Theorem hfsmvalt 9504

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

Assertion

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

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

Proof of Theorem hfsmvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8853	. . . 4 |- H~ e. V
2	1	opabex2 3607	. . 3 |- {<.x, y>. | (x e. H~ /\ y = ((S` x) + (T` x)))} e. V
3		fveq1 3720	. . . . . . 7 |- (f = S -> (f` x) = (S` x))
4	3	opreq1d 3972	. . . . . 6 |- (f = S -> ((f` x) + (g` x)) = ((S` x) + (g` x)))
5	4	eqeq2d 1485	. . . . 5 |- (f = S -> (y = ((f` x) + (g` x)) <-> y = ((S` x) + (g` x))))
6	5	anbi2d 615	. . . 4 |- (f = S -> ((x e. H~ /\ y = ((f` x) + (g` x))) <-> (x e. H~ /\ y = ((S` x) + (g` x)))))
7	6	opabbidv 2667	. . 3 |- (f = S -> {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))} = {<.x, y>. | (x e. H~ /\ y = ((S` x) + (g` x)))})
8		fveq1 3720	. . . . . . 7 |- (g = T -> (g` x) = (T` x))
9	8	opreq2d 3973	. . . . . 6 |- (g = T -> ((S` x) + (g` x)) = ((S` x) + (T` x)))
10	9	eqeq2d 1485	. . . . 5 |- (g = T -> (y = ((S` x) + (g` x)) <-> y = ((S` x) + (T` x))))
11	10	anbi2d 615	. . . 4 |- (g = T -> ((x e. H~ /\ y = ((S` x) + (g` x))) <-> (x e. H~ /\ y = ((S` x) + (T` x)))))
12	11	opabbidv 2667	. . 3 |- (g = T -> {<.x, y>. | (x e. H~ /\ y = ((S` x) + (g` x)))} = {<.x, y>. | (x e. H~ /\ y = ((S` x) + (T` x)))})
13		df-hfsum 9499	. . . 4 |- +fn = {<.<.f, g>., h>. | ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})}
14		axcnex 5254	. . . . . . . 8 |- CC e. V
15	14, 1	elmap 4331	. . . . . . 7 |- (f e. (CC ^m H~) <-> f:H~-->CC)
16	14, 1	elmap 4331	. . . . . . 7 |- (g e. (CC ^m H~) <-> g:H~-->CC)
17	15, 16	anbi12i 482	. . . . . 6 |- ((f e. (CC ^m H~) /\ g e. (CC ^m H~)) <-> (f:H~-->CC /\ g:H~-->CC))
18	17	anbi1i 481	. . . . 5 |- (((f e. (CC ^m H~) /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))}) <-> ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))}))
19	18	oprabbii 3994	. . . 4 |- {<.<.f, g>., h>. | ((f e. (CC ^m H~) /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})} = {<.<.f, g>., h>. | ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})}
20	13, 19	eqtr4 1497	. . 3 |- +fn = {<.<.f, g>., h>. | ((f e. (CC ^m H~) /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})}
21	2, 7, 12, 20	oprabval2 4025	. 2 |- ((S e. (CC ^m H~) /\ T e. (CC ^m H~)) -> (S +fn T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) + (T` x)))})
22	14, 1	elmap 4331	. 2 |- (S e. (CC ^m H~) <-> S:H~-->CC)
23	14, 1	elmap 4331	. 2 |- (T e. (CC ^m H~) <-> T:H~-->CC)
24	21, 22, 23	syl2anbr 456	1 |- ((S:H~-->CC /\ T:H~-->CC) -> (S +fn T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) + (T` x)))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  {copab 2663  -->wf 3175  ` cfv 3179  (class class class)co 3960  {copab2 3961   ^m cm 4319  CCcc 5219   + caddc 5224  H~chil 8772   +fn chfs 8794

This theorem is referenced by:  hfsvalt 9511

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612  ax-hilex 8853

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-opr 3962  df-oprab 3963  df-qs 4263  df-map 4321  df-ni 4987  df-nq 5025  df-np 5073  df-nr 5154  df-c 5227  df-hfsum 9499

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