Theorem islno 9753

Description: The predicate "is a linear operator."

Hypotheses

Ref	Expression
lnoval.1	|- X = (BaseSet` U)
lnoval.2	|- Y = (BaseSet` W)
lnoval.3	|- G = (+v` U)
lnoval.4	|- H = (+v` W)
lnoval.5	|- R = (.s` U)
lnoval.6	|- S = (.s` W)
lnoval.7	|- L = (U LnOp W)

Assertion

Ref	Expression
islno	|- ((U e. NrmCVec /\ W e. NrmCVec) -> (T e. L <-> (T:X-->Y /\ A.x e. X A.y e. CC A.z e. X (T` (xG(yRz))) = ((T` x)H(yS(T` z))))))

Distinct variable groups:   x,y,z,T   x,U,y,z   x,W,y,z   x,X,y,z

Proof of Theorem islno

Step	Hyp	Ref	Expression
1		lnoval.1	. . . 4 |- X = (BaseSet` U)
2		lnoval.2	. . . 4 |- Y = (BaseSet` W)
3		lnoval.3	. . . 4 |- G = (+v` U)
4		lnoval.4	. . . 4 |- H = (+v` W)
5		lnoval.5	. . . 4 |- R = (.s` U)
6		lnoval.6	. . . 4 |- S = (.s` W)
7		lnoval.7	. . . 4 |- L = (U LnOp W)
8	1, 2, 3, 4, 5, 6, 7	lnoval 9752	. . 3 |- ((U e. NrmCVec /\ W e. NrmCVec) -> L = {t | (t:X-->Y /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z))))})
9	8	eleq2d 1964	. 2 |- ((U e. NrmCVec /\ W e. NrmCVec) -> (T e. L <-> T e. {t | (t:X-->Y /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z))))}))
10		fvex 4689	. . . . . 6 |- (BaseSet` U) e. _V
11	1, 10	eqeltri 1967	. . . . 5 |- X e. _V
12		fex 4595	. . . . 5 |- ((T:X-->Y /\ X e. _V) -> T e. _V)
13	11, 12	mpan2 760	. . . 4 |- (T:X-->Y -> T e. _V)
14	13	adantr 425	. . 3 |- ((T:X-->Y /\ A.x e. X A.y e. CC A.z e. X (T` (xG(yRz))) = ((T` x)H(yS(T` z)))) -> T e. _V)
15		feq1 4551	. . . 4 |- (t = T -> (t:X-->Y <-> T:X-->Y))
16		fveq1 4680	. . . . . . 7 |- (t = T -> (t` (xG(yRz))) = (T` (xG(yRz))))
17		fveq1 4680	. . . . . . . 8 |- (t = T -> (t` x) = (T` x))
18		fveq1 4680	. . . . . . . . 9 |- (t = T -> (t` z) = (T` z))
19	18	opreq2d 4898	. . . . . . . 8 |- (t = T -> (yS(t` z)) = (yS(T` z)))
20	17, 19	opreq12d 4900	. . . . . . 7 |- (t = T -> ((t` x)H(yS(t` z))) = ((T` x)H(yS(T` z))))
21	16, 20	eqeq12d 1899	. . . . . 6 |- (t = T -> ((t` (xG(yRz))) = ((t` x)H(yS(t` z))) <-> (T` (xG(yRz))) = ((T` x)H(yS(T` z)))))
22	21	ralbidv 2123	. . . . 5 |- (t = T -> (A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z))) <-> A.z e. X (T` (xG(yRz))) = ((T` x)H(yS(T` z)))))
23	22	2ralbidv 2140	. . . 4 |- (t = T -> (A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z))) <-> A.x e. X A.y e. CC A.z e. X (T` (xG(yRz))) = ((T` x)H(yS(T` z)))))
24	15, 23	anbi12d 690	. . 3 |- (t = T -> ((t:X-->Y /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z)))) <-> (T:X-->Y /\ A.x e. X A.y e. CC A.z e. X (T` (xG(yRz))) = ((T` x)H(yS(T` z))))))
25	14, 24	elab3 2412	. 2 |- (T e. {t | (t:X-->Y /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z))))} <-> (T:X-->Y /\ A.x e. X A.y e. CC A.z e. X (T` (xG(yRz))) = ((T` x)H(yS(T` z)))))
26	9, 25	syl6bb 595	1 |- ((U e. NrmCVec /\ W e. NrmCVec) -> (T e. L <-> (T:X-->Y /\ A.x e. X A.y e. CC A.z e. X (T` (xG(yRz))) = ((T` x)H(yS(T` z))))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538   LnOp clno 9740

This theorem is referenced by:  lnolin 9754  lnof 9755  lnocoi 9757  0lno 9790  ipblnfi 9857

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-lno 9744

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