Theorem ellnopt 9741

Description: Property defining a linear Hilbert space operator.

Assertion

Ref	Expression
ellnopt	|- (T e. LinOp <-> (T:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z))))

Distinct variable group:   x,y,z,T

Proof of Theorem ellnopt

Step	Hyp	Ref	Expression
1		elisset 1814	. 2 |- (T e. LinOp -> T e. V)
2		ax-hilex 8824	. . . 4 |- H~ e. V
3		fex 3647	. . . 4 |- ((T:H~-->H~ /\ H~ e. V) -> T e. V)
4	2, 3	mpan2 695	. . 3 |- (T:H~-->H~ -> T e. V)
5	4	adantr 389	. 2 |- ((T:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z))) -> T e. V)
6		feq1 3616	. . . 4 |- (t = T -> (t:H~-->H~ <-> T:H~-->H~))
7		fveq1 3718	. . . . . . 7 |- (t = T -> (t` ((x .h y) +h z)) = (T` ((x .h y) +h z)))
8		fveq1 3718	. . . . . . . . 9 |- (t = T -> (t` y) = (T` y))
9	8	opreq2d 3971	. . . . . . . 8 |- (t = T -> (x .h (t` y)) = (x .h (T` y)))
10		fveq1 3718	. . . . . . . 8 |- (t = T -> (t` z) = (T` z))
11	9, 10	opreq12d 3973	. . . . . . 7 |- (t = T -> ((x .h (t` y)) +h (t` z)) = ((x .h (T` y)) +h (T` z)))
12	7, 11	eqeq12d 1487	. . . . . 6 |- (t = T -> ((t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z)) <-> (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z))))
13	12	ralbidv 1661	. . . . 5 |- (t = T -> (A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z)) <-> A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z))))
14	13	2ralbidv 1678	. . . 4 |- (t = T -> (A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z)) <-> A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z))))
15	6, 14	anbi12d 627	. . 3 |- (t = T -> ((t:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z))) <-> (T:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z)))))
16		df-lnop 9724	. . 3 |- LinOp = {t | (t:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x .h (t` y)) +h (t` z)))}
17	15, 16	elab2g 1897	. 2 |- (T e. V -> (T e. LinOp <-> (T:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z)))))
18	1, 5, 17	pm5.21nii 678	1 |- (T e. LinOp <-> (T:H~-->H~ /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x .h (T` y)) +h (T` z))))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643  Vcvv 1808  -->wf 3174  ` cfv 3178  (class class class)co 3958  CCcc 5215  H~chil 8743   +h cva 8744   .h csm 8745  LinOpclo 8771

This theorem is referenced by:  lnopft 9742  lnoplt 9795  unoplint 9801  hmoplint 9823  lnopm 9881  lnophs 9882  lnopco 9884  cnlnadjlem6 9961  adjlnopt 9975

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-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 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-hilex 8824

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-opr 3960  df-lnop 9724

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