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Theorem lnop0 22376
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 8675 . . . . . . . . 9  |-  1  e.  CC
2 ax-hv0cl 21413 . . . . . . . . 9  |-  0h  e.  ~H
31, 2hvmulcli 21424 . . . . . . . 8  |-  ( 1  .h  0h )  e. 
~H
4 ax-hvaddid 21414 . . . . . . . 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 21416 . . . . . . . 8  |-  ( 0h  e.  ~H  ->  (
1  .h  0h )  =  0h )
72, 6ax-mp 10 . . . . . . 7  |-  ( 1  .h  0h )  =  0h
85, 7eqtri 2273 . . . . . 6  |-  ( ( 1  .h  0h )  +h  0h )  =  0h
98fveq2i 5380 . . . . 5  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( T `  0h )
10 lnopl 22324 . . . . . . 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 669 . . . . . 6  |-  ( ( T  e.  LinOp  /\  1  e.  CC )  ->  ( T `  ( (
1  .h  0h )  +h  0h ) )  =  ( ( 1  .h  ( T `  0h ) )  +h  ( T `  0h )
) )
121, 11mpan2 655 . . . . 5  |-  ( T  e.  LinOp  ->  ( T `  ( ( 1  .h 
0h )  +h  0h ) )  =  ( ( 1  .h  ( T `  0h )
)  +h  ( T `
 0h ) ) )
139, 12syl5eqr 2299 . . . 4  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  ( ( 1  .h  ( T `  0h )
)  +h  ( T `
 0h ) ) )
14 lnopf 22269 . . . . . . 7  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
15 ffvelrn 5515 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  0h  e.  ~H )  -> 
( T `  0h )  e.  ~H )
162, 15mpan2 655 . . . . . . 7  |-  ( T : ~H --> ~H  ->  ( T `  0h )  e.  ~H )
1714, 16syl 17 . . . . . 6  |-  ( T  e.  LinOp  ->  ( T `  0h )  e.  ~H )
18 ax-hvmulid 21416 . . . . . 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 5725 . . . 4  |-  ( T  e.  LinOp  ->  ( (
1  .h  ( T `
 0h ) )  +h  ( T `  0h ) )  =  ( ( T `  0h )  +h  ( T `  0h ) ) )
2113, 20eqtrd 2285 . . 3  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  ( ( T `  0h )  +h  ( T `  0h ) ) )
2221oveq1d 5725 . 2  |-  ( T  e.  LinOp  ->  ( ( T `  0h )  -h  ( T `  0h ) )  =  ( ( ( T `  0h )  +h  ( T `  0h )
)  -h  ( T `
 0h ) ) )
23 hvsubid 21435 . . 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 21448 . . . 4  |-  ( ( ( T `  0h )  e.  ~H  /\  ( T `  0h )  e.  ~H )  ->  (
( ( T `  0h )  +h  ( T `  0h )
)  -h  ( T `
 0h ) )  =  ( T `  0h ) )
2625anidms 629 . . 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 2294 1  |-  ( T  e.  LinOp  ->  ( T `  0h )  =  0h )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   -->wf 4588   ` cfv 4592  (class class class)co 5710   CCcc 8615   1c1 8618   ~Hchil 21329    +h cva 21330    .h csm 21331   0hc0v 21334    -h cmv 21335   LinOpclo 21357
This theorem is referenced by:  lnopmul  22377  lnop0i  22380
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-hilex 21409  ax-hfvadd 21410  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvdistr2 21419  ax-hvmul0 21420
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752  df-sub 8919  df-neg 8920  df-hvsub 21381  df-lnop 22251
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