Theorem reparphtlem2 16064

Description: Lemma for reparpht 16065.

Hypotheses

Ref	Expression
reparphtlem2.1	|- G e. (II Cn II)
reparphtlem2.2	|- H = {<.<.x, y>., w>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ w = (((1 - y) x. (G` x)) + (y x. x)))}

Assertion

Ref	Expression
reparphtlem2	|- H e. ((II X.t II) Cn II)

Distinct variable group:   w,G,x,y

Proof of Theorem reparphtlem2

Step	Hyp	Ref	Expression
1		iitop 15867	. . . . 5 |- II e. Top
2	1, 1	txtopi 15909	. . . 4 |- (II X.t II) e. Top
3		eqid 1884	. . . . . 6 |- (abs o. - ) = (abs o. - )
4	3	cnmet 9182	. . . . 5 |- (abs o. - ) e. Met
5		eqid 1884	. . . . . 6 |- (Open` (abs o. - )) = (Open` (abs o. - ))
6	5	opntop 9147	. . . . 5 |- ((abs o. - ) e. Met -> (Open` (abs o. - )) e. Top)
7	4, 6	ax-mp 7	. . . 4 |- (Open` (abs o. - )) e. Top
8		iiuni 15868	. . . . 5 |- (0[,]1) = U.II
9		oprex 4907	. . . . . 6 |- ((1 - y) x. (G` x)) e. _V
10	9	a1i 8	. . . . 5 |- ((x e. (0[,]1) /\ y e. (0[,]1)) -> ((1 - y) x. (G` x)) e. _V)
11		oprex 4907	. . . . . 6 |- (y x. x) e. _V
12	11	a1i 8	. . . . 5 |- ((x e. (0[,]1) /\ y e. (0[,]1)) -> (y x. x) e. _V)
13		eqid 1884	. . . . 5 |- {<.<.x, y>., u>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ u = ((1 - y) x. (G` x)))} = {<.<.x, y>., u>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ u = ((1 - y) x. (G` x)))}
14		eqid 1884	. . . . 5 |- {<.<.x, y>., v>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ v = (y x. x))} = {<.<.x, y>., v>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ v = (y x. x))}
15		reparphtlem2.2	. . . . 5 |- H = {<.<.x, y>., w>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ w = (((1 - y) x. (G` x)) + (y x. x)))}
16		oprex 4907	. . . . . . 7 |- (1 - y) e. _V
17	16	a1i 8	. . . . . 6 |- ((x e. (0[,]1) /\ y e. (0[,]1)) -> (1 - y) e. _V)
18		fvex 4689	. . . . . . 7 |- (G` x) e. _V
19	18	a1i 8	. . . . . 6 |- ((x e. (0[,]1) /\ y e. (0[,]1)) -> (G` x) e. _V)
20		eqid 1884	. . . . . 6 |- {<.<.x, y>., v>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ v = (1 - y))} = {<.<.x, y>., v>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ v = (1 - y))}
21		eqid 1884	. . . . . 6 |- {<.<.x, y>., z>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ z = (G` x))} = {<.<.x, y>., z>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ z = (G` x))}
22	16	a1i 8	. . . . . . 7 |- (y e. (0[,]1) -> (1 - y) e. _V)
23		eqid 1884	. . . . . . 7 |- {<.y, u>. | (y e. (0[,]1) /\ u = (1 - y))} = {<.y, u>. | (y e. (0[,]1) /\ u = (1 - y))}
24		0re 6603	. . . . . . . . . . 11 |- 0 e. RR
25		1re 6598	. . . . . . . . . . 11 |- 1 e. RR
