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Theorem lnop0 22546
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 8795 . . . . . . . . 9  |-  1  e.  CC
2 ax-hv0cl 21583 . . . . . . . . 9  |-  0h  e.  ~H
31, 2hvmulcli 21594 . . . . . . . 8  |-  ( 1  .h  0h )  e. 
~H
4 ax-hvaddid 21584 . . . . . . . 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 21586 . . . . . . . 8  |-  ( 0h  e.  ~H  ->  (
1  .h  0h )  =  0h )
72, 6ax-mp 8 . . . . . . 7  |-  ( 1  .h  0h )  =  0h
85, 7eqtri 2303 . . . . . 6  |-  ( ( 1  .h  0h )  +h  0h )  =  0h
98fveq2i 5528 . . . . 5  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( T `  0h )
10 lnopl 22494 . . . . . . 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 666 . . . . . 6  |-  ( ( T  e.  LinOp  /\  1  e.  CC )  ->  ( T `  ( (
1  .h  0h )  +h  0h ) )  =  ( ( 1  .h  ( T `  0h ) )  +h  ( T `  0h )
) )
121, 11mpan2 652 . . . . 5  |-  ( T  e.  LinOp  ->  ( T `  ( ( 1  .h 
0h )  +h  0h ) )  =  ( ( 1  .h  ( T `  0h )
)  +h  ( T `
 0h ) ) )
139, 12syl5eqr 2329 . . . 4  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  ( ( 1  .h  ( T `  0h )
)  +h  ( T `
 0h ) ) )
14 lnopf 22439 . . . . . . 7  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
15 ffvelrn 5663 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  0h  e.  ~H )  -> 
( T `  0h )  e.  ~H )
162, 15mpan2 652 . . . . . . 7  |-  ( T : ~H --> ~H  ->  ( T `  0h )  e.  ~H )
1714, 16syl 15 . . . . . 6  |-  ( T  e.  LinOp  ->  ( T `  0h )  e.  ~H )
18 ax-hvmulid 21586 . . . . . 6  |-  ( ( T `  0h )  e.  ~H  ->  ( 1  .h  ( T `  0h ) )  =  ( T `  0h )
)
1917, 18syl 15 . . . . 5  |-  ( T  e.  LinOp  ->  ( 1  .h  ( T `  0h ) )  =  ( T `  0h )
)
2019oveq1d 5873 . . . 4  |-  ( T  e.  LinOp  ->  ( (
1  .h  ( T `
 0h ) )  +h  ( T `  0h ) )  =  ( ( T `  0h )  +h  ( T `  0h ) ) )
2113, 20eqtrd 2315 . . 3  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  ( ( T `  0h )  +h  ( T `  0h ) ) )
2221oveq1d 5873 . 2  |-  ( T  e.  LinOp  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  ( ( ( T `  0h )  +h  ( T `  0h )
)  -h  ( T `
 0h ) ) )
23 hvsubid 21605 . . 3  |-  ( ( T `  0h )  e.  ~H  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  0h )
2417, 23syl 15 . 2  |-  ( T  e.  LinOp  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  0h )
25 hvpncan 21618 . . . 4  |-  ( ( ( T `  0h )  e.  ~H  /\  ( T `  0h )  e.  ~H )  ->  (
( ( T `  0h )  +h  ( T `  0h )
)  -h  ( T `
 0h ) )  =  ( T `  0h ) )
2625anidms 626 . . 3  |-  ( ( T `  0h )  e.  ~H  ->  ( (
( T `  0h )  +h  ( T `  0h ) )  -h  ( T `  0h )
)  =  ( T `
 0h ) )
2717, 26syl 15 . 2  |-  ( T  e.  LinOp  ->  ( (
( T `  0h )  +h  ( T `  0h ) )  -h  ( T `  0h )
)  =  ( T `
 0h ) )
2822, 24, 273eqtr3rd 2324 1  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  0h )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   ~Hchil 21499    +h cva 21500    .h csm 21501   0hc0v 21504    -h cmv 21505   LinOpclo 21527
This theorem is referenced by:  lnopmul  22547  lnop0i  22550
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-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  ax-hilex 21579  ax-hfvadd 21580  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvdistr2 21589  ax-hvmul0 21590
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-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-hvsub 21551  df-lnop 22421
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