Theorem blocni 9805

Description: A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97.

Hypotheses

Ref	Expression
blocni.8	|- C = (IndMet` U)
blocni.d	|- D = (IndMet` W)
blocni.j	|- J = (Open` C)
blocni.k	|- K = (Open` D)
blocni.4	|- L = (U LnOp W)
blocni.5	|- B = (U BLnOp W)
blocni.u	|- U e. NrmCVec
blocni.w	|- W e. NrmCVec
blocni.l	|- T e. L

Assertion

Ref	Expression
blocni	|- (T e. (J Cn K) <-> T e. B)

Proof of Theorem blocni

Step	Hyp	Ref	Expression
1		blocni.u	. . . . . 6 |- U e. NrmCVec
2		eqid 1884	. . . . . . 7 |- (BaseSet` U) = (BaseSet` U)
3		eqid 1884	. . . . . . 7 |- (0v` U) = (0v` U)
4	2, 3	nvzcl 9587	. . . . . 6 |- (U e. NrmCVec -> (0v` U) e. (BaseSet` U))
5	1, 4	ax-mp 7	. . . . 5 |- (0v` U) e. (BaseSet` U)
6		blocni.8	. . . . . . . 8 |- C = (IndMet` U)
7	6	imsmet 9656	. . . . . . 7 |- (U e. NrmCVec -> C e. Met)
8	1, 7	ax-mp 7	. . . . . 6 |- C e. Met
9	2, 6, 1	imsbai 9654	. . . . . . 7 |- (BaseSet` U) = dom dom C
10		blocni.j	. . . . . . 7 |- J = (Open` C)
11	9, 10	uniopn2 9138	. . . . . 6 |- (C e. Met -> U.J = (BaseSet` U))
12	8, 11	ax-mp 7	. . . . 5 |- U.J = (BaseSet` U)
13	5, 12	eleqtrri 1970	. . . 4 |- (0v` U) e. U.J
14	10	opntop 9147	. . . . . 6 |- (C e. Met -> J e. Top)
15	8, 14	ax-mp 7	. . . . 5 |- J e. Top
16		blocni.w	. . . . . . 7 |- W e. NrmCVec
17		blocni.d	. . . . . . . 8 |- D = (IndMet` W)
18	17	imsmet 9656	. . . . . . 7 |- (W e. NrmCVec -> D e. Met)
19	16, 18	ax-mp 7	. . . . . 6 |- D e. Met
20		blocni.k	. . . . . . 7 |- K = (Open` D)
21	20	opntop 9147	. . . . . 6 |- (D e. Met -> K e. Top)
22	19, 21	ax-mp 7	. . . . 5 |- K e. Top
23		eqid 1884	. . . . . 6 |- U.J = U.J
24		eqid 1884	. . . . . 6 |- U.K = U.K
25	23, 24	cncnpi 9050	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (T e. (J Cn K) /\ (0v` U) e. U.J)) -> T e. ((J CnP K)` (0v` U)))
26	15, 22, 25	mpanl12 773	. . . 4 |- ((T e. (J Cn K) /\ (0v` U) e. U.J) -> T e. ((J CnP K)` (0v` U)))
27	13, 26	mpan2 760	. . 3 |- (T e. (J Cn K) -> T e. ((J CnP K)` (0v` U)))
28		blocni.4	. . . . 5 |- L = (U LnOp W)
29		blocni.5	. . . . 5 |- B = (U BLnOp W)
30		blocni.l	. . . . 5 |- T e. L
31	6, 17, 10, 20, 28, 29, 1, 16, 30, 2	blocnilem 9804	. . . 4 |- (((0v` U) e. (BaseSet` U) /\ T e. ((J CnP K)` (0v` U))) -> T e. B)
32	5, 31	mpan 759	. . 3 |- (T e. ((J CnP K)` (0v` U)) -> T e. B)
33	27, 32	syl 12	. 2 |- (T e. (J Cn K) -> T e. B)
34		eleq1 1957	. . 3 |- (T = (U 0op W) -> (T e. (J Cn K) <-> (U 0op W) e. (J Cn K)))
35		simpr 350	. . . . . . . . . . 11 |- (((T e. B /\ T =/= (U 0op W)) /\ y e. RR) -> y e. RR)
36		eqid 1884	. . . . . . . . . . . . . 14 |- (BaseSet` W) = (BaseSet` W)
37		eqid 1884	. . . . . . . . . . . . . 14 |- (UnormOpW) = (UnormOpW)
38	2, 36, 37, 29	nmblore 9786	. . . . . . . . . . . . 13 |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. B) -> ((UnormOpW)` T) e. RR)
39	1, 16, 38	mp3an12 1181	. . . . . . . . . . . 12 |- (T e. B -> ((UnormOpW)` T) e. RR)
40	39	ad2antrr 440	. . . . . . . . . . 11 |- (((T e. B /\ T =/= (U 0op W)) /\ y e. RR) -> ((UnormOpW)` T) e. RR)
41		eqid 1884	. . . . . . . . . . . . . . . 16 |- (U 0op W) = (U 0op W)
