Theorem ellnfnt 9812

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 1820	. 2 |- (T e. LinFn -> T e. V)
2		ax-hilex 8871	. . . 4 |- H~ e. V
3		fex 3659	. . . 4 |- ((T:H~-->CC /\ H~ e. V) -> T e. V)
4	2, 3	mpan2 698	. . 3 |- (T:H~-->CC -> T e. V)
5	4	adantr 391	. 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 3627	. . . 4 |- (t = T -> (t:H~-->CC <-> T:H~-->CC))
7		fveq1 3730	. . . . . . 7 |- (t = T -> (t` ((x .h y) +h z)) = (T` ((x .h y) +h z)))
8		fveq1 3730	. . . . . . . . 9 |- (t = T -> (t` y) = (T` y))
9	8	opreq2d 3983	. . . . . . . 8 |- (t = T -> (x x. (t` y)) = (x x. (T` y)))
10		fveq1 3730	. . . . . . . 8 |- (t = T -> (t` z) = (T` z))
11	9, 10	opreq12d 3985	. . . . . . 7 |- (t = T -> ((x x. (t` y)) + (t` z)) = ((x x. (T` y)) + (T` z)))
12	7, 11	eqeq12d 1492	. . . . . 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 1666	. . . . 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 1683	. . . 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 630	. . 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 9776	. . 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 1903	. 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 681	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 958   e. wcel 960  A.wral 1648  Vcvv 1814  -->wf 3185  ` cfv 3189  (class class class)co 3970  CCcc 5251   + caddc 5256   x. cmul 5258  H~chil 8790   +h cva 8791   .h csm 8792  LinFnclf 8825

This theorem is referenced by:  lnfnft 9813  lnfnlt 9857  bralnfnt 9874  0lnfn 9911  cnlnadjlem2 10003

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2699  ax-sep 2709  ax-pow 2749  ax-pr 2786  ax-un 2873  ax-hilex 8871

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-id 2842  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-f 3201  df-fv 3205  df-opr 3972  df-lnfn 9776

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