Theorem nrmsep2 11616

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 (_ 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 11614	. . . 4 |- (J e. Nrm -> J e. Top)
2		isnrm2 11613	. . . . 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 (_ o /\ (((cls` J)` o) i^i d) = (/)))))
3	2	biimpd 151	. . . 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 (_ o /\ (((cls` J)` o) i^i d) = (/)))))
4	1, 3	mpcom 49	. . 3 |- (J e. Nrm -> A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (c (_ o /\ (((cls` J)` o) i^i d) = (/))))
5		ineq1 2262	. . . . . . . . 9 |- (c = C -> (c i^i d) = (C i^i d))
6	5	eqeq1d 1526	. . . . . . . 8 |- (c = C -> ((c i^i d) = (/) <-> (C i^i d) = (/)))
7		sseq1 2134	. . . . . . . . . 10 |- (c = C -> (c (_ o <-> C (_ o))
8	7	anbi1d 620	. . . . . . . . 9 |- (c = C -> ((c (_ o /\ (((cls` J)` o) i^i d) = (/)) <-> (C (_ o /\ (((cls` J)` o) i^i d) = (/))))
9	8	rexbidv 1710	. . . . . . . 8 |- (c = C -> (E.o e. J (c (_ o /\ (((cls` J)` o) i^i d) = (/)) <-> E.o e. J (C (_ o /\ (((cls` J)` o) i^i d) = (/))))
10	6, 9	imbi12d 629	. . . . . . 7 |- (c = C -> (((c i^i d) = (/) -> E.o e. J (c (_ o /\ (((cls` J)` o) i^i d) = (/))) <-> ((C i^i d) = (/) -> E.o e. J (C (_ o /\ (((cls` J)` o) i^i d) = (/)))))
11		ineq2 2263	. . . . . . . . 9 |- (d = D -> (C i^i d) = (C i^i D))
12	11	eqeq1d 1526	. . . . . . . 8 |- (d = D -> ((C i^i d) = (/) <-> (C i^i D) = (/)))
13		ineq2 2263	. . . . . . . . . . 11 |- (d = D -> (((cls` J)` o) i^i d) = (((cls` J)` o) i^i D))
14	13	eqeq1d 1526	. . . . . . . . . 10 |- (d = D -> ((((cls` J)` o) i^i d) = (/) <-> (((cls` J)` o) i^i D) = (/)))
15	14	anbi2d 619	. . . . . . . . 9 |- (d = D -> ((C (_ o /\ (((cls` J)` o) i^i d) = (/)) <-> (C (_ o /\ (((cls` J)` o) i^i D) = (/))))
16	15	rexbidv 1710	. . . . . . . 8 |- (d = D -> (E.o e. J (C (_ o /\ (((cls` J)` o) i^i d) = (/)) <-> E.o e. J (C (_ o /\ (((cls` J)` o) i^i D) = (/))))
17	12, 16	imbi12d 629	. . . . . . 7 |- (d = D -> (((C i^i d) = (/) -> E.o e. J (C (_ o /\ (((cls` J)` o) i^i d) = (/))) <-> ((C i^i D) = (/) -> E.o e. J (C (_ o /\ (((cls` J)` o) i^i D) = (/)))))
18	10, 17	rcla42v 1926	. . . . . 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 (_ o /\ (((cls` J)` o) i^i d) = (/))) -> ((C i^i D) = (/) -> E.o e. J (C (_ o /\ (((cls` J)` o) i^i D) = (/)))))
19	18	com12 11	. . . . 5 |- (A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (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 (_ o /\ (((cls` J)` o) i^i D) = (/)))))
20	19	exp3a 374	. . . 4 |- (A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (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 (_ o /\ (((cls` J)` o) i^i D) = (/))))))
21	20	3impd 853	. . 3 |- (A.c e. (Clsd` J)A.d e. (Clsd` J)((c i^i d) = (/) -> E.o e. J (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 (_ o /\ (((cls` J)` o) i^i D) = (/))))
22	4, 21	syl 10	. 2 |- (J e. Nrm -> ((C e. (Clsd` J) /\ D e. (Clsd` J) /\ (C i^i D) = (/)) -> E.o e. J (C (_ o /\ (((cls` J)` o) i^i D) = (/))))
23	22	imp 348	1 |- ((J e. Nrm /\ (C e. (Clsd` J) /\ D e. (Clsd` J) /\ (C i^i D) = (/))) -> E.o e. J (C (_ o /\ (((cls` J)` o) i^i D) = (/)))

Syntax hints:   -> wi 3   /\ wa 221   /\ w3a 781   = wceq 992   e. wcel 994  A.wral 1691  E.wrex 1692   i^i cin 2098   (_ wss 2099  (/)c0 2332  ` cfv 3263  Topctop 7800  Clsdccld 7870  clsccl 7872  Nrmcnrm 11595

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-int 2601  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-fv 3279  df-top 7804  df-cld 7873  df-ntr 7874  df-cls 7875  df-nrm 11598

Copyright terms: Public domain