Theorem hocadddir 9705

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 3815	. . . . . 6 |- (x e. H~ -> (T` x) e. H~)
3		hods.1	. . . . . . 7 |- R:H~-->H~
4		hods.2	. . . . . . 7 |- S:H~-->H~
5		hosvaltOLD 9517	. . . . . . 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 708	. . . . . 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 9690	. . . . . 6 |- (x e. H~ -> ((R o. T)` x) = (R` (T` x)))
9	4, 1	hoco 9690	. . . . . 6 |- (x e. H~ -> ((S o. T)` x) = (S` (T` x)))
10	8, 9	opreq12d 3978	. . . . 5 |- (x e. H~ -> (((R o. T)` x) +h ((S o. T)` x)) = ((R` (T` x)) +h (S` (T` x))))
11	7, 10	eqtr4d 1510	. . . 4 |- (x e. H~ -> ((R +op S)` (T` x)) = (((R o. T)` x) +h ((S o. T)` x)))
12	3, 4	hoaddcl 9694	. . . . 5 |- (R +op S):H~-->H~
13	12, 1	hoco 9690	. . . 4 |- (x e. H~ -> (((R +op S) o. T)` x) = ((R +op S)` (T` x)))
14	3, 1	hocof 9692	. . . . 5 |- (R o. T):H~-->H~
15	4, 1	hocof 9692	. . . . 5 |- (S o. T):H~-->H~
16		hosvaltOLD 9517	. . . . 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 708	. . . 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 1517	. . 3 |- (x e. H~ -> (((R +op S) o. T)` x) = (((R o. T) +op (S o. T))` x))
19	18	rgen 1698	. 2 |- A.x e. H~ (((R +op S) o. T)` x) = (((R o. T) +op (S o. T))` x)
20	12, 1	hocof 9692	. . 3 |- ((R +op S) o. T):H~-->H~
21	14, 15	hoaddcl 9694	. . 3 |- ((R o. T) +op (S o. T)):H~-->H~
22	20, 21	hoeq 9687	. 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 956   e. wcel 958  A.wral 1645   o. ccom 3174  -->wf 3178  ` cfv 3182  (class class class)co 3963  H~chil 8788   +h cva 8789   +op chos 8807

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8869  ax-hfvadd 8870

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324  df-hosum 9506

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