Theorem lnoval 9752

Description: The set of linear operators between two normed complex vector spaces.

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
lnoval	|- ((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))))})

Distinct variable groups:   t,G   t,H   t,R   t,S   x,t,y,z,U   t,W,x,y,z   t,X,x,y,z   t,Y

Proof of Theorem lnoval

Step	Hyp	Ref	Expression
1		lnoval.1	. . . . 5 |- X = (BaseSet` U)
2		fvex 4689	. . . . 5 |- (BaseSet` U) e. _V
3	1, 2	eqeltri 1967	. . . 4 |- X e. _V
4		lnoval.2	. . . . 5 |- Y = (BaseSet` W)
5		fvex 4689	. . . . 5 |- (BaseSet` W) e. _V
6	4, 5	eqeltri 1967	. . . 4 |- Y e. _V
7		eqid 1884	. . . 4 |- {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 | (t:X-->Y /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z))))}
8	3, 6, 7	fabex 4597	. . 3 |- {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))))} e. _V
9		fveq2 4681	. . . . . . 7 |- (u = U -> (BaseSet` u) = (BaseSet` U))
10	9, 1	syl6eqr 1946	. . . . . 6 |- (u = U -> (BaseSet` u) = X)
11	10	feq2d 4557	. . . . 5 |- (u = U -> (t:(BaseSet` u)-->(BaseSet` w) <-> t:X-->(BaseSet` w)))
12		fveq2 4681	. . . . . . . . . . . . . 14 |- (u = U -> (.s` u) = (.s` U))
13		lnoval.5	. . . . . . . . . . . . . 14 |- R = (.s` U)
14	12, 13	syl6eqr 1946	. . . . . . . . . . . . 13 |- (u = U -> (.s` u) = R)
15	14	opreqd 4899	. . . . . . . . . . . 12 |- (u = U -> (y(.s` u)z) = (yRz))
16	15	opreq2d 4898	. . . . . . . . . . 11 |- (u = U -> (x(+v` u)(y(.s` u)z)) = (x(+v` u)(yRz)))
17		fveq2 4681	. . . . . . . . . . . . 13 |- (u = U -> (+v` u) = (+v` U))
18		lnoval.3	. . . . . . . . . . . . 13 |- G = (+v` U)
19	17, 18	syl6eqr 1946	. . . . . . . . . . . 12 |- (u = U -> (+v` u) = G)
20	19	opreqd 4899	. . . . . . . . . . 11 |- (u = U -> (x(+v` u)(yRz)) = (xG(yRz)))
21	16, 20	eqtrd 1925	. . . . . . . . . 10 |- (u = U -> (x(+v` u)(y(.s` u)z)) = (xG(yRz)))
22	21	fveq2d 4685	. . . . . . . . 9 |- (u = U -> (t` (x(+v` u)(y(.s` u)z))) = (t` (xG(yRz))))
23	22	eqeq1d 1892	. . . . . . . 8 |- (u = U -> ((t` (x(+v` u)(y(.s` u)z))) = ((t` x)(+v` w)(y(.s` w)(t` z))) <-> (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z)))))
24	10, 23	raleqbidv 2274	. . . . . . 7 |- (u = U -> (A.z e. (BaseSet` u)(t` (x(+v` u)(y(.s` u)z))) = ((t` x)(+v` w)(y(.s` w)(t` z))) <-> A.z e. X (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z)))))
25	24	ralbidv 2123	. . . . . 6 |- (u = U -> (A.y e. CC A.z e. (BaseSet` u)(t` (x(+v` u)(y(.s` u)z))) = ((t` x)(+v` w)(y(.s` w)(t` z))) <-> A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z)))))
26	10, 25	raleqbidv 2274	. . . . 5 |- (u = U -> (A.x e. (BaseSet` u)A.y e. CC A.z e. (BaseSet` u)(t` (x(+v` u)(y(.s` u)z))) = ((t` x)(+v` w)(y(.s` w)(t` z))) <-> A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z)))))
27	11, 26	anbi12d 690	. . . 4 |- (u = U -> ((t:(BaseSet` u)-->(BaseSet` w) /\ A.x e. (BaseSet` u)A.y e. CC A.z e. (BaseSet` u)(t` (x(+v` u)(y(.s` u)z))) = ((t` x)(+v` w)(y(.s` w)(t` z)))) <-> (t:X-->(BaseSet` w) /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z))))))
