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Theorem lnop0 23422
Description: The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnop0  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  0h )

Proof of Theorem lnop0
StepHypRef Expression
1 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
2 ax-hv0cl 22459 . . . . . . . . 9  |-  0h  e.  ~H
31, 2hvmulcli 22470 . . . . . . . 8  |-  ( 1  .h  0h )  e. 
~H
4 ax-hvaddid 22460 . . . . . . . 8  |-  ( ( 1  .h  0h )  e.  ~H  ->  ( (
1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
)
53, 4ax-mp 8 . . . . . . 7  |-  ( ( 1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
6 ax-hvmulid 22462 . . . . . . . 8  |-  ( 0h  e.  ~H  ->  (
1  .h  0h )  =  0h )
72, 6ax-mp 8 . . . . . . 7  |-  ( 1  .h  0h )  =  0h
85, 7eqtri 2424 . . . . . 6  |-  ( ( 1  .h  0h )  +h  0h )  =  0h
98fveq2i 5690 . . . . 5  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( T `  0h )
10 lnopl 23370 . . . . . . 7  |-  ( ( ( T  e.  LinOp  /\  1  e.  CC )  /\  ( 0h  e.  ~H  /\  0h  e.  ~H ) )  ->  ( T `  ( (
1  .h  0h )  +h  0h ) )  =  ( ( 1  .h  ( T `  0h ) )  +h  ( T `  0h )
) )
112, 2, 10mpanr12 667 . . . . . 6  |-  ( ( T  e.  LinOp  /\  1  e.  CC )  ->  ( T `  ( (
1  .h  0h )  +h  0h ) )  =  ( ( 1  .h  ( T `  0h ) )  +h  ( T `  0h )
) )
121, 11mpan2 653 . . . . 5  |-  ( T  e.  LinOp  ->  ( T `  ( ( 1  .h 
0h )  +h  0h ) )  =  ( ( 1  .h  ( T `  0h )
)  +h  ( T `
 0h ) ) )
139, 12syl5eqr 2450 . . . 4  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  ( ( 1  .h  ( T `  0h )
)  +h  ( T `
 0h ) ) )
14 lnopf 23315 . . . . . . 7  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
15 ffvelrn 5827 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  0h  e.  ~H )  -> 
( T `  0h )  e.  ~H )
162, 15mpan2 653 . . . . . . 7  |-  ( T : ~H --> ~H  ->  ( T `  0h )  e.  ~H )
1714, 16syl 16 . . . . . 6  |-  ( T  e.  LinOp  ->  ( T `  0h )  e.  ~H )
18 ax-hvmulid 22462 . . . . . 6  |-  ( ( T `  0h )  e.  ~H  ->  ( 1  .h  ( T `  0h ) )  =  ( T `  0h )
)
1917, 18syl 16 . . . . 5  |-  ( T  e.  LinOp  ->  ( 1  .h  ( T `  0h ) )  =  ( T `  0h )
)
2019oveq1d 6055 . . . 4  |-  ( T  e.  LinOp  ->  ( (
1  .h  ( T `
 0h ) )  +h  ( T `  0h ) )  =  ( ( T `  0h )  +h  ( T `  0h ) ) )
2113, 20eqtrd 2436 . . 3  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  ( ( T `  0h )  +h  ( T `  0h ) ) )
2221oveq1d 6055 . 2  |-  ( T  e.  LinOp  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  ( ( ( T `  0h )  +h  ( T `  0h )
)  -h  ( T `
 0h ) ) )
23 hvsubid 22481 . . 3  |-  ( ( T `  0h )  e.  ~H  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  0h )
2417, 23syl 16 . 2  |-  ( T  e.  LinOp  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  0h )
25 hvpncan 22494 . . . 4  |-  ( ( ( T `  0h )  e.  ~H  /\  ( T `  0h )  e.  ~H )  ->  (
( ( T `  0h )  +h  ( T `  0h )
)  -h  ( T `
 0h ) )  =  ( T `  0h ) )
2625anidms 627 . . 3  |-  ( ( T `  0h )  e.  ~H  ->  ( (
( T `  0h )  +h  ( T `  0h ) )  -h  ( T `  0h )
)  =  ( T `
 0h ) )
2717, 26syl 16 . 2  |-  ( T  e.  LinOp  ->  ( (
( T `  0h )  +h  ( T `  0h ) )  -h  ( T `  0h )
)  =  ( T `
 0h ) )
2822, 24, 273eqtr3rd 2445 1  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  0h )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   1c1 8947   ~Hchil 22375    +h cva 22376    .h csm 22377   0hc0v 22380    -h cmv 22381   LinOpclo 22403
This theorem is referenced by:  lnopmul  23423  lnop0i  23426
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hilex 22455  ax-hfvadd 22456  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvdistr2 22465  ax-hvmul0 22466
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-hvsub 22427  df-lnop 23297
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