Theorem ellnopt 9701

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 1808	. 2 |- (T e. LinOp -> T e. V)
2		ax-hilex 8790	. . . 4 |- H~ e. V
3		fex 3637	. . . 4 |- ((T:H~-->H~ /\ H~ e. V) -> T e. V)
4	2, 3	mpan2 694	. . 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 3606	. . . 4 |- (t = T -> (t:H~-->H~ <-> T:H~-->H~))
7		fveq1 3708	. . . . . . 7 |- (t = T -> (t` ((x .h y) +h z)) = (T` ((x .h y) +h z)))
8		fveq1 3708	. . . . . . . . 9 |- (t = T -> (t` y) = (T` y))
9	8	opreq2d 3961	. . . . . . . 8 |- (t = T -> (x .h (t` y)) = (x .h (T` y)))
10		fveq1 3708	. . . . . . . 8 |- (t = T -> (t` z) = (T` z))
11	9, 10	opreq12d 3963	. . . . . . 7 |- (t = T -> ((x .h (t` y)) +h (t` z)) = ((x .h (T` y)) +h (T` z)))
12	7, 11	eqeq12d 1481	. . . . . 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 1655	. . . . 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 1672	. . . 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 626	. . 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 9684	. . 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 1891	. 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 677	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 953   e. wcel 955  A.wral 1637  Vcvv 1802  -->wf 3168  ` cfv 3172  (class class class)co 3948  CCcc 5204  H~chil 8727   +h cva 8728   .h csm 8729  LinOpclo 8755

This theorem is referenced by:  lnopft 9702  lnoplt 9754  unoplint 9760  hmoplint 9782  lnopm 9840  lnophs 9841  lnopco 9843  cnlnadjlem6 9920  adjlnopt 9934

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-lnop 9684

Copyright terms: Public domain