MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lno0 Unicode version

Theorem lno0 21334
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 9813 . . . . 5  |-  -u 1  e.  CC
21a1i 10 . . . 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 21192 . . . . 5  |-  ( U  e.  NrmCVec  ->  Q  e.  X
)
653ad2ant1 976 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  Q  e.  X )
72, 6, 63jca 1132 . . 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 2283 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
10 eqid 2283 . . . 4  |-  ( +v
`  W )  =  ( +v `  W
)
11 eqid 2283 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
12 eqid 2283 . . . 4  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
13 lno0.7 . . . 4  |-  L  =  ( U  LnOp  W
)
143, 8, 9, 10, 11, 12, 13lnolin 21332 . . 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 649 . 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 21212 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  Q  e.  X )  ->  (
( -u 1 ( .s
OLD `  U ) Q ) ( +v
`  U ) Q )  =  Q )
175, 16mpdan 649 . . . 4  |-  ( U  e.  NrmCVec  ->  ( ( -u
1 ( .s OLD `  U ) Q ) ( +v `  U
) Q )  =  Q )
1817fveq2d 5529 . . 3  |-  ( U  e.  NrmCVec  ->  ( T `  ( ( -u 1
( .s OLD `  U
) Q ) ( +v `  U ) Q ) )  =  ( T `  Q
) )
19183ad2ant1 976 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  ( ( -u 1 ( .s OLD `  U ) Q ) ( +v `  U
) Q ) )  =  ( T `  Q ) )
20 simp2 956 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  W  e.  NrmCVec )
213, 8, 13lnof 21333 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> Y )
22 ffvelrn 5663 . . . 4  |-  ( ( T : X --> Y  /\  Q  e.  X )  ->  ( T `  Q
)  e.  Y )
2321, 6, 22syl2anc 642 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Q )  e.  Y )
24 lno0.z . . . 4  |-  Z  =  ( 0vec `  W
)
258, 10, 12, 24nvlinv 21212 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( T `  Q )  e.  Y )  ->  (
( -u 1 ( .s
OLD `  W )
( T `  Q
) ) ( +v
`  W ) ( T `  Q ) )  =  Z )
2620, 23, 25syl2anc 642 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
( -u 1 ( .s
OLD `  W )
( T `  Q
) ) ( +v
`  W ) ( T `  Q ) )  =  Z )
2715, 19, 263eqtr3d 2323 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Q )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   -ucneg 9038   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144    LnOp clno 21318
This theorem is referenced by:  lnomul  21338  nmlno0lem  21371  nmlnoubi  21374  lnon0  21376  nmblolbii  21377  blocnilem  21382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-lno 21322
  Copyright terms: Public domain W3C validator