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Theorem lno0 25444
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 10640 . . . . 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 25302 . . . . 5  |-  ( U  e.  NrmCVec  ->  Q  e.  X
)
653ad2ant1 1017 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  Q  e.  X )
72, 6, 63jca 1176 . . 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 2467 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
10 eqid 2467 . . . 4  |-  ( +v
`  W )  =  ( +v `  W
)
11 eqid 2467 . . . 4  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
12 eqid 2467 . . . 4  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
13 lno0.7 . . . 4  |-  L  =  ( U  LnOp  W
)
143, 8, 9, 10, 11, 12, 13lnolin 25442 . . 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 668 . 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 25322 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  Q  e.  X )  ->  (
( -u 1 ( .sOLD `  U ) Q ) ( +v
`  U ) Q )  =  Q )
175, 16mpdan 668 . . . 4  |-  ( U  e.  NrmCVec  ->  ( ( -u
1 ( .sOLD `  U ) Q ) ( +v `  U
) Q )  =  Q )
1817fveq2d 5870 . . 3  |-  ( U  e.  NrmCVec  ->  ( T `  ( ( -u 1
( .sOLD `  U ) Q ) ( +v `  U
) Q ) )  =  ( T `  Q ) )
19183ad2ant1 1017 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  ( ( -u 1 ( .sOLD `  U ) Q ) ( +v `  U
) Q ) )  =  ( T `  Q ) )
20 simp2 997 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  W  e.  NrmCVec )
213, 8, 13lnof 25443 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> Y )
2221, 6ffvelrnd 6023 . . 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 25322 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( T `  Q )  e.  Y )  ->  (
( -u 1 ( .sOLD `  W ) ( T `  Q
) ) ( +v
`  W ) ( T `  Q ) )  =  Z )
2520, 22, 24syl2anc 661 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
( -u 1 ( .sOLD `  W ) ( T `  Q
) ) ( +v
`  W ) ( T `  Q ) )  =  Z )
2615, 19, 253eqtr3d 2516 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Q )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6285   CCcc 9491   1c1 9494   -ucneg 9807   NrmCVeccnv 25250   +vcpv 25251   BaseSetcba 25252   .sOLDcns 25253   0veccn0v 25254    LnOp clno 25428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-ltxr 9634  df-sub 9808  df-neg 9809  df-grpo 24966  df-gid 24967  df-ginv 24968  df-ablo 25057  df-vc 25212  df-nv 25258  df-va 25261  df-ba 25262  df-sm 25263  df-0v 25264  df-nmcv 25266  df-lno 25432
This theorem is referenced by:  lnomul  25448  nmlno0lem  25481  nmlnoubi  25484  lnon0  25486  nmblolbii  25487  blocnilem  25492
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