Theorem lnfnaddi 28902

Description: Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
lnfnl.1	|- T e. LinFn

Assertion

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

Proof of Theorem lnfnaddi

Step	Hyp	Ref	Expression
1		ax-1cn 9994	. . 3 |- 1 e. CC
2		lnfnl.1	. . . 4 |- T e. LinFn
3	2	lnfnli 28899	. . 3 |- ( ( 1 e. CC /\ A e. ~H /\ B e. ~H ) -> ( T ` ( ( 1 .h A ) +h B ) ) = ( ( 1 x. ( T ` A ) ) + ( T ` B ) ) )
4	1, 3	mp3an1 1411	. 2 |- ( ( A e. ~H /\ B e. ~H ) -> ( T ` ( ( 1 .h A ) +h B ) ) = ( ( 1 x. ( T ` A ) ) + ( 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	lnfnfi 28900	. . . . . 6 |- T : ~H --> CC
10	9	ffvelrni 6358	. . . . 5 |- ( A e. ~H -> ( T ` A ) e. CC )
11	10	mulid2d 10058	. . . 4 |- ( A e. ~H -> ( 1 x. ( T ` A ) ) = ( T ` A ) )
12	11	adantr 481	. . 3 |- ( ( A e. ~H /\ B e. ~H ) -> ( 1 x. ( T ` A ) ) = ( T ` A ) )
13	12	oveq1d 6665	. 2 |- ( ( A e. ~H /\ B e. ~H ) -> ( ( 1 x. ( T ` A ) ) + ( T ` B ) ) = ( ( T ` A ) + ( T ` B ) ) )
14	4, 8, 13	3eqtr3d 2664	1 |- ( ( A e. ~H /\ B e. ~H ) -> ( T ` ( A +h B ) ) = ( ( T ` A ) + ( T ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   ~Hchil 27776   +h cva 27777   .h csm 27778   LinFnclf 27811

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-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-mulcom 10000  ax-mulass 10002  ax-distr 10003  ax-1rid 10006  ax-cnre 10009  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-lnfn 28707

This theorem is referenced by:  lnfnaddmuli  28904  nlelshi  28919