Theorem hosmvalt 9428

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 8790	. . . 4 |- H~ e. V
2	1	opabex2 3596	. . 3 |- {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))} e. V
3		fveq1 3708	. . . . . . 7 |- (f = S -> (f` x) = (S` x))
4	3	opreq1d 3960	. . . . . 6 |- (f = S -> ((f` x) +h (g` x)) = ((S` x) +h (g` x)))
5	4	eqeq2d 1478	. . . . 5 |- (f = S -> (y = ((f` x) +h (g` x)) <-> y = ((S` x) +h (g` x))))
6	5	anbi2d 614	. . . 4 |- (f = S -> ((x e. H~ /\ y = ((f` x) +h (g` x))) <-> (x e. H~ /\ y = ((S` x) +h (g` x)))))
7	6	opabbidv 2660	. . 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 3708	. . . . . . 7 |- (g = T -> (g` x) = (T` x))
9	8	opreq2d 3961	. . . . . 6 |- (g = T -> ((S` x) +h (g` x)) = ((S` x) +h (T` x)))
10	9	eqeq2d 1478	. . . . 5 |- (g = T -> (y = ((S` x) +h (g` x)) <-> y = ((S` x) +h (T` x))))
11	10	anbi2d 614	. . . 4 |- (g = T -> ((x e. H~ /\ y = ((S` x) +h (g` x))) <-> (x e. H~ /\ y = ((S` x) +h (T` x)))))
12	11	opabbidv 2660	. . 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 9423	. . . 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 4318	. . . . . . 7 |- (f e. (H~ ^m H~) <-> f:H~-->H~)
15	1, 1	elmap 4318	. . . . . . 7 |- (g e. (H~ ^m H~) <-> g:H~-->H~)
16	14, 15	anbi12i 481	. . . . . 6 |- ((f e. (H~ ^m H~) /\ g e. (H~ ^m H~)) <-> (f:H~-->H~ /\ g:H~-->H~))
17	16	anbi1i 480	. . . . 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 3982	. . . 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 1490	. . 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 4013	. 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 4318	. 2 |- (S e. (H~ ^m H~) <-> S:H~-->H~)
22	1, 1	elmap 4318	. 2 |- (T e. (H~ ^m H~) <-> T:H~-->H~)
23	20, 21, 22	syl2anbr 456	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 953   e. wcel 955  {copab 2656  -->wf 3168  ` cfv 3172  (class class class)co 3948  {copab2 3949   ^m cm 4306  H~chil 8727   +h cva 8728   +op chos 8746

This theorem is referenced by:  hosvalt 9433  hosvaltOLD 9434  hoaddclt 9601

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-oprab 3951  df-map 4308  df-hosum 9423

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