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Theorem lnop0 22539
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 8791 . . . . . . . . 9  |-  1  e.  CC
2 ax-hv0cl 21576 . . . . . . . . 9  |-  0h  e.  ~H
31, 2hvmulcli 21587 . . . . . . . 8  |-  ( 1  .h  0h )  e. 
~H
4 ax-hvaddid 21577 . . . . . . . 8  |-  ( ( 1  .h  0h )  e.  ~H  ->  ( (
1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
)
53, 4ax-mp 10 . . . . . . 7  |-  ( ( 1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
6 ax-hvmulid 21579 . . . . . . . 8  |-  ( 0h  e.  ~H  ->  (
1  .h  0h )  =  0h )
72, 6ax-mp 10 . . . . . . 7  |-  ( 1  .h  0h )  =  0h
85, 7eqtri 2305 . . . . . 6  |-  ( ( 1  .h  0h )  +h  0h )  =  0h
98fveq2i 5489 . . . . 5  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( T `  0h )
10 lnopl 22487 . . . . . . 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 668 . . . . . 6  |-  ( ( T  e.  LinOp  /\  1  e.  CC )  ->  ( T `  ( (
1  .h  0h )  +h  0h ) )  =  ( ( 1  .h  ( T `  0h ) )  +h  ( T `  0h )
) )
121, 11mpan2 654 . . . . 5  |-  ( T  e.  LinOp  ->  ( T `  ( ( 1  .h 
0h )  +h  0h ) )  =  ( ( 1  .h  ( T `  0h )
)  +h  ( T `
 0h ) ) )
139, 12syl5eqr 2331 . . . 4  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  ( ( 1  .h  ( T `  0h )
)  +h  ( T `
 0h ) ) )
14 lnopf 22432 . . . . . . 7  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
15 ffvelrn 5625 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  0h  e.  ~H )  -> 
( T `  0h )  e.  ~H )
162, 15mpan2 654 . . . . . . 7  |-  ( T : ~H --> ~H  ->  ( T `  0h )  e.  ~H )
1714, 16syl 17 . . . . . 6  |-  ( T  e.  LinOp  ->  ( T `  0h )  e.  ~H )
18 ax-hvmulid 21579 . . . . . 6  |-  ( ( T `  0h )  e.  ~H  ->  ( 1  .h  ( T `  0h ) )  =  ( T `  0h )
)
1917, 18syl 17 . . . . 5  |-  ( T  e.  LinOp  ->  ( 1  .h  ( T `  0h ) )  =  ( T `  0h )
)
2019oveq1d 5835 . . . 4  |-  ( T  e.  LinOp  ->  ( (
1  .h  ( T `
 0h ) )  +h  ( T `  0h ) )  =  ( ( T `  0h )  +h  ( T `  0h ) ) )
2113, 20eqtrd 2317 . . 3  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  ( ( T `  0h )  +h  ( T `  0h ) ) )
2221oveq1d 5835 . 2  |-  ( T  e.  LinOp  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  ( ( ( T `  0h )  +h  ( T `  0h )
)  -h  ( T `
 0h ) ) )
23 hvsubid 21598 . . 3  |-  ( ( T `  0h )  e.  ~H  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  0h )
2417, 23syl 17 . 2  |-  ( T  e.  LinOp  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  0h )
25 hvpncan 21611 . . . 4  |-  ( ( ( T `  0h )  e.  ~H  /\  ( T `  0h )  e.  ~H )  ->  (
( ( T `  0h )  +h  ( T `  0h )
)  -h  ( T `
 0h ) )  =  ( T `  0h ) )
2625anidms 628 . . 3  |-  ( ( T `  0h )  e.  ~H  ->  ( (
( T `  0h )  +h  ( T `  0h ) )  -h  ( T `  0h )
)  =  ( T `
 0h ) )
2717, 26syl 17 . 2  |-  ( T  e.  LinOp  ->  ( (
( T `  0h )  +h  ( T `  0h ) )  -h  ( T `  0h )
)  =  ( T `
 0h ) )
2822, 24, 273eqtr3rd 2326 1  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  0h )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   -->wf 5218   ` cfv 5222  (class class class)co 5820   CCcc 8731   1c1 8734   ~Hchil 21492    +h cva 21493    .h csm 21494   0hc0v 21497    -h cmv 21498   LinOpclo 21520
This theorem is referenced by:  lnopmul  22540  lnop0i  22543
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-hilex 21572  ax-hfvadd 21573  ax-hvass 21575  ax-hv0cl 21576  ax-hvaddid 21577  ax-hfvmul 21578  ax-hvmulid 21579  ax-hvdistr2 21582  ax-hvmul0 21583
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868  df-sub 9035  df-neg 9036  df-hvsub 21544  df-lnop 22414
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