Theorem islno 8410

Description: The predicate "is a linear operator."

Hypotheses

Ref	Expression
lnoval.1	|- X = (Base` U)
lnoval.2	|- Y = (Base` 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 = (Base` U)
2		lnoval.2	. . . 4 |- Y = (Base` 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 8409	. . 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 1544	. 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 3738	. . . . . 6 |- (Base` U) e. V
11	1, 10	eqeltr 1547	. . . . 5 |- X e. V
12		fex 3658	. . . . 5 |- ((T:X-->Y /\ X e. V) -> T e. V)
13	11, 12	mpan2 698	. . . 4 |- (T:X-->Y -> T e. V)
14	13	adantr 391	. . 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 3626	. . . 4 |- (t = T -> (t:X-->Y <-> T:X-->Y))
16		fveq1 3729	. . . . . . 7 |- (t = T -> (t` (xG(yRz))) = (T` (xG(yRz))))
17		fveq1 3729	. . . . . . . 8 |- (t = T -> (t` x) = (T` x))
18		fveq1 3729	. . . . . . . . 9 |- (t = T -> (t` z) = (T` z))
19	18	opreq2d 3982	. . . . . . . 8 |- (t = T -> (yS(t` z)) = (yS(T` z)))
20	17, 19	opreq12d 3984	. . . . . . 7 |- (t = T -> ((t` x)H(yS(t` z))) = ((T` x)H(yS(T` z))))
21	16, 20	eqeq12d 1492	. . . . . 6 |- (t = T -> ((t` (xG(yRz))) = ((t` x)H(yS(t` z))) <-> (T` (xG(yRz))) = ((T` x)H(yS(T` z)))))
22	21	ralbidv 1666	. . . . 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 1683	. . . 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 630	. . 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 1906	. 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 538	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 146   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  A.wral 1648  Vcvv 1814  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244  NrmCVeccnv 8199  +vcpv 8200  Basecba 8201  .scns 8202   LnOp clno 8397

This theorem is referenced by:  lnolin 8411  lnof 8412  lnocoi 8414  0lno 8446  ipblnfi 8512

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-9 967  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 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  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-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-lno 8401

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