26		iccssre 7565	. . . . . . . . . . 11 |- ((0 e. RR /\ 1 e. RR) -> (0[,]1) C_ RR)
27	24, 25, 26	mp2an 761	. . . . . . . . . 10 |- (0[,]1) C_ RR
28		axresscn 6420	. . . . . . . . . 10 |- RR C_ CC
29	27, 28	sstri 2626	. . . . . . . . 9 |- (0[,]1) C_ CC
30		resopab2 4256	. . . . . . . . 9 |- ((0[,]1) C_ CC -> ({<.y, u>. | (y e. CC /\ u = (1 - y))} |` (0[,]1)) = {<.y, u>. | (y e. (0[,]1) /\ u = (1 - y))})
31	29, 30	ax-mp 7	. . . . . . . 8 |- ({<.y, u>. | (y e. CC /\ u = (1 - y))} |` (0[,]1)) = {<.y, u>. | (y e. (0[,]1) /\ u = (1 - y))}
32		ax1cn 6422	. . . . . . . . . . . 12 |- 1 e. CC
33		eqid 1884	. . . . . . . . . . . . 13 |- {<.y, u>. | (y e. CC /\ u = (1 - y))} = {<.y, u>. | (y e. CC /\ u = (1 - y))}
34	33	sub2cncf 15886	. . . . . . . . . . . 12 |- (1 e. CC -> {<.y, u>. | (y e. CC /\ u = (1 - y))} e. (CC-cn->CC))
35	32, 34	ax-mp 7	. . . . . . . . . . 11 |- {<.y, u>. | (y e. CC /\ u = (1 - y))} e. (CC-cn->CC)
36	3, 5	cncfmet1 9184	. . . . . . . . . . 11 |- (CC-cn->CC) = ((Open` (abs o. - )) Cn (Open` (abs o. - )))
37	35, 36	eleqtri 1969	. . . . . . . . . 10 |- {<.y, u>. | (y e. CC /\ u = (1 - y))} e. ((Open` (abs o. - )) Cn (Open` (abs o. - )))
38	3	cnmetba 9181	. . . . . . . . . . . . . 14 |- CC = dom dom (abs o. - )
39	38, 5	uniopn2 9138	. . . . . . . . . . . . 13 |- ((abs o. - ) e. Met -> U.(Open` (abs o. - )) = CC)
40	39	eqcomd 1889	. . . . . . . . . . . 12 |- ((abs o. - ) e. Met -> CC = U.(Open` (abs o. - )))
41	4, 40	ax-mp 7	. . . . . . . . . . 11 |- CC = U.(Open` (abs o. - ))
42	41	cnres 15889	. . . . . . . . . 10 |- ((((Open` (abs o. - )) e. Top /\ (Open` (abs o. - )) e. Top) /\ ({<.y, u>. | (y e. CC /\ u = (1 - y))} e. ((Open` (abs o. - )) Cn (Open` (abs o. - ))) /\ (0[,]1) C_ CC)) -> ({<.y, u>. | (y e. CC /\ u = (1 - y))} |` (0[,]1)) e. ((subSp` <.(0[,]1), (Open` (abs o. - ))>.) Cn (Open` (abs o. - ))))
43	7, 7, 37, 29, 42	mp4an 15651	. . . . . . . . 9 |- ({<.y, u>. | (y e. CC /\ u = (1 - y))} |` (0[,]1)) e. ((subSp` <.(0[,]1), (Open` (abs o. - ))>.) Cn (Open` (abs o. - )))
44		dfii3 15870	. . . . . . . . . 10 |- II = (subSp` <.(0[,]1), (Open` (abs o. - ))>.)
45	44	opreq1i 4892	. . . . . . . . 9 |- (II Cn (Open` (abs o. - ))) = ((subSp` <.(0[,]1), (Open` (abs o. - ))>.) Cn (Open` (abs o. - )))
46	43, 45	eleqtrri 1970	. . . . . . . 8 |- ({<.y, u>. | (y e. CC /\ u = (1 - y))} |` (0[,]1)) e. (II Cn (Open` (abs o. - )))
47	31, 46	eqeltrri 1968	. . . . . . 7 |- {<.y, u>. | (y e. (0[,]1) /\ u = (1 - y))} e. (II Cn (Open` (abs o. - )))