42	37, 41, 28, 1, 16	nmlno0i 9794	. . . . . . . . . . . . . . 15 |- (T e. L -> (((UnormOpW)` T) = 0 <-> T = (U 0op W)))
43	30, 42	ax-mp 7	. . . . . . . . . . . . . 14 |- (((UnormOpW)` T) = 0 <-> T = (U 0op W))
44	43	biimpi 168	. . . . . . . . . . . . 13 |- (((UnormOpW)` T) = 0 -> T = (U 0op W))
45	44	necon3i 2042	. . . . . . . . . . . 12 |- (T =/= (U 0op W) -> ((UnormOpW)` T) =/= 0)
46	45	ad2antlr 441	. . . . . . . . . . 11 |- (((T e. B /\ T =/= (U 0op W)) /\ y e. RR) -> ((UnormOpW)` T) =/= 0)
47		redivcl 6978	. . . . . . . . . . 11 |- ((y e. RR /\ ((UnormOpW)` T) e. RR /\ ((UnormOpW)` T) =/= 0) -> (y / ((UnormOpW)` T)) e. RR)
48	35, 40, 46, 47	syl111anc 1100	. . . . . . . . . 10 |- (((T e. B /\ T =/= (U 0op W)) /\ y e. RR) -> (y / ((UnormOpW)` T)) e. RR)
49	48	ad2ant2r 445	. . . . . . . . 9 |- ((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) -> (y / ((UnormOpW)` T)) e. RR)
50		simpr 350	. . . . . . . . . . 11 |- (((T e. B /\ T =/= (U 0op W)) /\ (y e. RR /\ 0 < y)) -> (y e. RR /\ 0 < y))
51	37, 41, 28	nmlnogt0 9797	. . . . . . . . . . . . . . 15 |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) -> (T =/= (U 0op W) <-> 0 < ((UnormOpW)` T)))
52	1, 16, 30, 51	mp3an 1191	. . . . . . . . . . . . . 14 |- (T =/= (U 0op W) <-> 0 < ((UnormOpW)` T))
53	52	biimpi 168	. . . . . . . . . . . . 13 |- (T =/= (U 0op W) -> 0 < ((UnormOpW)` T))
54	39, 53	anim12i 360	. . . . . . . . . . . 12 |- ((T e. B /\ T =/= (U 0op W)) -> (((UnormOpW)` T) e. RR /\ 0 < ((UnormOpW)` T)))
55	54	adantr 425	. . . . . . . . . . 11 |- (((T e. B /\ T =/= (U 0op W)) /\ (y e. RR /\ 0 < y)) -> (((UnormOpW)` T) e. RR /\ 0 < ((UnormOpW)` T)))
56		divgt0 7037	. . . . . . . . . . 11 |- (((y e. RR /\ 0 < y) /\ (((UnormOpW)` T) e. RR /\ 0 < ((UnormOpW)` T))) -> 0 < (y / ((UnormOpW)` T)))
57	50, 55, 56	syl11anc 524	. . . . . . . . . 10 |- (((T e. B /\ T =/= (U 0op W)) /\ (y e. RR /\ 0 < y)) -> 0 < (y / ((UnormOpW)` T)))
58	57	adantlr 429	. . . . . . . . 9 |- ((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) -> 0 < (y / ((UnormOpW)` T)))
59	39	ad2antrr 440	. . . . . . . . . . . . . . 15 |- (((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) -> ((UnormOpW)` T) e. RR)
60	59	ad2antrr 440	. . . . . . . . . . . . . 14 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> ((UnormOpW)` T) e. RR)
61	9	metcl 9088	. . . . . . . . . . . . . . . . 17 |- ((C e. Met /\ x e. (BaseSet` U) /\ w e. (BaseSet` U)) -> (xCw) e. RR)
62	8, 61	mp3an1 1178	. . . . . . . . . . . . . . . 16 |- ((x e. (BaseSet` U) /\ w e. (BaseSet` U)) -> (xCw) e. RR)
63	62	adantlr 429	. . . . . . . . . . . . . . 15 |- (((x e. (BaseSet` U) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> (xCw) e. RR)
64	63	adantlll 432	. . . . . . . . . . . . . 14 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> (xCw) e. RR)
65		simplrl 454	. . . . . . . . . . . . . 14 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> y e. RR)
66	53	ad2antlr 441	. . . . . . . . . . . . . . 15 |- (((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) -> 0 < ((UnormOpW)` T))
