Theorem hosmvalt 9513

Description: Value of the sum of two Hilbert space operators.

Assertion

Ref	Expression
hosmvalt	|- ((S:H~-->H~ /\ T:H~-->H~) -> (S +op T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))})

Distinct variable groups:   x,y,S   x,T,y

Proof of Theorem hosmvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8871	. . . 4 |- H~ e. V
2	1	opabex2 3617	. . 3 |- {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))} e. V
3		fveq1 3730	. . . . . . 7 |- (f = S -> (f` x) = (S` x))
4	3	opreq1d 3982	. . . . . 6 |- (f = S -> ((f` x) +h (g` x)) = ((S` x) +h (g` x)))
5	4	eqeq2d 1489	. . . . 5 |- (f = S -> (y = ((f` x) +h (g` x)) <-> y = ((S` x) +h (g` x))))
6	5	anbi2d 618	. . . 4 |- (f = S -> ((x e. H~ /\ y = ((f` x) +h (g` x))) <-> (x e. H~ /\ y = ((S` x) +h (g` x)))))
7	6	opabbidv 2676	. . 3 |- (f = S -> {<.x, y>. | (x e. H~ /\ y = ((f` x) +h (g` x)))} = {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (g` x)))})
8		fveq1 3730	. . . . . . 7 |- (g = T -> (g` x) = (T` x))
9	8	opreq2d 3983	. . . . . 6 |- (g = T -> ((S` x) +h (g` x)) = ((S` x) +h (T` x)))
10	9	eqeq2d 1489	. . . . 5 |- (g = T -> (y = ((S` x) +h (g` x)) <-> y = ((S` x) +h (T` x))))
11	10	anbi2d 618	. . . 4 |- (g = T -> ((x e. H~ /\ y = ((S` x) +h (g` x))) <-> (x e. H~ /\ y = ((S` x) +h (T` x)))))
12	11	opabbidv 2676	. . 3 |- (g = T -> {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (g` x)))} = {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))})
13		df-hosum 9508	. . . 4 |- +op = {<.<.f, g>., h>. | ((f:H~-->H~ /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) +h (g` x)))})}
14	1, 1	elmap 4341	. . . . . . 7 |- (f e. (H~ ^m H~) <-> f:H~-->H~)
15	1, 1	elmap 4341	. . . . . . 7 |- (g e. (H~ ^m H~) <-> g:H~-->H~)
16	14, 15	anbi12i 484	. . . . . 6 |- ((f e. (H~ ^m H~) /\ g e. (H~ ^m H~)) <-> (f:H~-->H~ /\ g:H~-->H~))
17	16	anbi1i 483	. . . . 5 |- (((f e. (H~ ^m H~) /\ g e. (H~ ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) +h (g` x)))}) <-> ((f:H~-->H~ /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) +h (g` x)))}))
18	17	oprabbii 4004	. . . 4 |- {<.<.f, g>., h>. | ((f e. (H~ ^m H~) /\ g e. (H~ ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) +h (g` x)))})} = {<.<.f, g>., h>. | ((f:H~-->H~ /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) +h (g` x)))})}
19	13, 18	eqtr4 1501	. . 3 |- +op = {<.<.f, g>., h>. | ((f e. (H~ ^m H~) /\ g e. (H~ ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) +h (g` x)))})}
20	2, 7, 12, 19	oprabval2 4035	. 2 |- ((S e. (H~ ^m H~) /\ T e. (H~ ^m H~)) -> (S +op T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))})
21	1, 1	elmap 4341	. 2 |- (S e. (H~ ^m H~) <-> S:H~-->H~)
22	1, 1	elmap 4341	. 2 |- (T e. (H~ ^m H~) <-> T:H~-->H~)
23	20, 21, 22	syl2anbr 458	1 |- ((S:H~-->H~ /\ T:H~-->H~) -> (S +op T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {copab 2672  -->wf 3185  ` cfv 3189  (class class class)co 3970  {copab2 3971   ^m cm 4329  H~chil 8790   +h cva 8791   +op chos 8809

This theorem is referenced by:  hosvalt 9518  hosvaltOLD 9519  hoaddclt 9686

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 2699  ax-sep 2709  ax-pow 2749  ax-pr 2786  ax-un 2873  ax-hilex 8871

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-rex 1653  df-v 1815  df-sbc 1945  df-csb 2006  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-id 2842  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-f 3201  df-fv 3205  df-opr 3972  df-oprab 3973  df-map 4331  df-hosum 9508

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