48	8, 8, 1, 1, 7, 22, 23, 20, 47	cnoprab2 15922	. . . . . 6 |- {<.<.x, y>., v>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ v = (1 - y))} e. ((II X.t II) Cn (Open` (abs o. - )))
49	18	a1i 8	. . . . . . 7 |- (x e. (0[,]1) -> (G` x) e. _V)
50		reparphtlem2.1	. . . . . . . . . 10 |- G e. (II Cn II)
51	8, 8	cnf 9038	. . . . . . . . . 10 |- ((II e. Top /\ II e. Top /\ G e. (II Cn II)) -> G:(0[,]1)-->(0[,]1))
52	1, 1, 50, 51	mp3an 1191	. . . . . . . . 9 |- G:(0[,]1)-->(0[,]1)
53		ffn 4562	. . . . . . . . 9 |- (G:(0[,]1)-->(0[,]1) -> G Fn (0[,]1))
54	52, 53	ax-mp 7	. . . . . . . 8 |- G Fn (0[,]1)
55		dffn5 4717	. . . . . . . 8 |- (G Fn (0[,]1) <-> G = {<.x, u>. | (x e. (0[,]1) /\ u = (G` x))})
56	54, 55	mpbi 206	. . . . . . 7 |- G = {<.x, u>. | (x e. (0[,]1) /\ u = (G` x))}
57	44	opreq2i 4893	. . . . . . . . 9 |- (II Cn II) = (II Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
58	50, 57	eleqtri 1969	. . . . . . . 8 |- G e. (II Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
59	41	cnss 15892	. . . . . . . 8 |- (((II e. Top /\ (Open` (abs o. - )) e. Top) /\ ((0[,]1) C_ CC /\ G e. (II Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.)))) -> G e. (II Cn (Open` (abs o. - ))))
60	1, 7, 29, 58, 59	mp4an 15651	. . . . . . 7 |- G e. (II Cn (Open` (abs o. - )))
61	8, 8, 1, 1, 7, 49, 56, 21, 60	cnoprab1 15921	. . . . . 6 |- {<.<.x, y>., z>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ z = (G` x))} e. ((II X.t II) Cn (Open` (abs o. - )))
62	3, 5	mulcntx 15929	. . . . . 6 |- x. e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
63	8, 8, 1, 1, 7, 7, 7, 17, 19, 20, 21, 13, 48, 61, 62	cnoproprabco 15919	. . . . 5 |- {<.<.x, y>., u>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ u = ((1 - y) x. (G` x)))} e. ((II X.t II) Cn (Open` (abs o. - )))
64		ssid 2634	. . . . . 6 |- CC C_ CC
65		iimulcl 15874	. . . . . . . 8 |- ((y e. (0[,]1) /\ x e. (0[,]1)) -> (y x. x) e. (0[,]1))
66	29	sseli 2617	. . . . . . . 8 |- ((y x. x) e. (0[,]1) -> (y x. x) e. CC)
67	65, 66	syl 12	. . . . . . 7 |- ((y e. (0[,]1) /\ x e. (0[,]1)) -> (y x. x) e. CC)
68	67	ancoms 484	. . . . . 6 |- ((x e. (0[,]1) /\ y e. (0[,]1)) -> (y x. x) e. CC)
69		axmulopr 6418	. . . . . . . . 9 |- x. :(CC X. CC)-->CC
70		ffn 4562	. . . . . . . . 9 |- ( x. :(CC X. CC)-->CC -> x. Fn (CC X. CC))
71	69, 70	ax-mp 7	. . . . . . . 8 |- x. Fn (CC X. CC)
72		fnoprv 4946	. . . . . . . 8 |- ( x. Fn (CC X. CC) <-> x. = {<.<.x, y>., v>. | ((x e. CC /\ y e. CC) /\ v = (x x. y))})
73	71, 72	mpbi 206	. . . . . . 7 |- x. = {<.<.x, y>., v>. | ((x e. CC /\ y e. CC) /\ v = (x x. y))}
74		mulcom 6459	. . . . . . . . . 10 |- ((x e. CC /\ y e. CC) -> (x x. y) = (y x. x))
