Theorem nrmsep2 15555

Description: In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other.

Assertion

Ref	Expression
nrmsep2	|- ((J e. Nrm /\ (C e. (Clsd` J) /\ D e. (Clsd` J) /\ (C i^i D) = (/))) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i D) = (/)))

Distinct variable groups:   C,o   D,o   o,J

Proof of Theorem nrmsep2

Step	Hyp	Ref	Expression
1		nrmtop 15553	. . . 4 |- (J e. Nrm -> J e. Top)
2		isnrm2 15552	. . . . 5 |- (J e. Top -> (J e. Nrm <-> A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (c C_ o /\ (((cls` J)` o) i^i d) = (/)))))
3	2	biimpd 170	. . . 4 |- (J e. Top -> (J e. Nrm -> A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (c C_ o /\ (((cls` J)` o) i^i d) = (/)))))
4	1, 3	mpcom 60	. . 3 |- (J e. Nrm -> A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (c C_ o /\ (((cls` J)` o) i^i d) = (/))))
5		ineq1 2789	. . . . . . . . 9 |- (c = C -> (c i^i d) = (C i^i d))
6	5	eqeq1d 1892	. . . . . . . 8 |- (c = C -> ((c i^i d) = (/) <-> (C i^i d) = (/)))
7		sseq1 2637	. . . . . . . . . 10 |- (c = C -> (c C_ o <-> C C_ o))
8	7	anbi1d 679	. . . . . . . . 9 |- (c = C -> ((c C_ o /\ (((cls` J)` o) i^i d) = (/)) <-> (C C_ o /\ (((cls` J)` o) i^i d) = (/))))
9	8	rexbidv 2124	. . . . . . . 8 |- (c = C -> (E.o e. J (c C_ o /\ (((cls` J)` o) i^i d) = (/)) <-> E.o e. J (C C_ o /\ (((cls` J)` o) i^i d) = (/))))
10	6, 9	imbi12d 688	. . . . . . 7 |- (c = C -> (((c i^i d) = (/) -> E.o e. J (c C_ o /\ (((cls` J)` o) i^i d) = (/))) <-> ((C i^i d) = (/) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i d) = (/)))))
11		ineq2 2790	. . . . . . . . 9 |- (d = D -> (C i^i d) = (C i^i D))
12	11	eqeq1d 1892	. . . . . . . 8 |- (d = D -> ((C i^i d) = (/) <-> (C i^i D) = (/)))
13		ineq2 2790	. . . . . . . . . . 11 |- (d = D -> (((cls` J)` o) i^i d) = (((cls` J)` o) i^i D))
14	13	eqeq1d 1892	. . . . . . . . . 10 |- (d = D -> ((((cls` J)` o) i^i d) = (/) <-> (((cls` J)` o) i^i D) = (/)))
15	14	anbi2d 678	. . . . . . . . 9 |- (d = D -> ((C C_ o /\ (((cls` J)` o) i^i d) = (/)) <-> (C C_ o /\ (((cls` J)` o) i^i D) = (/))))
16	15	rexbidv 2124	. . . . . . . 8 |- (d = D -> (E.o e. J (C C_ o /\ (((cls` J)` o) i^i d) = (/)) <-> E.o e. J (C C_ o /\ (((cls` J)` o) i^i D) = (/))))
17	12, 16	imbi12d 688	. . . . . . 7 |- (d = D -> (((C i^i d) = (/) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i d) = (/))) <-> ((C i^i D) = (/) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i D) = (/)))))
18	10, 17	rcla42v 2384	. . . . . 6 |- ((C e. (Clsd` J) /\ D e. (Clsd` J)) -> (A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (c C_ o /\ (((cls` J)` o) i^i d) = (/))) -> ((C i^i D) = (/) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i D) = (/)))))
19	18	com12 14	. . . . 5 |- (A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (c C_ o /\ (((cls` J)` o) i^i d) = (/))) -> ((C e. (Clsd` J) /\ D e. (Clsd` J)) -> ((C i^i D) = (/) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i D) = (/)))))
20	19	exp3a 405	. . . 4 |- (A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (c C_ o /\ (((cls` J)` o) i^i d) = (/))) -> (C e. (Clsd` J) -> (D e. (Clsd` J) -> ((C i^i D) = (/) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i D) = (/))))))
21	20	3impd 1082	. . 3 |- (A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (c C_ o /\ (((cls` J)` o) i^i d) = (/))) -> ((C e. (Clsd` J) /\ D e. (Clsd` J) /\ (C i^i D) = (/)) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i D) = (/))))
22	4, 21	syl 12	. 2 |- (J e. Nrm -> ((C e. (Clsd` J) /\ D e. (Clsd` J) /\ (C i^i D) = (/)) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i D) = (/))))
23	22	imp 377	1 |- ((J e. Nrm /\ (C e. (Clsd` J) /\ D e. (Clsd` J) /\ (C i^i D) = (/))) -> E.o e. J (C C_ o /\ (((cls` J)` o) i^i D) = (/)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875  ` cfv 3998  Topctop 8857  Clsdccld 8936  clsccl 8938  Nrmcnrm 15534

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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-ral 2109  df-rex 2110  df-rab 2112  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-int 3215  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-fv 4014  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941  df-nrm 15537

Copyright terms: Public domain