Theorem metcnp4 9248

Description: Two ways to say a mapping from metric C to metric D is continuous at point P. Theorem 14-4.3 of [Gleason] p. 240.

Hypotheses

Ref	Expression
metcnp4.1	|- X = dom dom C
metcnp4.3	|- Y = dom dom D
metcnp4.c	|- J = (Open` C)
metcnp4.d	|- K = (Open` D)
metcnp4.5	|- G = {<.j, y>. | (j e. NN /\ y = (F` (f` j)))}

Assertion

Ref	Expression
metcnp4	|- ((C e. Met /\ D e. Met /\ P e. X) -> (F e. ((J CnP K)` P) <-> (F:X-->Y /\ A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P)))))

Distinct variable groups:   C,f   D,f   f,j,y,F   P,f   f,X,j   f,Y,j,y

Proof of Theorem metcnp4

Step	Hyp	Ref	Expression
1		metcnp4.1	. . 3 |- X = dom dom C
2		metcnp4.c	. . 3 |- J = (Open` C)
3		metcnp4.3	. . 3 |- Y = dom dom D
4		metcnp4.d	. . 3 |- K = (Open` D)
5	1, 2, 3, 4	metcnp2 9166	. 2 |- ((C e. Met /\ D e. Met /\ P e. X) -> (F e. ((J CnP K)` P) <-> (F:X-->Y /\ A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))))))
6		metcnp4.5	. . . . . . . . . . 11 |- G = {<.j, y>. | (j e. NN /\ y = (F` (f` j)))}
7	1, 3, 2, 4, 6	metcnp4lem2 9247	. . . . . . . . . 10 |- (((C e. Met /\ P e. X) /\ F:X-->Y) -> ((f:NN-->X /\ f(~~>m` C)P) -> (A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))) -> A.x e. RR (0 < x -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x)))))
8	7	3adantl2 1033	. . . . . . . . 9 |- (((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) -> ((f:NN-->X /\ f(~~>m` C)P) -> (A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))) -> A.x e. RR (0 < x -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x)))))
9	8	imp 377	. . . . . . . 8 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ (f:NN-->X /\ f(~~>m` C)P)) -> (A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))) -> A.x e. RR (0 < x -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x))))
10		simpll2 916	. . . . . . . . . 10 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ f:NN-->X) -> D e. Met)
11		ffvelrn 4787	. . . . . . . . . . . . 13 |- ((F:X-->Y /\ P e. X) -> (F` P) e. Y)
12	11	ancoms 484	. . . . . . . . . . . 12 |- ((P e. X /\ F:X-->Y) -> (F` P) e. Y)
13	12	3ad2antl3 1040	. . . . . . . . . . 11 |- (((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) -> (F` P) e. Y)
14	13	adantr 425	. . . . . . . . . 10 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ f:NN-->X) -> (F` P) e. Y)
15		ffvelrn 4787	. . . . . . . . . . . . . . 15 |- ((F:X-->Y /\ (f` j) e. X) -> (F` (f` j)) e. Y)
16		ffvelrn 4787	. . . . . . . . . . . . . . 15 |- ((f:NN-->X /\ j e. NN) -> (f` j) e. X)
17	15, 16	sylan2 500	. . . . . . . . . . . . . 14 |- ((F:X-->Y /\ (f:NN-->X /\ j e. NN)) -> (F` (f` j)) e. Y)
18	17	anassrs 489	. . . . . . . . . . . . 13 |- (((F:X-->Y /\ f:NN-->X) /\ j e. NN) -> (F` (f` j)) e. Y)
19	18	r19.21aiva 2176	. . . . . . . . . . . 12 |- ((F:X-->Y /\ f:NN-->X) -> A.j e. NN (F` (f` j)) e. Y)
20	6	fopab2 4796	. . . . . . . . . . . 12 |- (A.j e. NN (F` (f` j)) e. Y <-> G:NN-->Y)
21	19, 20	sylib 215	. . . . . . . . . . 11 |- ((F:X-->Y /\ f:NN-->X) -> G:NN-->Y)
22	21	adantll 428	. . . . . . . . . 10 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ f:NN-->X) -> G:NN-->Y)
23		1z 7368	. . . . . . . . . . 11 |- 1 e. ZZ
24		nnuz 7608	. . . . . . . . . . 11 |- NN = (ZZ>=` 1)
25	3, 23, 24	lmbrf2 9209	. . . . . . . . . 10 |- ((D e. Met /\ (F` P) e. Y /\ G:NN-->Y) -> (G(~~>m` D)(F` P) <-> A.x e. RR (0 < x -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x))))
26	10, 14, 22, 25	syl111anc 1100	. . . . . . . . 9 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ f:NN-->X) -> (G(~~>m` D)(F` P) <-> A.x e. RR (0 < x -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x))))
27	26	adantrr 431	. . . . . . . 8 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ (f:NN-->X /\ f(~~>m` C)P)) -> (G(~~>m` D)(F` P) <-> A.x e. RR (0 < x -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x))))
28	9, 27	sylibrd 221	. . . . . . 7 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ (f:NN-->X /\ f(~~>m` C)P)) -> (A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))) -> G(~~>m` D)(F` P)))
