Theorem hoadddi 28662

Description: Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Assertion

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

Proof of Theorem hoadddi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> A e. CC )
2		ffvelrn 6357	. . . . . . 7 |- ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
3	2	3ad2antl2 1224	. . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( T ` x ) e. ~H )
4		ffvelrn 6357	. . . . . . 7 |- ( ( U : ~H --> ~H /\ x e. ~H ) -> ( U ` x ) e. ~H )
5	4	3ad2antl3 1225	. . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( U ` x ) e. ~H )
6		ax-hvdistr1 27865	. . . . . 6 |- ( ( A e. CC /\ ( T ` x ) e. ~H /\ ( U ` x ) e. ~H ) -> ( A .h ( ( T ` x ) +h ( U ` x ) ) ) = ( ( A .h ( T ` x ) ) +h ( A .h ( U ` x ) ) ) )
7	1, 3, 5, 6	syl3anc 1326	. . . . 5 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( T ` x ) +h ( U ` x ) ) ) = ( ( A .h ( T ` x ) ) +h ( A .h ( U ` x ) ) ) )
8		hosval 28599	. . . . . . . 8 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H /\ x e. ~H ) -> ( ( T +op U ) ` x ) = ( ( T ` x ) +h ( U ` x ) ) )
9	8	oveq2d 6666	. . . . . . 7 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H /\ x e. ~H ) -> ( A .h ( ( T +op U ) ` x ) ) = ( A .h ( ( T ` x ) +h ( U ` x ) ) ) )
10	9	3expa 1265	. . . . . 6 |- ( ( ( T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( T +op U ) ` x ) ) = ( A .h ( ( T ` x ) +h ( U ` x ) ) ) )
11	10	3adantl1 1217	. . . . 5 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( T +op U ) ` x ) ) = ( A .h ( ( T ` x ) +h ( U ` x ) ) ) )
12		homval 28600	. . . . . . . 8 |- ( ( A e. CC /\ T : ~H --> ~H /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
13	12	3expa 1265	. . . . . . 7 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
14	13	3adantl3 1219	. . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op T ) ` x ) = ( A .h ( T ` x ) ) )
15		homval 28600	. . . . . . . 8 |- ( ( A e. CC /\ U : ~H --> ~H /\ x e. ~H ) -> ( ( A .op U ) ` x ) = ( A .h ( U ` x ) ) )
16	15	3expa 1265	. . . . . . 7 |- ( ( ( A e. CC /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op U ) ` x ) = ( A .h ( U ` x ) ) )
17	16	3adantl2 1218	. . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op U ) ` x ) = ( A .h ( U ` x ) ) )
18	14, 17	oveq12d 6668	. . . . 5 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) ` x ) +h ( ( A .op U ) ` x ) ) = ( ( A .h ( T ` x ) ) +h ( A .h ( U ` x ) ) ) )
19	7, 11, 18	3eqtr4d 2666	. . . 4 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( A .h ( ( T +op U ) ` x ) ) = ( ( ( A .op T ) ` x ) +h ( ( A .op U ) ` x ) ) )
20		hoaddcl 28617	. . . . . . 7 |- ( ( T : ~H --> ~H /\ U : ~H --> ~H ) -> ( T +op U ) : ~H --> ~H )
21	20	anim2i 593	. . . . . 6 |- ( ( A e. CC /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( A e. CC /\ ( T +op U ) : ~H --> ~H ) )
22	21	3impb 1260	. . . . 5 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( A e. CC /\ ( T +op U ) : ~H --> ~H ) )
23		homval 28600	. . . . . 6 |- ( ( A e. CC /\ ( T +op U ) : ~H --> ~H /\ x e. ~H ) -> ( ( A .op ( T +op U ) ) ` x ) = ( A .h ( ( T +op U ) ` x ) ) )
24	23	3expa 1265	. . . . 5 |- ( ( ( A e. CC /\ ( T +op U ) : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op ( T +op U ) ) ` x ) = ( A .h ( ( T +op U ) ` x ) ) )
