Theorem hocadddir 9700

Description: Distributive law for Hilbert space operator sum.

Hypotheses

Ref	Expression
hods.1	|- R:H~-->H~
hods.2	|- S:H~-->H~
hods.3	|- T:H~-->H~

Assertion

Ref	Expression
hocadddir	|- ((R +op S) o. T) = ((R o. T) +op (S o. T))

Proof of Theorem hocadddir

Step	Hyp	Ref	Expression
1		hods.3	. . . . . . 7 |- T:H~-->H~
2	1	ffvelrni 3821	. . . . . 6 |- (x e. H~ -> (T` x) e. H~)
3		hods.1	. . . . . . 7 |- R:H~-->H~
4		hods.2	. . . . . . 7 |- S:H~-->H~
5		hosvaltOLD 9512	. . . . . . 7 |- (((R:H~-->H~ /\ S:H~-->H~) /\ (T` x) e. H~) -> ((R +op S)` (T` x)) = ((R` (T` x)) +h (S` (T` x))))
6	3, 4, 5	mpanl12 710	. . . . . 6 |- ((T` x) e. H~ -> ((R +op S)` (T` x)) = ((R` (T` x)) +h (S` (T` x))))
7	2, 6	syl 10	. . . . 5 |- (x e. H~ -> ((R +op S)` (T` x)) = ((R` (T` x)) +h (S` (T` x))))
8	3, 1	hoco 9685	. . . . . 6 |- (x e. H~ -> ((R o. T)` x) = (R` (T` x)))
9	4, 1	hoco 9685	. . . . . 6 |- (x e. H~ -> ((S o. T)` x) = (S` (T` x)))
10	8, 9	opreq12d 3984	. . . . 5 |- (x e. H~ -> (((R o. T)` x) +h ((S o. T)` x)) = ((R` (T` x)) +h (S` (T` x))))
11	7, 10	eqtr4d 1513	. . . 4 |- (x e. H~ -> ((R +op S)` (T` x)) = (((R o. T)` x) +h ((S o. T)` x)))
12	3, 4	hoaddcl 9689	. . . . 5 |- (R +op S):H~-->H~
13	12, 1	hoco 9685	. . . 4 |- (x e. H~ -> (((R +op S) o. T)` x) = ((R +op S)` (T` x)))
14	3, 1	hocof 9687	. . . . 5 |- (R o. T):H~-->H~
15	4, 1	hocof 9687	. . . . 5 |- (S o. T):H~-->H~
16		hosvaltOLD 9512	. . . . 5 |- ((((R o. T):H~-->H~ /\ (S o. T):H~-->H~) /\ x e. H~) -> (((R o. T) +op (S o. T))` x) = (((R o. T)` x) +h ((S o. T)` x)))
17	14, 15, 16	mpanl12 710	. . . 4 |- (x e. H~ -> (((R o. T) +op (S o. T))` x) = (((R o. T)` x) +h ((S o. T)` x)))
18	11, 13, 17	3eqtr4d 1520	. . 3 |- (x e. H~ -> (((R +op S) o. T)` x) = (((R o. T) +op (S o. T))` x))
19	18	rgen 1701	. 2 |- A.x e. H~ (((R +op S) o. T)` x) = (((R o. T) +op (S o. T))` x)
20	12, 1	hocof 9687	. . 3 |- ((R +op S) o. T):H~-->H~
21	14, 15	hoaddcl 9689	. . 3 |- ((R o. T) +op (S o. T)):H~-->H~
22	20, 21	hoeq 9682	. 2 |- (A.x e. H~ (((R +op S) o. T)` x) = (((R o. T) +op (S o. T))` x) <-> ((R +op S) o. T) = ((R o. T) +op (S o. T)))
23	19, 22	mpbi 189	1 |- ((R +op S) o. T) = ((R o. T) +op (S o. T))

Syntax hints:   = wceq 958   e. wcel 960  A.wral 1648   o. ccom 3180  -->wf 3184  ` cfv 3188  (class class class)co 3969  H~chil 8783   +h cva 8784   +op chos 8802

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864  ax-hfvadd 8865

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-map 4330  df-hosum 9501

Copyright terms: Public domain