Theorem lnopaddi 28830

Description: Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)

Hypothesis

Ref	Expression
lnopl.1	|- T e. LinOp

Assertion

Ref	Expression
lnopaddi	|- ( ( A e. ~H /\ B e. ~H ) -> ( T ` ( A +h B ) ) = ( ( T ` A ) +h ( T ` B ) ) )

Proof of Theorem lnopaddi

Step	Hyp	Ref	Expression
1		ax-1cn 9994	. . 3 |- 1 e. CC
2		lnopl.1	. . . 4 |- T e. LinOp
3	2	lnopli 28827	. . 3 |- ( ( 1 e. CC /\ A e. ~H /\ B e. ~H ) -> ( T ` ( ( 1 .h A ) +h B ) ) = ( ( 1 .h ( T ` A ) ) +h ( T ` B ) ) )
4	1, 3	mp3an1 1411	. 2 |- ( ( A e. ~H /\ B e. ~H ) -> ( T ` ( ( 1 .h A ) +h B ) ) = ( ( 1 .h ( T ` A ) ) +h ( T ` B ) ) )
5		ax-hvmulid 27863	. . . . 5 |- ( A e. ~H -> ( 1 .h A ) = A )
6	5	oveq1d 6665	. . . 4 |- ( A e. ~H -> ( ( 1 .h A ) +h B ) = ( A +h B ) )
7	6	fveq2d 6195	. . 3 |- ( A e. ~H -> ( T ` ( ( 1 .h A ) +h B ) ) = ( T ` ( A +h B ) ) )
8	7	adantr 481	. 2 |- ( ( A e. ~H /\ B e. ~H ) -> ( T ` ( ( 1 .h A ) +h B ) ) = ( T ` ( A +h B ) ) )
9	2	lnopfi 28828	. . . . . 6 |- T : ~H --> ~H
10	9	ffvelrni 6358	. . . . 5 |- ( A e. ~H -> ( T ` A ) e. ~H )
11		ax-hvmulid 27863	. . . . 5 |- ( ( T ` A ) e. ~H -> ( 1 .h ( T ` A ) ) = ( T ` A ) )
12	10, 11	syl 17	. . . 4 |- ( A e. ~H -> ( 1 .h ( T ` A ) ) = ( T ` A ) )
13	12	adantr 481	. . 3 |- ( ( A e. ~H /\ B e. ~H ) -> ( 1 .h ( T ` A ) ) = ( T ` A ) )
14	13	oveq1d 6665	. 2 |- ( ( A e. ~H /\ B e. ~H ) -> ( ( 1 .h ( T ` A ) ) +h ( T ` B ) ) = ( ( T ` A ) +h ( T ` B ) ) )
15	4, 8, 14	3eqtr3d 2664	1 |- ( ( A e. ~H /\ B e. ~H ) -> ( T ` ( A +h B ) ) = ( ( T ` A ) +h ( T ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   ~Hchil 27776   +h cva 27777   .h csm 27778   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-1cn 9994  ax-hilex 27856  ax-hvmulid 27863

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-lnop 28700

This theorem is referenced by:  lnopaddmuli  28832  lnophsi  28860  lnopeq0lem1  28864  lnophmlem2  28876  imaelshi  28917  cnlnadjlem2  28927