67	66	ad2antrr 440	. . . . . . . . . . . . . 14 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> 0 < ((UnormOpW)` T))
68		ltmuldiv2OLD 7048	. . . . . . . . . . . . . 14 |- (((((UnormOpW)` T) e. RR /\ (xCw) e. RR /\ y e. RR) /\ 0 < ((UnormOpW)` T)) -> ((((UnormOpW)` T) x. (xCw)) < y <-> (xCw) < (y / ((UnormOpW)` T))))
69	60, 64, 65, 67, 68	syl31anc 1103	. . . . . . . . . . . . 13 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> ((((UnormOpW)` T) x. (xCw)) < y <-> (xCw) < (y / ((UnormOpW)` T))))
70	69	biimpar 461	. . . . . . . . . . . 12 |- ((((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) /\ (xCw) < (y / ((UnormOpW)` T))) -> (((UnormOpW)` T) x. (xCw)) < y)
71	2, 36, 6, 17, 37, 29, 1, 16	blometi 9803	. . . . . . . . . . . . . . . . 17 |- ((T e. B /\ x e. (BaseSet` U) /\ w e. (BaseSet` U)) -> ((T` x)D(T` w)) <_ (((UnormOpW)` T) x. (xCw)))
72	71	3expa 1067	. . . . . . . . . . . . . . . 16 |- (((T e. B /\ x e. (BaseSet` U)) /\ w e. (BaseSet` U)) -> ((T` x)D(T` w)) <_ (((UnormOpW)` T) x. (xCw)))
73	72	adantllr 433	. . . . . . . . . . . . . . 15 |- ((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ w e. (BaseSet` U)) -> ((T` x)D(T` w)) <_ (((UnormOpW)` T) x. (xCw)))
74	73	adantlr 429	. . . . . . . . . . . . . 14 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> ((T` x)D(T` w)) <_ (((UnormOpW)` T) x. (xCw)))
75	36, 17, 16	imsbai 9654	. . . . . . . . . . . . . . . . . . . 20 |- (BaseSet` W) = dom dom D
76	75	metcl 9088	. . . . . . . . . . . . . . . . . . 19 |- ((D e. Met /\ (T` x) e. (BaseSet` W) /\ (T` w) e. (BaseSet` W)) -> ((T` x)D(T` w)) e. RR)
77	19, 76	mp3an1 1178	. . . . . . . . . . . . . . . . . 18 |- (((T` x) e. (BaseSet` W) /\ (T` w) e. (BaseSet` W)) -> ((T` x)D(T` w)) e. RR)
78	2, 36, 28	lnof 9755	. . . . . . . . . . . . . . . . . . . 20 |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) -> T:(BaseSet` U)-->(BaseSet` W))
79	1, 16, 30, 78	mp3an 1191	. . . . . . . . . . . . . . . . . . 19 |- T:(BaseSet` U)-->(BaseSet` W)
80	79	ffvelrni 4788	. . . . . . . . . . . . . . . . . 18 |- (x e. (BaseSet` U) -> (T` x) e. (BaseSet` W))
81	79	ffvelrni 4788	. . . . . . . . . . . . . . . . . 18 |- (w e. (BaseSet` U) -> (T` w) e. (BaseSet` W))
82	77, 80, 81	syl2an 503	. . . . . . . . . . . . . . . . 17 |- ((x e. (BaseSet` U) /\ w e. (BaseSet` U)) -> ((T` x)D(T` w)) e. RR)
83	82	adantll 428	. . . . . . . . . . . . . . . 16 |- ((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ w e. (BaseSet` U)) -> ((T` x)D(T` w)) e. RR)
84	83	adantlr 429	. . . . . . . . . . . . . . 15 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> ((T` x)D(T` w)) e. RR)
85		remulcl 6457	. . . . . . . . . . . . . . . . . . 19 |- ((((UnormOpW)` T) e. RR /\ (xCw) e. RR) -> (((UnormOpW)` T) x. (xCw)) e. RR)
86	85, 39, 62	syl2an 503	. . . . . . . . . . . . . . . . . 18 |- ((T e. B /\ (x e. (BaseSet` U) /\ w e. (BaseSet` U))) -> (((UnormOpW)` T) x. (xCw)) e. RR)
87	86	anassrs 489	. . . . . . . . . . . . . . . . 17 |- (((T e. B /\ x e. (BaseSet` U)) /\ w e. (BaseSet` U)) -> (((UnormOpW)` T) x. (xCw)) e. RR)
