaseSet `  W )
lno0.5  |-  Q  =  ( 0vec `  U
)
lno0.z  |-  Z  =  ( 0vec `  W
)
lno0.7  |-  L  =  ( U  LnOp  W
)
Assertion
Ref Expression
lno0  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Q )  =  Z )

Proof of Theorem lno0
StepHypRef Expression
1 neg1cn 10000 . . . . 5  |-  -u 1  e.  CC
21a1i 11 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  -u 1  e.  CC )
3 lno0.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
4 lno0.5 . . . . . 6  |-  Q  =  ( 0vec `  U
)
53, 4nvzcl 21964 . . . . 5  |-  ( U  e.  NrmCVec  ->  Q  e.  X
)
653ad2ant1 978 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  Q  e.  X )
72, 6, 63jca 1134 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( -u 1  e.  CC  /\  Q  e.  X  /\  Q  e.  X )
)
8 lno0.2 . . . 4  |-  Y  =  ( BaseSet `  W )
9 eqid 2388 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
10 eqid 2388 . . . 4  |-  ( +v
`  W )  =  ( +v `  W
)
11 eqid 2388 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
12 eqid 2388 . . . 4  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
13 lno0.7 . . . 4  |-  L  =  ( U  LnOp  W
)
143, 8, 9, 10, 11, 12, 13lnolin 22104 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( -u 1  e.  CC  /\  Q  e.  X  /\  Q  e.  X )
)  ->  ( T `  ( ( -u 1
( .s OLD `  U
) Q ) ( +v `  U ) Q ) )  =  ( ( -u 1
( .s OLD `  W
) ( T `  Q ) ) ( +v `  W ) ( T `  Q
) ) )
157, 14mpdan 650 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  ( ( -u 1 ( .s OLD `  U ) Q ) ( +v `  U
) Q ) )  =  ( ( -u
1 ( .s OLD `  W ) ( T `
 Q ) ) ( +v `  W
) ( T `  Q ) ) )
163, 9, 11, 4nvlinv 21984 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  Q  e.  X )  ->  (
( -u 1 ( .s
OLD `  U ) Q ) ( +v
`  U ) Q )  =  Q )
175, 16mpdan 650 . . . 4  |-  ( U  e.  NrmCVec  ->  ( ( -u
1 ( .s OLD `  U ) Q ) ( +v `  U
) Q )  =  Q )
1817fveq2d 5673 . . 3  |-  ( U  e.  NrmCVec  ->  ( T `  ( ( -u 1
( .s OLD `  U
) Q ) ( +v `  U ) Q ) )  =  ( T `  Q
) )
19183ad2ant1 978 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  ( ( -u 1 ( .s OLD `  U ) Q ) ( +v `  U
) Q ) )  =  ( T `  Q ) )
20 simp2 958 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  W  e.  NrmCVec )
213, 8, 13lnof 22105 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> Y )
2221, 6ffvelrnd 5811 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Q )  e.  Y )
23 lno0.z . . . 4  |-  Z  =  ( 0vec `  W
)
248, 10, 12, 23nvlinv 21984 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( T `  Q )  e.  Y )  ->  (
( -u 1 ( .s
OLD `  W )
( T `  Q
) ) ( +v
`  W ) ( T `  Q ) )  =  Z )
2520, 22, 24syl2anc 643 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
( -u 1 ( .s
OLD `  W )
( T `  Q
) ) ( +v
`  W ) ( T `  Q ) )  =  Z )
2615, 19, 253eqtr3d 2428 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Q )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   CCcc 8922   1c1 8925   -ucneg 9225   NrmCVeccnv 21912   +vcpv 21913   BaseSetcba 21914   .s
OLDcns 21915   0veccn0v 21916    LnOp clno 22090
This theorem is referenced by:  lnomul  22110  nmlno0lem  22143  nmlnoubi  22146  lnon0  22148  nmblolbii  22149  blocnilem  22154
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-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  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-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-ltxr 9059  df-sub 9226  df-neg 9227  df-grpo 21628  df-gid 21629  df-ginv 21630  df-ablo 21719  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-nmcv 21928  df-lno 22094
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