Theorem hosmval 11144

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

Assertion

Ref	Expression
hosmval	|- ((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 hosmval

Step	Hyp	Ref	Expression
1		ax-hilex 10501	. . . 4 |- ~H e. _V
2	1	opabex2 4539	. . 3 |- {<.x, y>. | (x e. ~H /\ y = ((S` x) +h (T` x)))} e. _V
3		fveq1 4680	. . . . . . 7 |- (f = S -> (f` x) = (S` x))
4	3	opreq1d 4897	. . . . . 6 |- (f = S -> ((f` x) +h (g` x)) = ((S` x) +h (g` x)))
5	4	eqeq2d 1895	. . . . 5 |- (f = S -> (y = ((f` x) +h (g` x)) <-> y = ((S` x) +h (g` x))))
6	5	anbi2d 678	. . . 4 |- (f = S -> ((x e. ~H /\ y = ((f` x) +h (g` x))) <-> (x e. ~H /\ y = ((S` x) +h (g` x)))))
7	6	opabbidv 3401	. . 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 4680	. . . . . . 7 |- (g = T -> (g` x) = (T` x))
9	8	opreq2d 4898	. . . . . 6 |- (g = T -> ((S` x) +h (g` x)) = ((S` x) +h (T` x)))
10	9	eqeq2d 1895	. . . . 5 |- (g = T -> (y = ((S` x) +h (g` x)) <-> y = ((S` x) +h (T` x))))
11	10	anbi2d 678	. . . 4 |- (g = T -> ((x e. ~H /\ y = ((S` x) +h (g` x))) <-> (x e. ~H /\ y = ((S` x) +h (T` x)))))
12	11	opabbidv 3401	. . 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 11139	. . . 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 5393	. . . . . . 7 |- (f e. (~H ^m ~H) <-> f:~H-->~H)
15	1, 1	elmap 5393	. . . . . . 7 |- (g e. (~H ^m ~H) <-> g:~H-->~H)
16	14, 15	anbi12i 540	. . . . . 6 |- ((f e. (~H ^m ~H) /\ g e. (~H ^m ~H)) <-> (f:~H-->~H /\ g:~H-->~H))
17	16	anbi1i 539	. . . . 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 4923	. . . 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	eqtr4i 1911	. . 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 4957	. 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 5393	. 2 |- (S e. (~H ^m ~H) <-> S:~H-->~H)
22	1, 1	elmap 5393	. 2 |- (T e. (~H ^m ~H) <-> T:~H-->~H)
23	20, 21, 22	syl2anbr 505	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 240   = wceq 1298   e. wcel 1300  {copab 3395  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885   ^m cmap 5381  ~Hchil 10420   +h cva 10421   +op chos 10439

This theorem is referenced by:  hosval 11149  hosvalOLD 11150  hoaddcl 11321

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-hosum 11139

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