Theorem hoaddclt 9684

Description: The sum of Hilbert space operators is an operator.

Assertion

Ref	Expression
hoaddclt	|- ((S:H~-->H~ /\ T:H~-->H~) -> (S +op T):H~-->H~)

Proof of Theorem hoaddclt

Step	Hyp	Ref	Expression
1		hvaddclt 8882	. . . . . 6 |- (((S` x) e. H~ /\ (T` x) e. H~) -> ((S` x) +h (T` x)) e. H~)
2		ffvelrn 3814	. . . . . 6 |- ((S:H~-->H~ /\ x e. H~) -> (S` x) e. H~)
3		ffvelrn 3814	. . . . . 6 |- ((T:H~-->H~ /\ x e. H~) -> (T` x) e. H~)
4	1, 2, 3	syl2an 454	. . . . 5 |- (((S:H~-->H~ /\ x e. H~) /\ (T:H~-->H~ /\ x e. H~)) -> ((S` x) +h (T` x)) e. H~)
5	4	anandirs 513	. . . 4 |- (((S:H~-->H~ /\ T:H~-->H~) /\ x e. H~) -> ((S` x) +h (T` x)) e. H~)
6	5	r19.21aiva 1714	. . 3 |- ((S:H~-->H~ /\ T:H~-->H~) -> A.x e. H~ ((S` x) +h (T` x)) e. H~)
7		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)))}
8	7	fopab2 3823	. . 3 |- (A.x e. H~ ((S` x) +h (T` x)) e. H~ <-> {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))}:H~-->H~)
9	6, 8	sylib 198	. 2 |- ((S:H~-->H~ /\ T:H~-->H~) -> {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))}:H~-->H~)
10		hosmvalt 9511	. . 3 |- ((S:H~-->H~ /\ T:H~-->H~) -> (S +op T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))})
11	10	feq1d 3624	. 2 |- ((S:H~-->H~ /\ T:H~-->H~) -> ((S +op T):H~-->H~ <-> {<.x, y>. | (x e. H~ /\ y = ((S` x) +h (T` x)))}:H~-->H~))
12	9, 11	mpbird 196	1 |- ((S:H~-->H~ /\ T:H~-->H~) -> (S +op T):H~-->H~)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  {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:  hoaddcl 9694  hoadd4t 9710  hoadddit 9729  hoadddirt 9730  hosub4t 9739  hoaddsubasst 9741  ho2timest 9745  hmopst 9945  adjaddt 10026

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  ax-hfvadd 8870

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-ral 1649  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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