88	87	adantllr 433	. . . . . . . . . . . . . . . 16 |- ((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ w e. (BaseSet` U)) -> (((UnormOpW)` T) x. (xCw)) e. RR)
89	88	adantlr 429	. . . . . . . . . . . . . . 15 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> (((UnormOpW)` T) x. (xCw)) e. RR)
90		lelttr 6693	. . . . . . . . . . . . . . 15 |- ((((T` x)D(T` w)) e. RR /\ (((UnormOpW)` T) x. (xCw)) e. RR /\ y e. RR) -> ((((T` x)D(T` w)) <_ (((UnormOpW)` T) x. (xCw)) /\ (((UnormOpW)` T) x. (xCw)) < y) -> ((T` x)D(T` w)) < y))
91	84, 89, 65, 90	syl111anc 1100	. . . . . . . . . . . . . 14 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> ((((T` x)D(T` w)) <_ (((UnormOpW)` T) x. (xCw)) /\ (((UnormOpW)` T) x. (xCw)) < y) -> ((T` x)D(T` w)) < y))
92	74, 91	mpand 765	. . . . . . . . . . . . 13 |- (((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) -> ((((UnormOpW)` T) x. (xCw)) < y -> ((T` x)D(T` w)) < y))
93	92	imp 377	. . . . . . . . . . . 12 |- ((((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) /\ (((UnormOpW)` T) x. (xCw)) < y) -> ((T` x)D(T` w)) < y)
94	70, 93	syldan 516	. . . . . . . . . . 11 |- ((((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) /\ w e. (BaseSet` U)) /\ (xCw) < (y / ((UnormOpW)` T))) -> ((T` x)D(T` w)) < y)
95	94	exp31 407	. . . . . . . . . 10 |- ((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) -> (w e. (BaseSet` U) -> ((xCw) < (y / ((UnormOpW)` T)) -> ((T` x)D(T` w)) < y)))
96	95	r19.21aiv 2175	. . . . . . . . 9 |- ((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) -> A.w e. (BaseSet` U)((xCw) < (y / ((UnormOpW)` T)) -> ((T` x)D(T` w)) < y))
97		breq2 3342	. . . . . . . . . . 11 |- (z = (y / ((UnormOpW)` T)) -> (0 < z <-> 0 < (y / ((UnormOpW)` T))))
98		breq2 3342	. . . . . . . . . . . . 13 |- (z = (y / ((UnormOpW)` T)) -> ((xCw) < z <-> (xCw) < (y / ((UnormOpW)` T))))
99	98	imbi1d 675	. . . . . . . . . . . 12 |- (z = (y / ((UnormOpW)` T)) -> (((xCw) < z -> ((T` x)D(T` w)) < y) <-> ((xCw) < (y / ((UnormOpW)` T)) -> ((T` x)D(T` w)) < y)))
100	99	ralbidv 2123	. . . . . . . . . . 11 |- (z = (y / ((UnormOpW)` T)) -> (A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y) <-> A.w e. (BaseSet` U)((xCw) < (y / ((UnormOpW)` T)) -> ((T` x)D(T` w)) < y)))
101	97, 100	anbi12d 690	. . . . . . . . . 10 |- (z = (y / ((UnormOpW)` T)) -> ((0 < z /\ A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y)) <-> (0 < (y / ((UnormOpW)` T)) /\ A.w e. (BaseSet` U)((xCw) < (y / ((UnormOpW)` T)) -> ((T` x)D(T` w)) < y))))
102	101	rcla4ev 2381	. . . . . . . . 9 |- (((y / ((UnormOpW)` T)) e. RR /\ (0 < (y / ((UnormOpW)` T)) /\ A.w e. (BaseSet` U)((xCw) < (y / ((UnormOpW)` T)) -> ((T` x)D(T` w)) < y))) -> E.z e. RR (0 < z /\ A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y)))
103	49, 58, 96, 102	syl12anc 1098	. . . . . . . 8 |- ((((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) /\ (y e. RR /\ 0 < y)) -> E.z e. RR (0 < z /\ A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y)))
104	103	exp32 408	. . . . . . 7 |- (((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) -> (y e. RR -> (0 < y -> E.z e. RR (0 < z /\ A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y)))))
