Theorem ellnfnt 9801

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 1815	. 2 |- (T e. LinFn -> T e. V)
2		ax-hilex 8853	. . . 4 |- H~ e. V
3		fex 3649	. . . 4 |- ((T:H~-->CC /\ H~ e. V) -> T e. V)
4	2, 3	mpan2 695	. . 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 3617	. . . 4 |- (t = T -> (t:H~-->CC <-> T:H~-->CC))
7		fveq1 3720	. . . . . . 7 |- (t = T -> (t` ((x .h y) +h z)) = (T` ((x .h y) +h z)))
8		fveq1 3720	. . . . . . . . 9 |- (t = T -> (t` y) = (T` y))
9	8	opreq2d 3973	. . . . . . . 8 |- (t = T -> (x x. (t` y)) = (x x. (T` y)))
10		fveq1 3720	. . . . . . . 8 |- (t = T -> (t` z) = (T` z))
11	9, 10	opreq12d 3975	. . . . . . 7 |- (t = T -> ((x x. (t` y)) + (t` z)) = ((x x. (T` y)) + (T` z)))
12	7, 11	eqeq12d 1488	. . . . . 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 1662	. . . . 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 1679	. . . 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 627	. . 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 9765	. . 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 1898	. 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 678	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 955   e. wcel 957  A.wral 1644  Vcvv 1809  -->wf 3175  ` cfv 3179  (class class class)co 3960  CCcc 5219   + caddc 5224   x. cmul 5226  H~chil 8772   +h cva 8773   .h csm 8774  LinFnclf 8807

This theorem is referenced by:  lnfnft 9802  lnfnlt 9846  bralnfnt 9863  0lnfn 9900  cnlnadjlem2 9992

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 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-hilex 8853

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-opr 3962  df-lnfn 9765

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