29	28	ex 402	. . . . . 6 |- (((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) -> ((f:NN-->X /\ f(~~>m` C)P) -> (A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))) -> G(~~>m` D)(F` P))))
30	29	com23 36	. . . . 5 |- (((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) -> (A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))) -> ((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P))))
31	30	19.21adv 1666	. . . 4 |- (((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) -> (A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))) -> A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P))))
32		simprl 450	. . . . . . . . . . . . . . 15 |- (((C e. Met /\ P e. X) /\ (f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k))) -> f:NN-->X)
33	1	lmnn 9213	. . . . . . . . . . . . . . 15 |- (((C e. Met /\ P e. X) /\ (f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k))) -> f(~~>m` C)P)
34	32, 33	jca 310	. . . . . . . . . . . . . 14 |- (((C e. Met /\ P e. X) /\ (f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k))) -> (f:NN-->X /\ f(~~>m` C)P))
35	34	ex 402	. . . . . . . . . . . . 13 |- ((C e. Met /\ P e. X) -> ((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> (f:NN-->X /\ f(~~>m` C)P)))
36	35	3adant2 895	. . . . . . . . . . . 12 |- ((C e. Met /\ D e. Met /\ P e. X) -> ((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> (f:NN-->X /\ f(~~>m` C)P)))
37	36	ad2antrr 440	. . . . . . . . . . 11 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ (x e. RR /\ 0 < x)) -> ((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> (f:NN-->X /\ f(~~>m` C)P)))
38	3, 23, 24	lmcvg2 9211	. . . . . . . . . . . . . . . . . . . 20 |- (((D e. Met /\ (F` P) e. Y /\ G(~~>m` D)(F` P)) /\ (x e. RR /\ 0 < x)) -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x))
39		nnre 7112	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (k e. NN -> k e. RR)
40		leid 6701	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (k e. RR -> k <_ k)
41	39, 40	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- (k e. NN -> k <_ k)
42		breq2 3342	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (m = k -> (k <_ m <-> k <_ k))
43		fveq2 4681	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (m = k -> (G` m) = (G` k))
44	43	opreq1d 4897	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (m = k -> ((G` m)D(F` P)) = ((G` k)D(F` P)))
45	44	breq1d 3348	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (m = k -> (((G` m)D(F` P)) < x <-> ((G` k)D(F` P)) < x))
46	42, 45	imbi12d 688	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (m = k -> ((k <_ m -> ((G` m)D(F` P)) < x) <-> (k <_ k -> ((G` k)D(F` P)) < x)))
47	46	rcla4v 2376	. . . . . . . . . . . . . . . . . . . . . . 23 |- (k e. NN -> (A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x) -> (k <_ k -> ((G` k)D(F` P)) < x)))
48	41, 47	mpid 58	. . . . . . . . . . . . . . . . . . . . . 22 |- (k e. NN -> (A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x) -> ((G` k)D(F` P)) < x))
49	1, 3, 2, 4, 6	metcnp4lem1 9246	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (k e. NN -> (G` k) = (F` (f` k)))
50	49	opreq1d 4897	. . . . . . . . . . . . . . . . . . . . . . 23 |- (k e. NN -> ((G` k)D(F` P)) = ((F` (f` k))D(F` P)))
51	50	breq1d 3348	. . . . . . . . . . . . . . . . . . . . . 22 |- (k e. NN -> (((G` k)D(F` P)) < x <-> ((F` (f` k))D(F` P)) < x))
52	48, 51	sylibd 219	. . . . . . . . . . . . . . . . . . . . 21 |- (k e. NN -> (A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x) -> ((F` (f` k))D(F` P)) < x))
53	52	reximia 2196	. . . . . . . . . . . . . . . . . . . 20 |- (E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x) -> E.k e. NN ((F` (f` k))D(F` P)) < x)
54	38, 53	syl 12	. . . . . . . . . . . . . . . . . . 19 |- (((D e. Met /\ (F` P) e. Y /\ G(~~>m` D)(F` P)) /\ (x e. RR /\ 0 < x)) -> E.k e. NN ((F` (f` k))D(F` P)) < x)
55	54	ex 402	. . . . . . . . . . . . . . . . . 18 |- ((D e. Met /\ (F` P) e. Y /\ G(~~>m` D)(F` P)) -> ((x e. RR /\ 0 < x) -> E.k e. NN ((F` (f` k))D(F` P)) < x))