75	74	eqeq2d 1895	. . . . . . . . 9 |- ((x e. CC /\ y e. CC) -> (v = (x x. y) <-> v = (y x. x)))
76	75	pm5.32i 707	. . . . . . . 8 |- (((x e. CC /\ y e. CC) /\ v = (x x. y)) <-> ((x e. CC /\ y e. CC) /\ v = (y x. x)))
77	76	oprabbii 4923	. . . . . . 7 |- {<.<.x, y>., v>. | ((x e. CC /\ y e. CC) /\ v = (x x. y))} = {<.<.x, y>., v>. | ((x e. CC /\ y e. CC) /\ v = (y x. x))}
78	73, 77	eqtri 1908	. . . . . 6 |- x. = {<.<.x, y>., v>. | ((x e. CC /\ y e. CC) /\ v = (y x. x))}
79	41	subspid 10249	. . . . . . . 8 |- ((Open` (abs o. - )) e. Top -> (subSp` <.CC, (Open` (abs o. - ))>.) = (Open` (abs o. - )))
80	79	eqcomd 1889	. . . . . . 7 |- ((Open` (abs o. - )) e. Top -> (Open` (abs o. - )) = (subSp` <.CC, (Open` (abs o. - ))>.))
81	7, 80	ax-mp 7	. . . . . 6 |- (Open` (abs o. - )) = (subSp` <.CC, (Open` (abs o. - ))>.)
82	41, 41, 41, 29, 29, 64, 68, 78, 14, 7, 7, 7, 62, 44, 44, 81	cnresoprab2 15916	. . . . 5 |- {<.<.x, y>., v>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ v = (y x. x))} e. ((II X.t II) Cn (Open` (abs o. - )))
83	3, 5	addcntx 15927	. . . . 5 |- + e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
84	8, 8, 1, 1, 7, 7, 7, 10, 12, 13, 14, 15, 63, 82, 83	cnoproprabco 15919	. . . 4 |- H e. ((II X.t II) Cn (Open` (abs o. - )))
85	2, 7, 84	3pm3.2i 1048	. . 3 |- ((II X.t II) e. Top /\ (Open` (abs o. - )) e. Top /\ H e. ((II X.t II) Cn (Open` (abs o. - ))))
86		oprex 4907	. . . . . . . 8 |- (((1 - v) x. (G` u)) + (v x. u)) e. _V
87		fveq2 4681	. . . . . . . . . 10 |- (x = u -> (G` x) = (G` u))
88	87	opreq2d 4898	. . . . . . . . 9 |- (x = u -> ((1 - y) x. (G` x)) = ((1 - y) x. (G` u)))
89		opreq2 4890	. . . . . . . . 9 |- (x = u -> (y x. x) = (y x. u))
90	88, 89	opreq12d 4900	. . . . . . . 8 |- (x = u -> (((1 - y) x. (G` x)) + (y x. x)) = (((1 - y) x. (G` u)) + (y x. u)))
91		opreq2 4890	. . . . . . . . . 10 |- (y = v -> (1 - y) = (1 - v))
92	91	opreq1d 4897	. . . . . . . . 9 |- (y = v -> ((1 - y) x. (G` u)) = ((1 - v) x. (G` u)))
93		opreq1 4889	. . . . . . . . 9 |- (y = v -> (y x. u) = (v x. u))
94	92, 93	opreq12d 4900	. . . . . . . 8 |- (y = v -> (((1 - y) x. (G` u)) + (y x. u)) = (((1 - v) x. (G` u)) + (v x. u)))
95	86, 90, 94, 15	oprabval2 4957	. . . . . . 7 |- ((u e. (0[,]1) /\ v e. (0[,]1)) -> (uHv) = (((1 - v) x. (G` u)) + (v x. u)))
96		lincmb01icc 15879	. . . . . . . . . 10 |- ((0 e. RR /\ 1 e. RR) -> (((G` u) e. (0[,]1) /\ u e. (0[,]1) /\ v e. (0[,]1)) -> (((1 - v) x. (G` u)) + (v x. u)) e. (0[,]1)))
97	24, 25, 96	mp2an 761	. . . . . . . . 9 |- (((G` u) e. (0[,]1) /\ u e. (0[,]1) /\ v e. (0[,]1)) -> (((1 - v) x. (G` u)) + (v x. u)) e. (0[,]1))