25	22, 24	sylan 488	. . . 4 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op ( T +op U ) ) ` x ) = ( A .h ( ( T +op U ) ` x ) ) )
26		homulcl 28618	. . . . . . 7 |- ( ( A e. CC /\ T : ~H --> ~H ) -> ( A .op T ) : ~H --> ~H )
27		homulcl 28618	. . . . . . 7 |- ( ( A e. CC /\ U : ~H --> ~H ) -> ( A .op U ) : ~H --> ~H )
28	26, 27	anim12i 590	. . . . . 6 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( A e. CC /\ U : ~H --> ~H ) ) -> ( ( A .op T ) : ~H --> ~H /\ ( A .op U ) : ~H --> ~H ) )
29	28	3impdi 1381	. . . . 5 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) : ~H --> ~H /\ ( A .op U ) : ~H --> ~H ) )
30		hosval 28599	. . . . . 6 |- ( ( ( A .op T ) : ~H --> ~H /\ ( A .op U ) : ~H --> ~H /\ x e. ~H ) -> ( ( ( A .op T ) +op ( A .op U ) ) ` x ) = ( ( ( A .op T ) ` x ) +h ( ( A .op U ) ` x ) ) )
31	30	3expa 1265	. . . . 5 |- ( ( ( ( A .op T ) : ~H --> ~H /\ ( A .op U ) : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) +op ( A .op U ) ) ` x ) = ( ( ( A .op T ) ` x ) +h ( ( A .op U ) ` x ) ) )
32	29, 31	sylan 488	. . . 4 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( ( A .op T ) +op ( A .op U ) ) ` x ) = ( ( ( A .op T ) ` x ) +h ( ( A .op U ) ` x ) ) )
33	19, 25, 32	3eqtr4d 2666	. . 3 |- ( ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) /\ x e. ~H ) -> ( ( A .op ( T +op U ) ) ` x ) = ( ( ( A .op T ) +op ( A .op U ) ) ` x ) )
34	33	ralrimiva 2966	. 2 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> A. x e. ~H ( ( A .op ( T +op U ) ) ` x ) = ( ( ( A .op T ) +op ( A .op U ) ) ` x ) )
35		homulcl 28618	. . . . 5 |- ( ( A e. CC /\ ( T +op U ) : ~H --> ~H ) -> ( A .op ( T +op U ) ) : ~H --> ~H )
36	20, 35	sylan2 491	. . . 4 |- ( ( A e. CC /\ ( T : ~H --> ~H /\ U : ~H --> ~H ) ) -> ( A .op ( T +op U ) ) : ~H --> ~H )
37	36	3impb 1260	. . 3 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( A .op ( T +op U ) ) : ~H --> ~H )
38		hoaddcl 28617	. . . . 5 |- ( ( ( A .op T ) : ~H --> ~H /\ ( A .op U ) : ~H --> ~H ) -> ( ( A .op T ) +op ( A .op U ) ) : ~H --> ~H )
39	26, 27, 38	syl2an 494	. . . 4 |- ( ( ( A e. CC /\ T : ~H --> ~H ) /\ ( A e. CC /\ U : ~H --> ~H ) ) -> ( ( A .op T ) +op ( A .op U ) ) : ~H --> ~H )
40	39	3impdi 1381	. . 3 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( ( A .op T ) +op ( A .op U ) ) : ~H --> ~H )
41		hoeq 28619	. . 3 |- ( ( ( A .op ( T +op U ) ) : ~H --> ~H /\ ( ( A .op T ) +op ( A .op U ) ) : ~H --> ~H ) -> ( A. x e. ~H ( ( A .op ( T +op U ) ) ` x ) = ( ( ( A .op T ) +op ( A .op U ) ) ` x ) <-> ( A .op ( T +op U ) ) = ( ( A .op T ) +op ( A .op U ) ) ) )
42	37, 40, 41	syl2anc 693	. 2 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( A. x e. ~H ( ( A .op ( T +op U ) ) ` x ) = ( ( ( A .op T ) +op ( A .op U ) ) ` x ) <-> ( A .op ( T +op U ) ) = ( ( A .op T ) +op ( A .op U ) ) ) )
43	34, 42	mpbid 222	1 |- ( ( A e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( A .op ( T +op U ) ) = ( ( A .op T ) +op ( A .op U ) ) )

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   ~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-hilex 27856  ax-hfvadd 27857  ax-hfvmul 27862  ax-hvdistr1 27865

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:  hosubdi  28667  honegdi  28668  ho2times  28678  opsqrlem6  29004