56	55	3expia 1069	. . . . . . . . . . . . . . . . 17 |- ((D e. Met /\ (F` P) e. Y) -> (G(~~>m` D)(F` P) -> ((x e. RR /\ 0 < x) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
57	56	com23 36	. . . . . . . . . . . . . . . 16 |- ((D e. Met /\ (F` P) e. Y) -> ((x e. RR /\ 0 < x) -> (G(~~>m` D)(F` P) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
58	57, 11	sylan2 500	. . . . . . . . . . . . . . 15 |- ((D e. Met /\ (F:X-->Y /\ P e. X)) -> ((x e. RR /\ 0 < x) -> (G(~~>m` D)(F` P) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
59	58	anassrs 489	. . . . . . . . . . . . . 14 |- (((D e. Met /\ F:X-->Y) /\ P e. X) -> ((x e. RR /\ 0 < x) -> (G(~~>m` D)(F` P) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
60	59	an1rs 547	. . . . . . . . . . . . 13 |- (((D e. Met /\ P e. X) /\ F:X-->Y) -> ((x e. RR /\ 0 < x) -> (G(~~>m` D)(F` P) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
61	60	3adantl1 1032	. . . . . . . . . . . 12 |- (((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) -> ((x e. RR /\ 0 < x) -> (G(~~>m` D)(F` P) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
62	61	imp 377	. . . . . . . . . . 11 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ (x e. RR /\ 0 < x)) -> (G(~~>m` D)(F` P) -> E.k e. NN ((F` (f` k))D(F` P)) < x))
63	37, 62	imim12d 69	. . . . . . . . . 10 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ (x e. RR /\ 0 < x)) -> (((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P)) -> ((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
64	63	alimdv 1668	. . . . . . . . 9 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ (x e. RR /\ 0 < x)) -> (A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P)) -> A.f((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
65		impexp 374	. . . . . . . . . . . . 13 |- (((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> E.k e. NN ((F` (f` k))D(F` P)) < x) <-> (f:NN-->X -> (A.k e. NN ((f` k)CP) < (1 / k) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
66		r19.35 2231	. . . . . . . . . . . . . 14 |- (E.k e. NN (((f` k)CP) < (1 / k) -> ((F` (f` k))D(F` P)) < x) <-> (A.k e. NN ((f` k)CP) < (1 / k) -> E.k e. NN ((F` (f` k))D(F` P)) < x))
67	66	imbi2i 202	. . . . . . . . . . . . 13 |- ((f:NN-->X -> E.k e. NN (((f` k)CP) < (1 / k) -> ((F` (f` k))D(F` P)) < x)) <-> (f:NN-->X -> (A.k e. NN ((f` k)CP) < (1 / k) -> E.k e. NN ((F` (f` k))D(F` P)) < x)))
68	65, 67	bitr4i 193	. . . . . . . . . . . 12 |- (((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> E.k e. NN ((F` (f` k))D(F` P)) < x) <-> (f:NN-->X -> E.k e. NN (((f` k)CP) < (1 / k) -> ((F` (f` k))D(F` P)) < x)))
69	68	albii 1346	. . . . . . . . . . 11 |- (A.f((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> E.k e. NN ((F` (f` k))D(F` P)) < x) <-> A.f(f:NN-->X -> E.k e. NN (((f` k)CP) < (1 / k) -> ((F` (f` k))D(F` P)) < x)))
70		nnex 7116	. . . . . . . . . . . 12 |- NN e. _V
71		opreq1 4889	. . . . . . . . . . . . . 14 |- (w = (f` k) -> (wCP) = ((f` k)CP))
72	71	breq1d 3348	. . . . . . . . . . . . 13 |- (w = (f` k) -> ((wCP) < (1 / k) <-> ((f` k)CP) < (1 / k)))
73		fveq2 4681	. . . . . . . . . . . . . . 15 |- (w = (f` k) -> (F` w) = (F` (f` k)))
74	73	opreq1d 4897	. . . . . . . . . . . . . 14 |- (w = (f` k) -> ((F` w)D(F` P)) = ((F` (f` k))D(F` P)))
75	74	breq1d 3348	. . . . . . . . . . . . 13 |- (w = (f` k) -> (((F` w)D(F` P)) < x <-> ((F` (f` k))D(F` P)) < x))
76	72, 75	imbi12d 688	. . . . . . . . . . . 12 |- (w = (f` k) -> (((wCP) < (1 / k) -> ((F` w)D(F` P)) < x) <-> (((f` k)CP) < (1 / k) -> ((F` (f` k))D(F` P)) < x)))
77	70, 76	ac6n 5919	. . . . . . . . . . 11 |- (A.f(f:NN-->X -> E.k e. NN (((f` k)CP) < (1 / k) -> ((F` (f` k))D(F` P)) < x)) -> E.k e. NN A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x))
78	69, 77	sylbi 216	. . . . . . . . . 10 |- (A.f((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> E.k e. NN ((F` (f` k))D(F` P)) < x) -> E.k e. NN A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x))
79		nnrecre 7136	. . . . . . . . . . . . 13 |- (k e. NN -> (1 / k) e. RR)
80	79	adantr 425	. . . . . . . . . . . 12 |- ((k e. NN /\ A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x)) -> (1 / k) e. RR)