28	27	abbidv 2008	. . 3 |- (u = U -> {t | (t:(BaseSet` u)-->(BaseSet` w) /\ A.x e. (BaseSet` u)A.y e. CC A.z e. (BaseSet` u)(t` (x(+v` u)(y(.s` u)z))) = ((t` x)(+v` w)(y(.s` w)(t` z))))} = {t | (t:X-->(BaseSet` w) /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z))))})
29		fveq2 4681	. . . . . . 7 |- (w = W -> (BaseSet` w) = (BaseSet` W))
30	29, 4	syl6eqr 1946	. . . . . 6 |- (w = W -> (BaseSet` w) = Y)
31		feq3 4553	. . . . . 6 |- ((BaseSet` w) = Y -> (t:X-->(BaseSet` w) <-> t:X-->Y))
32	30, 31	syl 12	. . . . 5 |- (w = W -> (t:X-->(BaseSet` w) <-> t:X-->Y))
33		fveq2 4681	. . . . . . . . . . . 12 |- (w = W -> (.s` w) = (.s` W))
34		lnoval.6	. . . . . . . . . . . 12 |- S = (.s` W)
35	33, 34	syl6eqr 1946	. . . . . . . . . . 11 |- (w = W -> (.s` w) = S)
36	35	opreqd 4899	. . . . . . . . . 10 |- (w = W -> (y(.s` w)(t` z)) = (yS(t` z)))
37	36	opreq2d 4898	. . . . . . . . 9 |- (w = W -> ((t` x)(+v` w)(y(.s` w)(t` z))) = ((t` x)(+v` w)(yS(t` z))))
38		fveq2 4681	. . . . . . . . . . 11 |- (w = W -> (+v` w) = (+v` W))
39		lnoval.4	. . . . . . . . . . 11 |- H = (+v` W)
40	38, 39	syl6eqr 1946	. . . . . . . . . 10 |- (w = W -> (+v` w) = H)
41	40	opreqd 4899	. . . . . . . . 9 |- (w = W -> ((t` x)(+v` w)(yS(t` z))) = ((t` x)H(yS(t` z))))
42	37, 41	eqtrd 1925	. . . . . . . 8 |- (w = W -> ((t` x)(+v` w)(y(.s` w)(t` z))) = ((t` x)H(yS(t` z))))
43	42	eqeq2d 1895	. . . . . . 7 |- (w = W -> ((t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z))) <-> (t` (xG(yRz))) = ((t` x)H(yS(t` z)))))
44	43	ralbidv 2123	. . . . . 6 |- (w = W -> (A.z e. X (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z))) <-> A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z)))))
45	44	2ralbidv 2140	. . . . 5 |- (w = W -> (A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z))) <-> A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z)))))
46	32, 45	anbi12d 690	. . . 4 |- (w = W -> ((t:X-->(BaseSet` w) /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(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))))))
47	46	abbidv 2008	. . 3 |- (w = W -> {t | (t:X-->(BaseSet` w) /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)(+v` w)(y(.s` w)(t` z))))} = {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))))})
48		df-lno 9744	. . 3 |- LnOp = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = {t | (t:(BaseSet` u)-->(BaseSet` w) /\ A.x e. (BaseSet` u)A.y e. CC A.z e. (BaseSet` u)(t` (x(+v` u)(y(.s` u)z))) = ((t` x)(+v` w)(y(.s` w)(t` z))))})}
49	8, 28, 47, 48	oprabval2 4957	. 2 |- ((U e. NrmCVec /\ W e. NrmCVec) -> (U LnOp W) = {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))))})
50		lnoval.7	. 2 |- L = (U LnOp W)
51	49, 50	syl5eq 1940	1 |- ((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))))})

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:  islno 9753  hhlnoi 11463

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-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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