Theorem lnop0 28825

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

Step	Hyp	Ref	Expression
1		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
2		ax-hv0cl 27860	. . . . . . . . 9 |- 0h e. ~H
3	1, 2	hvmulcli 27871	. . . . . . . 8 |- ( 1 .h 0h ) e. ~H
4		ax-hvaddid 27861	. . . . . . . 8 |- ( ( 1 .h 0h ) e. ~H -> ( ( 1 .h 0h ) +h 0h ) = ( 1 .h 0h ) )
5	3, 4	ax-mp 5	. . . . . . 7 |- ( ( 1 .h 0h ) +h 0h ) = ( 1 .h 0h )
6		ax-hvmulid 27863	. . . . . . . 8 |- ( 0h e. ~H -> ( 1 .h 0h ) = 0h )
7	2, 6	ax-mp 5	. . . . . . 7 |- ( 1 .h 0h ) = 0h
8	5, 7	eqtri 2644	. . . . . 6 |- ( ( 1 .h 0h ) +h 0h ) = 0h
9	8	fveq2i 6194	. . . . 5 |- ( T ` ( ( 1 .h 0h ) +h 0h ) ) = ( T ` 0h )
10		lnopl 28773	. . . . . . 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 ) ) )
11	2, 2, 10	mpanr12 721	. . . . . 6 |- ( ( T e. LinOp /\ 1 e. CC ) -> ( T ` ( ( 1 .h 0h ) +h 0h ) ) = ( ( 1 .h ( T ` 0h ) ) +h ( T ` 0h ) ) )
12	1, 11	mpan2 707	. . . . 5 |- ( T e. LinOp -> ( T ` ( ( 1 .h 0h ) +h 0h ) ) = ( ( 1 .h ( T ` 0h ) ) +h ( T ` 0h ) ) )
13	9, 12	syl5eqr 2670	. . . 4 |- ( T e. LinOp -> ( T ` 0h ) = ( ( 1 .h ( T ` 0h ) ) +h ( T ` 0h ) ) )
14		lnopf 28718	. . . . . . 7 |- ( T e. LinOp -> T : ~H --> ~H )
15		ffvelrn 6357	. . . . . . . 8 |- ( ( T : ~H --> ~H /\ 0h e. ~H ) -> ( T ` 0h ) e. ~H )
16	2, 15	mpan2 707	. . . . . . 7 |- ( T : ~H --> ~H -> ( T ` 0h ) e. ~H )
17	14, 16	syl 17	. . . . . 6 |- ( T e. LinOp -> ( T ` 0h ) e. ~H )
18		ax-hvmulid 27863	. . . . . 6 |- ( ( T ` 0h ) e. ~H -> ( 1 .h ( T ` 0h ) ) = ( T ` 0h ) )
19	17, 18	syl 17	. . . . 5 |- ( T e. LinOp -> ( 1 .h ( T ` 0h ) ) = ( T ` 0h ) )
20	19	oveq1d 6665	. . . 4 |- ( T e. LinOp -> ( ( 1 .h ( T ` 0h ) ) +h ( T ` 0h ) ) = ( ( T ` 0h ) +h ( T ` 0h ) ) )
21	13, 20	eqtrd 2656	. . 3 |- ( T e. LinOp -> ( T ` 0h ) = ( ( T ` 0h ) +h ( T ` 0h ) ) )
22	21	oveq1d 6665	. 2 |- ( T e. LinOp -> ( ( T ` 0h ) -h ( T ` 0h ) ) = ( ( ( T ` 0h ) +h ( T ` 0h ) ) -h ( T ` 0h ) ) )
23		hvsubid 27883	. . 3 |- ( ( T ` 0h ) e. ~H -> ( ( T ` 0h ) -h ( T ` 0h ) ) = 0h )
24	17, 23	syl 17	. 2 |- ( T e. LinOp -> ( ( T ` 0h ) -h ( T ` 0h ) ) = 0h )
25		hvpncan 27896	. . . 4 |- ( ( ( T ` 0h ) e. ~H /\ ( T ` 0h ) e. ~H ) -> ( ( ( T ` 0h ) +h ( T ` 0h ) ) -h ( T ` 0h ) ) = ( T ` 0h ) )
26	25	anidms 677	. . 3 |- ( ( T ` 0h ) e. ~H -> ( ( ( T ` 0h ) +h ( T ` 0h ) ) -h ( T ` 0h ) ) = ( T ` 0h ) )
27	17, 26	syl 17	. 2 |- ( T e. LinOp -> ( ( ( T ` 0h ) +h ( T ` 0h ) ) -h ( T ` 0h ) ) = ( T ` 0h ) )
28	22, 24, 27	3eqtr3rd 2665	1 |- ( T e. LinOp -> ( T ` 0h ) = 0h )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   ~Hchil 27776   +h cva 27777   .h csm 27778   0hc0v 27781   -h cmv 27782   LinOpclo 27804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-hilex 27856  ax-hfvadd 27857  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvdistr2 27866  ax-hvmul0 27867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-hvsub 27828  df-lnop 28700

This theorem is referenced by:  lnopmul  28826  lnop0i  28829