Theorem islno 8414

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 8413	. . 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 1541	. 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 3732	. . . . . 6 |- (Base` U) e. V
11	1, 10	eqeltr 1544	. . . . 5 |- X e. V
12		fex 3652	. . . . 5 |- ((T:X-->Y /\ X e. V) -> T e. V)
13	11, 12	mpan2 696	. . . 4 |- (T:X-->Y -> T e. V)
14	13	adantr 389	. . 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 3620	. . . 4 |- (t = T -> (t:X-->Y <-> T:X-->Y))
16		fveq1 3723	. . . . . . 7 |- (t = T -> (t` (xG(yRz))) = (T` (xG(yRz))))
17		fveq1 3723	. . . . . . . 8 |- (t = T -> (t` x) = (T` x))
18		fveq1 3723	. . . . . . . . 9 |- (t = T -> (t` z) = (T` z))
19	18	opreq2d 3976	. . . . . . . 8 |- (t = T -> (yS(t` z)) = (yS(T` z)))
20	17, 19	opreq12d 3978	. . . . . . 7 |- (t = T -> ((t` x)H(yS(t` z))) = ((T` x)H(yS(T` z))))
21	16, 20	eqeq12d 1489	. . . . . 6 |- (t = T -> ((t` (xG(yRz))) = ((t` x)H(yS(t` z))) <-> (T` (xG(yRz))) = ((T` x)H(yS(T` z)))))
22	21	ralbidv 1663	. . . . 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 1680	. . . 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 628	. . 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 1903	. 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 536	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 956   e. wcel 958  {cab 1463  A.wral 1645  Vcvv 1811  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206   LnOp clno 8401

This theorem is referenced by:  lnolin 8415  lnof 8416  lnocoi 8418  0lno 8450  ipblnfi 8516

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-9 965  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-sbc 1942  df-csb 2002  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-oprab 3966  df-lno 8405

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