Theorem cnfnc 11491

Description: Basic continuity property of a continuous functional.

Assertion

Ref	Expression
cnfnc	|- (((T e. ConFn /\ A e. ~H) /\ (B e. RR /\ 0 < B)) -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B)))

Distinct variable groups:   x,y,A   x,B,y   x,T,y

Proof of Theorem cnfnc

Step	Hyp	Ref	Expression
1		elcnfn 11446	. . . 4 |- (T e. ConFn <-> (T:~H-->CC /\ A.z e. ~H A.w e. RR (0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w)))))
2	1	simprbi 353	. . 3 |- (T e. ConFn -> A.z e. ~H A.w e. RR (0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w))))
3		opreq2 4890	. . . . . . . . . . . . 13 |- (z = A -> (y -h z) = (y -h A))
4	3	fveq2d 4685	. . . . . . . . . . . 12 |- (z = A -> (normh` (y -h z)) = (normh` (y -h A)))
5	4	breq1d 3348	. . . . . . . . . . 11 |- (z = A -> ((normh` (y -h z)) < x <-> (normh` (y -h A)) < x))
6		fveq2 4681	. . . . . . . . . . . . . 14 |- (z = A -> (T` z) = (T` A))
7	6	opreq2d 4898	. . . . . . . . . . . . 13 |- (z = A -> ((T` y) - (T` z)) = ((T` y) - (T` A)))
8	7	fveq2d 4685	. . . . . . . . . . . 12 |- (z = A -> (abs` ((T` y) - (T` z))) = (abs` ((T` y) - (T` A))))
9	8	breq1d 3348	. . . . . . . . . . 11 |- (z = A -> ((abs` ((T` y) - (T` z))) < w <-> (abs` ((T` y) - (T` A))) < w))
10	5, 9	imbi12d 688	. . . . . . . . . 10 |- (z = A -> (((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w) <-> ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w)))
11	10	ralbidv 2123	. . . . . . . . 9 |- (z = A -> (A.y e. ~H ((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w) <-> A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w)))
12	11	anbi2d 678	. . . . . . . 8 |- (z = A -> ((0 < x /\ A.y e. ~H ((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w)) <-> (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w))))
13	12	rexbidv 2124	. . . . . . 7 |- (z = A -> (E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w)) <-> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w))))
14	13	imbi2d 674	. . . . . 6 |- (z = A -> ((0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w))) <-> (0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w)))))
15	14	ralbidv 2123	. . . . 5 |- (z = A -> (A.w e. RR (0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w))) <-> A.w e. RR (0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w)))))
16	15	rcla4cv 2377	. . . 4 |- (A.z e. ~H A.w e. RR (0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w))) -> (A e. ~H -> A.w e. RR (0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w)))))
17		breq2 3342	. . . . . 6 |- (w = B -> (0 < w <-> 0 < B))
18		breq2 3342	. . . . . . . . . 10 |- (w = B -> ((abs` ((T` y) - (T` A))) < w <-> (abs` ((T` y) - (T` A))) < B))
19	18	imbi2d 674	. . . . . . . . 9 |- (w = B -> (((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w) <-> ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B)))
20	19	ralbidv 2123	. . . . . . . 8 |- (w = B -> (A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w) <-> A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B)))
21	20	anbi2d 678	. . . . . . 7 |- (w = B -> ((0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w)) <-> (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B))))
22	21	rexbidv 2124	. . . . . 6 |- (w = B -> (E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w)) <-> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B))))
23	17, 22	imbi12d 688	. . . . 5 |- (w = B -> ((0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w))) <-> (0 < B -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B)))))
24	23	rcla4cv 2377	. . . 4 |- (A.w e. RR (0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < w))) -> (B e. RR -> (0 < B -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B)))))
25	16, 24	syl6 25	. . 3 |- (A.z e. ~H A.w e. RR (0 < w -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h z)) < x -> (abs` ((T` y) - (T` z))) < w))) -> (A e. ~H -> (B e. RR -> (0 < B -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B))))))
26	2, 25	syl 12	. 2 |- (T e. ConFn -> (A e. ~H -> (B e. RR -> (0 < B -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B))))))
27	26	imp43 397	1 |- (((T e. ConFn /\ A e. ~H) /\ (B e. RR /\ 0 < B)) -> E.x e. RR (0 < x /\ A.y e. ~H ((normh` (y -h A)) < x -> (abs` ((T` y) - (T` A))) < B)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   - cmin 6445   < clt 6653  abscabs 8000  ~Hchil 10420   -h cmv 10424  normhcno 10426  ConFnccnf 10454

This theorem is referenced by:  nmcfnexlem2 11618  nlelchi 11631

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-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-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-cnfn 11410

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