Theorem ellnfnt 9810

Description: Property defining a linear functional.

Assertion

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

Distinct variable group:   x,y,z,T

Proof of Theorem ellnfnt

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (T e. LinFn -> T e. V)
2		ax-hilex 8869	. . . 4 |- H~ e. V
3		fex 3652	. . . 4 |- ((T:H~-->CC /\ H~ e. V) -> T e. V)
4	2, 3	mpan2 696	. . 3 |- (T:H~-->CC -> T e. V)
5	4	adantr 389	. 2 |- ((T:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x x. (T` y)) + (T` z))) -> T e. V)
6		feq1 3620	. . . 4 |- (t = T -> (t:H~-->CC <-> T:H~-->CC))
7		fveq1 3723	. . . . . . 7 |- (t = T -> (t` ((x .h y) +h z)) = (T` ((x .h y) +h z)))
8		fveq1 3723	. . . . . . . . 9 |- (t = T -> (t` y) = (T` y))
9	8	opreq2d 3976	. . . . . . . 8 |- (t = T -> (x x. (t` y)) = (x x. (T` y)))
10		fveq1 3723	. . . . . . . 8 |- (t = T -> (t` z) = (T` z))
11	9, 10	opreq12d 3978	. . . . . . 7 |- (t = T -> ((x x. (t` y)) + (t` z)) = ((x x. (T` y)) + (T` z)))
12	7, 11	eqeq12d 1489	. . . . . 6 |- (t = T -> ((t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z)) <-> (T` ((x .h y) +h z)) = ((x x. (T` y)) + (T` z))))
13	12	ralbidv 1663	. . . . 5 |- (t = T -> (A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z)) <-> A.z e. H~ (T` ((x .h y) +h z)) = ((x x. (T` y)) + (T` z))))
14	13	2ralbidv 1680	. . . 4 |- (t = T -> (A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z)) <-> A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x x. (T` y)) + (T` z))))
15	6, 14	anbi12d 628	. . 3 |- (t = T -> ((t:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z))) <-> (T:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x x. (T` y)) + (T` z)))))
16		df-lnfn 9774	. . 3 |- LinFn = {t | (t:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (t` ((x .h y) +h z)) = ((x x. (t` y)) + (t` z)))}
17	15, 16	elab2g 1900	. 2 |- (T e. V -> (T e. LinFn <-> (T:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x x. (T` y)) + (T` z)))))
18	1, 5, 17	pm5.21nii 679	1 |- (T e. LinFn <-> (T:H~-->CC /\ A.x e. CC A.y e. H~ A.z e. H~ (T` ((x .h y) +h z)) = ((x x. (T` y)) + (T` z))))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232   + caddc 5237   x. cmul 5239  H~chil 8788   +h cva 8789   .h csm 8790  LinFnclf 8823

This theorem is referenced by:  lnfnft 9811  lnfnlt 9855  bralnfnt 9872  0lnfn 9909  cnlnadjlem2 10001

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-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-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-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-lnfn 9774

Copyright terms: Public domain