105	104	r19.21aiv 2175	. . . . . 6 |- (((T e. B /\ T =/= (U 0op W)) /\ x e. (BaseSet` U)) -> A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y))))
106	105	r19.21aiva 2176	. . . . 5 |- ((T e. B /\ T =/= (U 0op W)) -> A.x e. (BaseSet` U)A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y))))
107	106, 79	jctil 316	. . . 4 |- ((T e. B /\ T =/= (U 0op W)) -> (T:(BaseSet` U)-->(BaseSet` W) /\ A.x e. (BaseSet` U)A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y)))))
108	9, 10, 75, 20	metcn 9167	. . . . 5 |- ((C e. Met /\ D e. Met) -> (T e. (J Cn K) <-> (T:(BaseSet` U)-->(BaseSet` W) /\ A.x e. (BaseSet` U)A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y))))))
109	8, 19, 108	mp2an 761	. . . 4 |- (T e. (J Cn K) <-> (T:(BaseSet` U)-->(BaseSet` W) /\ A.x e. (BaseSet` U)A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. (BaseSet` U)((xCw) < z -> ((T` x)D(T` w)) < y)))))
110	107, 109	sylibr 217	. . 3 |- ((T e. B /\ T =/= (U 0op W)) -> T e. (J Cn K))
111		eqid 1884	. . . . . . 7 |- (0v` W) = (0v` W)
112	2, 111, 41	0ofval 9787	. . . . . 6 |- ((U e. NrmCVec /\ W e. NrmCVec) -> (U 0op W) = ((BaseSet` U) X. {(0v` W)}))
113	1, 16, 112	mp2an 761	. . . . 5 |- (U 0op W) = ((BaseSet` U) X. {(0v` W)})
114	8, 19	pm3.2i 307	. . . . . 6 |- (C e. Met /\ D e. Met)
115	36, 111	nvzcl 9587	. . . . . . . 8 |- (W e. NrmCVec -> (0v` W) e. (BaseSet` W))
116	16, 115	ax-mp 7	. . . . . . 7 |- (0v` W) e. (BaseSet` W)
117		fvex 4689	. . . . . . . 8 |- (0v` W) e. _V
118	117	fconst 4602	. . . . . . 7 |- ((BaseSet` U) X. {(0v` W)}):(BaseSet` U)-->{(0v` W)}
119	116, 118	pm3.2i 307	. . . . . 6 |- ((0v` W) e. (BaseSet` W) /\ ((BaseSet` U) X. {(0v` W)}):(BaseSet` U)-->{(0v` W)})
120	9, 75, 10, 20	metcnconst 9163	. . . . . 6 |- (((C e. Met /\ D e. Met) /\ ((0v` W) e. (BaseSet` W) /\ ((BaseSet` U) X. {(0v` W)}):(BaseSet` U)-->{(0v` W)})) -> ((BaseSet` U) X. {(0v` W)}) e. (J Cn K))
121	114, 119, 120	mp2an 761	. . . . 5 |- ((BaseSet` U) X. {(0v` W)}) e. (J Cn K)
122	113, 121	eqeltri 1967	. . . 4 |- (U 0op W) e. (J Cn K)
123	122	a1i 8	. . 3 |- (T e. B -> (U 0op W) e. (J Cn K))
124	34, 110, 123	pm2.61ne 2087	. 2 |- (T e. B -> T e. (J Cn K))
125	33, 124	impbii 174	1 |- (T e. (J Cn K) <-> T e. B)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {csn 3044  U.cuni 3177   class class class wbr 3338   X. cxp 3984  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   x. cmul 6391   / cdiv 6447   <_ cle 6448   < clt 6653  Topctop 8857   Cn ccn 9028   CnP ccnp 9029  Metcme 9066  Opencopn 9069  NrmCVeccnv 9535  BaseSetcba 9537  0vcn0v 9539  IndMetcims 9542   LnOp clno 9740  normOpcnmo 9741   BLnOp cblo 9742   0op c0o 9743

This theorem is referenced by:  lnocni 9806  blocn 9807

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  ax-inf2 5731

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-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-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-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552  df-lno 9744  df-nmo 9745  df-blo 9746  df-0o 9747

Copyright terms: Public domain