Theorem hodmvalt 9453

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

Assertion

Ref	Expression
hodmvalt	|- ((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 hodmvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8808	. . . 4 |- H~ e. V
2	1	opabex2 3602	. . 3 |- {<.x, y>. | (x e. H~ /\ y = ((S` x) -h (T` x)))} e. V
3		fveq1 3714	. . . . . . 7 |- (f = S -> (f` x) = (S` x))
4	3	opreq1d 3966	. . . . . 6 |- (f = S -> ((f` x) -h (g` x)) = ((S` x) -h (g` x)))
5	4	eqeq2d 1483	. . . . 5 |- (f = S -> (y = ((f` x) -h (g` x)) <-> y = ((S` x) -h (g` x))))
6	5	anbi2d 615	. . . 4 |- (f = S -> ((x e. H~ /\ y = ((f` x) -h (g` x))) <-> (x e. H~ /\ y = ((S` x) -h (g` x)))))
7	6	opabbidv 2665	. . 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 3714	. . . . . . 7 |- (g = T -> (g` x) = (T` x))
9	8	opreq2d 3967	. . . . . 6 |- (g = T -> ((S` x) -h (g` x)) = ((S` x) -h (T` x)))
10	9	eqeq2d 1483	. . . . 5 |- (g = T -> (y = ((S` x) -h (g` x)) <-> y = ((S` x) -h (T` x))))
11	10	anbi2d 615	. . . 4 |- (g = T -> ((x e. H~ /\ y = ((S` x) -h (g` x))) <-> (x e. H~ /\ y = ((S` x) -h (T` x)))))
12	11	opabbidv 2665	. . 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-hodif 9448	. . . 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 4324	. . . . . . 7 |- (f e. (H~ ^m H~) <-> f:H~-->H~)
15	1, 1	elmap 4324	. . . . . . 7 |- (g e. (H~ ^m H~) <-> g:H~-->H~)
16	14, 15	anbi12i 482	. . . . . 6 |- ((f e. (H~ ^m H~) /\ g e. (H~ ^m H~)) <-> (f:H~-->H~ /\ g:H~-->H~))
17	16	anbi1i 481	. . . . 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 3988	. . . 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 1495	. . 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 4019	. 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 4324	. 2 |- (S e. (H~ ^m H~) <-> S:H~-->H~)
22	1, 1	elmap 4324	. 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 954   e. wcel 956  {copab 2661  -->wf 3173  ` cfv 3177  (class class class)co 3954  {copab2 3955   ^m cm 4312  H~chil 8727   -h cmv 8731   -op chod 8748

This theorem is referenced by:  hodvalt 9459  hodvaltOLD 9460  hosubcl 9635

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-hilex 8808

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-opr 3956  df-oprab 3957  df-map 4314  df-hodif 9448

Copyright terms: Public domain