Theorem metcnp4lem2 7969

Description: Lemma for metcnp4 7970.

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
metcnp4lem2	|- (((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)))))

Distinct variable groups:   f,k,m,w,x,z,C   D,f,k,m,w,x,z   f,j,y,F,k,m,w,x,z   k,G,m,x,z   w,J,x,z   x,K   P,f,k,m,w,x,z   f,X,j,k,m,w,x,z   f,Y,j,k,m,w,x,y,z

Proof of Theorem metcnp4lem2

Step	Hyp	Ref	Expression
1		metcnp4.1	. . . . . . . . . . . 12 |- X = dom dom C
2		1z 6159	. . . . . . . . . . . 12 |- 1 e. ZZ
3		nnuz 6439	. . . . . . . . . . . 12 |- NN = (ZZ>` 1)
4	1, 2, 3	lmcvg 7932	. . . . . . . . . . 11 |- (((C e. Met /\ P e. X /\ f(~~>m` C)P) /\ (z e. RR /\ 0 < z)) -> E.k e. NN A.m e. NN (k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z)))
5	4	ex 373	. . . . . . . . . 10 |- ((C e. Met /\ P e. X /\ f(~~>m` C)P) -> ((z e. RR /\ 0 < z) -> E.k e. NN A.m e. NN (k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z))))
6	5	3expa 833	. . . . . . . . 9 |- (((C e. Met /\ P e. X) /\ f(~~>m` C)P) -> ((z e. RR /\ 0 < z) -> E.k e. NN A.m e. NN (k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z))))
7	6	ad2ant2rl 411	. . . . . . . 8 |- ((((C e. Met /\ P e. X) /\ F:X-->Y) /\ (f:NN-->X /\ f(~~>m` C)P)) -> ((z e. RR /\ 0 < z) -> E.k e. NN A.m e. NN (k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z))))
8		metcnp4.3	. . . . . . . . . . . . . . . . . . 19 |- Y = dom dom D
9		metcnp4.c	. . . . . . . . . . . . . . . . . . 19 |- J = (Open` C)
10		metcnp4.d	. . . . . . . . . . . . . . . . . . 19 |- K = (Open` D)
11		metcnp4.5	. . . . . . . . . . . . . . . . . . 19 |- G = {<.j, y>. | (j e. NN /\ y = (F` (f` j)))}
12	1, 8, 9, 10, 11	metcnp4lem1 7968	. . . . . . . . . . . . . . . . . 18 |- (m e. NN -> (G` m) = (F` (f` m)))
13	12	opreq1d 3975	. . . . . . . . . . . . . . . . 17 |- (m e. NN -> ((G` m)D(F` P)) = ((F` (f` m))D(F` P)))
14	13	ad2antlr 405	. . . . . . . . . . . . . . . 16 |- ((((F:X-->Y /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)) /\ m e. NN) /\ ((f` m) e. X /\ ((f` m)CP) < z)) -> ((G` m)D(F` P)) = ((F` (f` m))D(F` P)))
15		opreq1 3968	. . . . . . . . . . . . . . . . . . . . . 22 |- (w = (f` m) -> (wCP) = ((f` m)CP))
16	15	breq1d 2629	. . . . . . . . . . . . . . . . . . . . 21 |- (w = (f` m) -> ((wCP) < z <-> ((f` m)CP) < z))
17		fveq2 3724	. . . . . . . . . . . . . . . . . . . . . . 23 |- (w = (f` m) -> (F` w) = (F` (f` m)))
18	17	opreq1d 3975	. . . . . . . . . . . . . . . . . . . . . 22 |- (w = (f` m) -> ((F` w)D(F` P)) = ((F` (f` m))D(F` P)))
19	18	breq1d 2629	. . . . . . . . . . . . . . . . . . . . 21 |- (w = (f` m) -> (((F` w)D(F` P)) < x <-> ((F` (f` m))D(F` P)) < x))
20	16, 19	imbi12d 626	. . . . . . . . . . . . . . . . . . . 20 |- (w = (f` m) -> (((wCP) < z -> ((F` w)D(F` P)) < x) <-> (((f` m)CP) < z -> ((F` (f` m))D(F` P)) < x)))
21	20	rcla4cv 1874	. . . . . . . . . . . . . . . . . . 19 |- (A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x) -> ((f` m) e. X -> (((f` m)CP) < z -> ((F` (f` m))D(F` P)) < x)))
22	21	imp32 363	. . . . . . . . . . . . . . . . . 18 |- ((A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x) /\ ((f` m) e. X /\ ((f` m)CP) < z)) -> ((F` (f` m))D(F` P)) < x)
23	22	adantll 392	. . . . . . . . . . . . . . . . 17 |- (((F:X-->Y /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)) /\ ((f` m) e. X /\ ((f` m)CP) < z)) -> ((F` (f` m))D(F` P)) < x)
24	23	adantlr 393	. . . . . . . . . . . . . . . 16 |- ((((F:X-->Y /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)) /\ m e. NN) /\ ((f` m) e. X /\ ((f` m)CP) < z)) -> ((F` (f` m))D(F` P)) < x)
25	14, 24	eqbrtrd 2635	. . . . . . . . . . . . . . 15 |- ((((F:X-->Y /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)) /\ m e. NN) /\ ((f` m) e. X /\ ((f` m)CP) < z)) -> ((G` m)D(F` P)) < x)
26	25	ex 373	. . . . . . . . . . . . . 14 |- (((F:X-->Y /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)) /\ m e. NN) -> (((f` m) e. X /\ ((f` m)CP) < z) -> ((G` m)D(F` P)) < x))
27	26	imim2d 25	. . . . . . . . . . . . 13 |- (((F:X-->Y /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)) /\ m e. NN) -> ((k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z)) -> (k <_ m -> ((G` m)D(F` P)) < x)))
28	27	r19.20dva 1709	. . . . . . . . . . . 12 |- ((F:X-->Y /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)) -> (A.m e. NN (k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z)) -> A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x)))
29	28	r19.22sdv 1738	. . . . . . . . . . 11 |- ((F:X-->Y /\ A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x)) -> (E.k e. NN A.m e. NN (k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z)) -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x)))
30	29	ex 373	. . . . . . . . . 10 |- (F:X-->Y -> (A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x) -> (E.k e. NN A.m e. NN (k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z)) -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x))))
31	30	com23 32	. . . . . . . . 9 |- (F:X-->Y -> (E.k e. NN A.m e. NN (k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z)) -> (A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x) -> E.k e. NN A.m e. NN (k <_ m -> ((G` m)D(F` P)) < x))))
32	31	ad2antlr 405	. . . . . . . 8 |- ((((C e. Met /\ P e. X) /\ F:X-->Y) /\ (f:NN-->X /\ f(~~>m` C)P)) -> (E.k e. NN A.m e. NN (k <_ m -> ((f` m) e. X /\ ((f` m)CP) < z)) -> (A.w e. X ((wCP) < z -> ((F` w)D(F` P)) < x) -> E.k e. NN A.