81		nnrecgt0 7137	. . . . . . . . . . . . 13 |- (k e. NN -> 0 < (1 / k))
82	81	anim1i 361	. . . . . . . . . . . 12 |- ((k e. NN /\ A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x)) -> (0 < (1 / k) /\ A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x)))
83		breq2 3342	. . . . . . . . . . . . . 14 |- (z = (1 / k) -> (0 < z <-> 0 < (1 / k)))
84		breq2 3342	. . . . . . . . . . . . . . . 16 |- (z = (1 / k) -> ((wCP) < z <-> (wCP) < (1 / k)))
85	84	imbi1d 675	. . . . . . . . . . . . . . 15 |- (z = (1 / k) -> (((wCP) < z -> ((F` w)D(F` P)) < x) <-> ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x)))
86	85	ralbidv 2123	. . . . . . . . . . . . . 14 |- (z = (1 / k) -> (A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x) <-> A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x)))
87	83, 86	anbi12d 690	. . . . . . . . . . . . 13 |- (z = (1 / k) -> ((0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)) <-> (0 < (1 / k) /\ A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x))))
88	87	rcla4ev 2381	. . . . . . . . . . . 12 |- (((1 / k) e. RR /\ (0 < (1 / k) /\ A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x))) -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)))
89	80, 82, 88	syl11anc 524	. . . . . . . . . . 11 |- ((k e. NN /\ A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x)) -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)))
90	89	r19.23aiva 2212	. . . . . . . . . 10 |- (E.k e. NN A.w e. X ((wCP) < (1 / k) -> ((F` w)D(F` P)) < x) -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)))
91	78, 90	syl 12	. . . . . . . . 9 |- (A.f((f:NN-->X /\ A.k e. NN ((f` k)CP) < (1 / k)) -> E.k e. NN ((F` (f` k))D(F` P)) < x) -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)))
92	64, 91	syl6 25	. . . . . . . 8 |- ((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ (x e. RR /\ 0 < x)) -> (A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P)) -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))))
93	92	imp 377	. . . . . . 7 |- (((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ (x e. RR /\ 0 < x)) /\ A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P))) -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)))
94	93	an1rs 547	. . . . . 6 |- (((((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) /\ A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P))) /\ (x e. RR /\ 0 < x)) -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)))
95	94	exp43 415	. . . . 5 |- (((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) -> (A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P)) -> (x e. RR -> (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))))))
96	95	r19.21adv 2181	. . . 4 |- (((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) -> (A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P)) -> A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)))))
97	31, 96	impbid 574	. . 3 |- (((C e. Met /\ D e. Met /\ P e. X) /\ F:X-->Y) -> (A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x))) <-> A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P))))
98	97	pm5.32da 711	. 2 |- ((C e. Met /\ D e. Met /\ P e. X) -> ((F:X-->Y /\ A.x e. RR (0 < x -> E.z e. RR (0 < z /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)))) <-> (F:X-->Y /\ A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P)))))
99	5, 98	bitrd 587	1 |- ((C e. Met /\ D e. Met /\ P e. X) -> (F e. ((J CnP K)` P) <-> (F:X-->Y /\ A.f((f:NN-->X /\ f(~~>m` C)P) -> G(~~>m` D)(F` P)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338  {copab 3395  dom cdm 3986  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386  1c1 6387   / cdiv 6447   <_ cle 6448  NNcn 6449   < clt 6653   CnP ccnp 9029  Metcme 9066  Opencopn 9069  ~~>mclm 9197

This theorem is referenced by:  metcn4 9249

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-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-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-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-top 8861  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200

Copyright terms: Public domain