Theorem hosvalt 9516

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

Assertion

Ref	Expression
hosvalt	|- ((S:H~-->H~ /\ T:H~-->H~ /\ A e. H~) -> ((S +op T)` A) = ((S` A) +h (T` A)))

Proof of Theorem hosvalt

Step	Hyp	Ref	Expression
1		hosmvalt 9511	. . . 4 |- ((S:H~-->H~ /\ T:H~-->H~) -> (S +op T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))})
2	1	fveq1d 3726	. . 3 |- ((S:H~-->H~ /\ T:H~-->H~) -> ((S +op T)` A) = ({<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))}` A))
3		fveq2 3724	. . . . 5 |- (x = A -> (S` x) = (S` A))
4		fveq2 3724	. . . . 5 |- (x = A -> (T` x) = (T` A))
5	3, 4	opreq12d 3978	. . . 4 |- (x = A -> ((S` x) +h (T` x)) = ((S` A) +h (T` A)))
6		eqid 1475	. . . 4 |- {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))} = {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))}
7		oprex 3983	. . . 4 |- ((S` A) +h (T` A)) e. V
8	5, 6, 7	fvopab4 3780	. . 3 |- (A e. H~ -> ({<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))}` A) = ((S` A) +h (T` A)))
9	2, 8	sylan9eq 1527	. 2 |- (((S:H~-->H~ /\ T:H~-->H~) /\ A e. H~) -> ((S +op T)` A) = ((S` A) +h (T` A)))
10	9	3impa 828	1 |- ((S:H~-->H~ /\ T:H~-->H~ /\ A e. H~) -> ((S +op T)` A) = ((S` A) +h (T` A)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  {copab 2666  -->wf 3178  ` cfv 3182  (class class class)co 3963  H~chil 8788   +h cva 8789   +op chos 8807

This theorem is referenced by:  hoaddid1 9712  honegsub 9722  hoadddit 9729  hoadddirt 9730

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324  df-hosum 9506

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