98	97	3expa 1067	. . . . . . . 8 |- ((((G` u) e. (0[,]1) /\ u e. (0[,]1)) /\ v e. (0[,]1)) -> (((1 - v) x. (G` u)) + (v x. u)) e. (0[,]1))
99	52	ffvelrni 4788	. . . . . . . . 9 |- (u e. (0[,]1) -> (G` u) e. (0[,]1))
100	99	ancri 321	. . . . . . . 8 |- (u e. (0[,]1) -> ((G` u) e. (0[,]1) /\ u e. (0[,]1)))
101	98, 100	sylan 497	. . . . . . 7 |- ((u e. (0[,]1) /\ v e. (0[,]1)) -> (((1 - v) x. (G` u)) + (v x. u)) e. (0[,]1))
102	95, 101	eqeltrd 1971	. . . . . 6 |- ((u e. (0[,]1) /\ v e. (0[,]1)) -> (uHv) e. (0[,]1))
103	102	rgen2a 2160	. . . . 5 |- A.u e. (0[,]1)A.v e. (0[,]1)(uHv) e. (0[,]1)
104		fveq2 4681	. . . . . . . 8 |- (z = <.u, v>. -> (H` z) = (H` <.u, v>.))
105		df-opr 4886	. . . . . . . 8 |- (uHv) = (H` <.u, v>.)
106	104, 105	syl6eqr 1946	. . . . . . 7 |- (z = <.u, v>. -> (H` z) = (uHv))
107	106	eleq1d 1963	. . . . . 6 |- (z = <.u, v>. -> ((H` z) e. (0[,]1) <-> (uHv) e. (0[,]1)))
108	107	ralxp 4041	. . . . 5 |- (A.z e. ((0[,]1) X. (0[,]1))(H` z) e. (0[,]1) <-> A.u e. (0[,]1)A.v e. (0[,]1)(uHv) e. (0[,]1))
109	103, 108	mpbir 207	. . . 4 |- A.z e. ((0[,]1) X. (0[,]1))(H` z) e. (0[,]1)
110	29, 109	pm3.2i 307	. . 3 |- ((0[,]1) C_ CC /\ A.z e. ((0[,]1) X. (0[,]1))(H` z) e. (0[,]1))
111	1, 1, 8, 8	txunii 15910	. . . 4 |- ((0[,]1) X. (0[,]1)) = U.(II X.t II)
112	111, 41	cnimass 15888	. . 3 |- ((((II X.t II) e. Top /\ (Open` (abs o. - )) e. Top /\ H e. ((II X.t II) Cn (Open` (abs o. - )))) /\ ((0[,]1) C_ CC /\ A.z e. ((0[,]1) X. (0[,]1))(H` z) e. (0[,]1))) -> H e. ((II X.t II) Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.)))
113	85, 110, 112	mp2an 761	. 2 |- H e. ((II X.t II) Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
114	44	opreq2i 4893	. 2 |- ((II X.t II) Cn II) = ((II X.t II) Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
115	113, 114	eleqtrri 1970	1 |- H e. ((II X.t II) Cn II)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  <.cop 3046  U.cuni 3177  {copab 3395   X. cxp 3984   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  [,]cicc 7527  abscabs 8000  -cn->ccncf 8524  Topctop 8857   X.t ctx 8930   Cn ccn 9028  Metcme 9066  Opencopn 9069  subSpcsubsp 10242  IIcii 15865

This theorem is referenced by:  reparpht 16065

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-icc 7531  df-uz 7587  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-cncf 8525  df-top 8861  df-topsp 8862  df-bases 8863  df-topgen 8864  df-tx 8931  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-subsp 10243  df-ii 15866

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