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Theorem lno0 26242
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 10713 . . . . 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 26100 . . . . 5  |-  ( U  e.  NrmCVec  ->  Q  e.  X
)
653ad2ant1 1026 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  Q  e.  X )
72, 6, 63jca 1185 . . 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 2429 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
10 eqid 2429 . . . 4  |-  ( +v
`  W )  =  ( +v `  W
)
11 eqid 2429 . . . 4  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
12 eqid 2429 . . . 4  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
13 lno0.7 . . . 4  |-  L  =  ( U  LnOp  W
)
143, 8, 9, 10, 11, 12, 13lnolin 26240 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( -u 1  e.  CC  /\  Q  e.  X  /\  Q  e.  X )
)  ->  ( T `  ( ( -u 1
( .sOLD `  U ) Q ) ( +v `  U
) Q ) )  =  ( ( -u
1 ( .sOLD `  W ) ( T `
 Q ) ) ( +v `  W
) ( T `  Q ) ) )
157, 14mpdan 672 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  ( ( -u 1 ( .sOLD `  U ) Q ) ( +v `  U
) Q ) )  =  ( ( -u
1 ( .sOLD `  W ) ( T `
 Q ) ) ( +v `  W
) ( T `  Q ) ) )
163, 9, 11, 4nvlinv 26120 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  Q  e.  X )  ->  (
( -u 1 ( .sOLD `  U ) Q ) ( +v
`  U ) Q )  =  Q )
175, 16mpdan 672 . . . 4  |-  ( U  e.  NrmCVec  ->  ( ( -u
1 ( .sOLD `  U ) Q ) ( +v `  U
) Q )  =  Q )
1817fveq2d 5885 . . 3  |-  ( U  e.  NrmCVec  ->  ( T `  ( ( -u 1
( .sOLD `  U ) Q ) ( +v `  U
) Q ) )  =  ( T `  Q ) )
19183ad2ant1 1026 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  ( ( -u 1 ( .sOLD `  U ) Q ) ( +v `  U
) Q ) )  =  ( T `  Q ) )
20 simp2 1006 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  W  e.  NrmCVec )
213, 8, 13lnof 26241 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> Y )
2221, 6ffvelrnd 6038 . . 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 26120 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( T `  Q )  e.  Y )  ->  (
( -u 1 ( .sOLD `  W ) ( T `  Q
) ) ( +v
`  W ) ( T `  Q ) )  =  Z )
2520, 22, 24syl2anc 665 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
( -u 1 ( .sOLD `  W ) ( T `  Q
) ) ( +v
`  W ) ( T `  Q ) )  =  Z )
2615, 19, 253eqtr3d 2478 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Q )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   CCcc 9536   1c1 9539   -ucneg 9860   NrmCVeccnv 26048   +vcpv 26049   BaseSetcba 26050   .sOLDcns 26051   0veccn0v 26052    LnOp clno 26226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-ltxr 9679  df-sub 9861  df-neg 9862  df-grpo 25764  df-gid 25765  df-ginv 25766  df-ablo 25855  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-nmcv 26064  df-lno 26230
This theorem is referenced by:  lnomul  26246  nmlno0lem  26279  nmlnoubi  26282  lnon0  26284  nmblolbii  26285  blocnilem  26290
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