Theorem hoadddir 28663

Description: Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
hoadddir	|- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A + B ) .op T ) = ( ( A .op T ) +op ( B .op T ) ) )

Proof of Theorem hoadddir

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcl 10018	. . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
2	1	anim1i 592	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ T : ~H --> ~H ) -> ( ( A + B ) e. CC /\ T : ~H --> ~H ) )
3	2	3impa 1259	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A + B ) e. CC /\ T : ~H --> ~H ) )
4		homval 28600	. . . . . . 7 |- ( ( ( A + B ) e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( ( A + B ) .op T ) ` x ) = ( ( A + B ) .h ( T ` x ) ) )
5	4	3expa 1265	. . . . . 6 |- ( ( ( ( A + B ) e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A + B ) .op T ) ` x ) = ( ( A + B ) .h ( T ` x ) ) )
6	3, 5	sylan 488	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A + B ) .op T ) ` x ) = ( ( A + B ) .h ( T ` x ) ) )
7		homval 28600	. . . . . . . . 9 |- ( ( A e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
8	7	3expa 1265	. . . . . . . 8 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
9	8	3adantl2 1218	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
10		homval 28600	. . . . . . . . 9 |- ( ( B e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( B .op T ) ` x ) = ( B .h ( T ` x ) ) )
11	10	3expa 1265	. . . . . . . 8 |- ( ( ( B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( B .op T ) ` x ) = ( B .h ( T ` x ) ) )
12	11	3adantl1 1217	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( B .op T ) ` x ) = ( B .h ( T ` x ) ) )
13	9, 12	oveq12d 6668	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) ` x ) +h ( ( B .op T ) ` x ) ) = ( ( A .h ( T ` x ) ) +h ( B .h ( T ` x ) ) ) )
14		ffvelrn 6357	. . . . . . . . . 10 |- ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
15		ax-hvdistr2 27866	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ ( T ` x ) e. ~H ) -> ( ( A + B ) .h ( T ` x ) ) = ( ( A .h ( T ` x ) ) +h ( B .h ( T ` x ) ) ) )
16	14, 15	syl3an3 1361	. . . . . . . . 9 |- ( ( A e. CC /\ B e. CC /\ ( T : ~H --> ~H /\ x e. ~H ) ) -> ( ( A + B ) .h ( T ` x ) ) = ( ( A .h ( T ` x ) ) +h ( B .h ( T ` x ) ) ) )
17	16	3exp 1264	. . . . . . . 8 |- ( A e. CC -> ( B e. CC -> ( ( T : ~H --> ~H /\ x e. ~H ) -> ( ( A + B ) .h ( T ` x ) ) = ( ( A .h ( T ` x ) ) +h ( B .h ( T ` x ) ) ) ) ) )
18	17	exp4a 633	. . . . . . 7 |- ( A e. CC -> ( B e. CC -> ( T : ~H --> ~H -> ( x e. ~H -> ( ( A + B ) .h ( T ` x ) ) = ( ( A .h ( T ` x ) ) +h ( B .h ( T ` x ) ) ) ) ) ) )
19	18	3imp1 1280	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( A + B ) .h ( T ` x ) ) = ( ( A .h ( T ` x ) ) +h ( B .h ( T ` x ) ) ) )
20	13, 19	eqtr4d 2659	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) ` x ) +h ( ( B .op T ) ` x ) ) = ( ( A + B ) .h ( T ` x ) ) )
21	6, 20	eqtr4d 2659	. . . 4 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A + B ) .op T ) ` x ) = ( ( ( A .op T ) ` x ) +h ( ( B .op T ) ` x ) ) )
22		homulcl 28618	. . . . . . 7 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )
23		homulcl 28618	. . . . . . 7 |- ( ( B e. CC /\ T : ~H --> ~H ) -> ( B .op T ) : ~H --> ~H )
24	22, 23	anim12i 590	. . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( B e. CC /\ T : ~H --> ~H ) ) -> ( ( A .op T ) : ~H --> ~H /\ ( B .op T ) : ~H --> ~H ) )
25	24	3impdir 1382	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A .op T ) : ~H --> ~H /\ ( B .op T ) : ~H --> ~H ) )
26		hosval 28599	. . . . . 6 |- ( ( ( A .op T ) : ~H --> ~H /\ ( B .op T ) : ~H --> ~H /\ x e. ~H ) -> ( ( ( A .op T ) +op ( B .op T ) ) ` x ) = ( ( ( A .op T ) ` x ) +h ( ( B .op T ) ` x ) ) )
27	26	3expa 1265	. . . . 5 |- ( ( ( ( A .op T ) : ~H --> ~H /\ ( B .op T ) : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) +op ( B .op T ) ) ` x ) = ( ( ( A .op T ) ` x ) +h ( ( B .op T ) ` x ) ) )
28	25, 27	sylan 488	. . . 4 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) +op ( B .op T ) ) ` x ) = ( ( ( A .op T ) ` x ) +h ( ( B .op T ) ` x ) ) )
29	21, 28	eqtr4d 2659	. . 3 |- ( ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A + B ) .op T ) ` x ) = ( ( ( A .op T ) +op ( B .op T ) ) ` x ) )
30	29	ralrimiva 2966	. 2 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> A. x e. ~H ( ( ( A + B ) .op T ) ` x ) = ( ( ( A .op T ) +op ( B .op T ) ) ` x ) )
31		homulcl 28618	. . . 4 |- ( ( ( A + B ) e. CC /\ T : ~H --> ~H ) -> ( ( A + B ) .op T ) : ~H --> ~H )
32	1, 31	stoic3 1701	. . 3 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A + B ) .op T ) : ~H --> ~H )
33		hoaddcl 28617	. . . . 5 |- ( ( ( A .op T ) : ~H --> ~H /\ ( B .op T ) : ~H --> ~H ) -> ( ( A .op T ) +op ( B .op T ) ) : ~H --> ~H )
34	22, 23, 33	syl2an 494	. . . 4 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( B e. CC /\ T : ~H --> ~H ) ) -> ( ( A .op T ) +op ( B .op T ) ) : ~H --> ~H )
35	34	3impdir 1382	. . 3 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A .op T ) +op ( B .op T ) ) : ~H --> ~H )
36		hoeq 28619	. . 3 |- ( ( ( ( A + B ) .op T ) : ~H --> ~H /\ ( ( A .op T ) +op ( B .op T ) ) : ~H --> ~H ) -> ( A. x e. ~H ( ( ( A + B ) .op T ) ` x ) = ( ( ( A .op T ) +op ( B .op T ) ) ` x ) <-> ( ( A + B ) .op T ) = ( ( A .op T ) +op ( B .op T ) ) ) )
37	32, 35, 36	syl2anc 693	. 2 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( A. x e. ~H ( ( ( A + B ) .op T ) ` x ) = ( ( ( A .op T ) +op ( B .op T ) ) ` x ) <-> ( ( A + B ) .op T ) = ( ( A .op T ) +op ( B .op T ) ) ) )
38	30, 37	mpbid 222	1 |- ( ( A e. CC /\ B e. CC /\ T : ~H --> ~H ) -> ( ( A + B ) .op T ) = ( ( A .op T ) +op ( B .op T ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   + caddc 9939   ~Hchil 27776   +h cva 27777   .h csm 27778   +op chos 27795   .op chot 27796

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-addcl 9996  ax-hilex 27856  ax-hfvadd 27857  ax-hfvmul 27862  ax-hvdistr2 27866

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-ne 2795  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-hosum 28589  df-homul 28590

This theorem is referenced by:  ho2times  28678