Theorem mreclatdemoBAD 20096

Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 16418. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6263 update): This proof uses the old df-clat 16339 and references the required instance of mreclatBAD 16418 as a hypothesis. When mreclatBAD 16418 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below.

Hypothesis

Ref	Expression
mreclatBAD.	|- ( ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) e. (Moore ` U. ( TopOpen ` W ) ) -> (toInc ` ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) ) e. CLat )

Assertion

Ref	Expression
mreclatdemoBAD	|- ( W e. ( TopSp i^i LMod ) -> (toInc ` ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) ) e. CLat )

Proof of Theorem mreclatdemoBAD

Step	Hyp	Ref	Expression
1		fvex 5887	. . . . 5 |- ( TopOpen ` W ) e. _V
2	1	uniex 6597	. . . 4 |- U. ( TopOpen ` W ) e. _V
3		mremre 15495	. . . 4 |- ( U. ( TopOpen ` W ) e. _V -> (Moore ` U. ( TopOpen ` W ) ) e. (Moore ` ~P U. ( TopOpen ` W ) ) )
4	2, 3	mp1i 13	. . 3 |- ( W e. ( TopSp i^i LMod ) -> (Moore ` U. ( TopOpen ` W ) ) e. (Moore ` ~P U. ( TopOpen ` W ) ) )
5		inss2 3683	. . . . . 6 |- ( TopSp i^i LMod ) C_ LMod
6	5	sseli 3460	. . . . 5 |- ( W e. ( TopSp i^i LMod ) -> W e. LMod )
7		eqid 2422	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
8		eqid 2422	. . . . . 6 |- ( LSubSp ` W ) = ( LSubSp ` W )
9	7, 8	lssmre 18174	. . . . 5 |- ( W e. LMod -> ( LSubSp ` W ) e. (Moore ` ( Base ` W ) ) )
10	6, 9	syl 17	. . . 4 |- ( W e. ( TopSp i^i LMod ) -> ( LSubSp ` W ) e. (Moore ` ( Base ` W ) ) )
11		inss1 3682	. . . . . 6 |- ( TopSp i^i LMod ) C_ TopSp
12	11	sseli 3460	. . . . 5 |- ( W e. ( TopSp i^i LMod ) -> W e. TopSp )
13		eqid 2422	. . . . . . 7 |- ( TopOpen ` W ) = ( TopOpen ` W )
14	7, 13	tpsuni 19937	. . . . . 6 |- ( W e. TopSp -> ( Base ` W ) = U. ( TopOpen ` W ) )
15	14	fveq2d 5881	. . . . 5 |- ( W e. TopSp -> (Moore ` ( Base ` W ) ) = (Moore ` U. ( TopOpen ` W ) ) )
16	12, 15	syl 17	. . . 4 |- ( W e. ( TopSp i^i LMod ) -> (Moore ` ( Base ` W ) ) = (Moore ` U. ( TopOpen ` W ) ) )
17	10, 16	eleqtrd 2512	. . 3 |- ( W e. ( TopSp i^i LMod ) -> ( LSubSp ` W ) e. (Moore ` U. ( TopOpen ` W ) ) )
18	13	tpstop 19938	. . . 4 |- ( W e. TopSp -> ( TopOpen ` W ) e. Top )
19		eqid 2422	. . . . 5 |- U. ( TopOpen ` W ) = U. ( TopOpen ` W )
20	19	cldmre 20078	. . . 4 |- ( ( TopOpen ` W ) e. Top -> ( Clsd ` ( TopOpen ` W ) ) e. (Moore ` U. ( TopOpen ` W ) ) )
21	12, 18, 20	3syl 18	. . 3 |- ( W e. ( TopSp i^i LMod ) -> ( Clsd ` ( TopOpen ` W ) ) e. (Moore ` U. ( TopOpen ` W ) ) )
22		mreincl 15490	. . 3 |- ( ( (Moore ` U. ( TopOpen ` W ) ) e. (Moore ` ~P U. ( TopOpen ` W ) ) /\ ( LSubSp ` W ) e. (Moore ` U. ( TopOpen ` W ) ) /\ ( Clsd ` ( TopOpen ` W ) ) e. (Moore ` U. ( TopOpen ` W ) ) ) -> ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) e. (Moore ` U. ( TopOpen ` W ) ) )
23	4, 17, 21, 22	syl3anc 1264	. 2 |- ( W e. ( TopSp i^i LMod ) -> ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) e. (Moore ` U. ( TopOpen ` W ) ) )
24		mreclatBAD.	. 2 |- ( ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) e. (Moore ` U. ( TopOpen ` W ) ) -> (toInc ` ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) ) e. CLat )
25	23, 24	syl 17	1 |- ( W e. ( TopSp i^i LMod ) -> (toInc ` ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) ) e. CLat )

Syntax hints:   -> wi 4   = wceq 1437   e. wcel 1868   _Vcvv 3081   i^i cin 3435   ~Pcpw 3979   U.cuni 4216   ` cfv 5597   Basecbs 15106   TopOpenctopn 15305  Moorecmre 15473   CLatccla 16338  toInccipo 16382   LModclmod 18076   LSubSpclss 18140   Topctop 19901   TopSpctps 19903   Clsdccld 20015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15109  df-slot 15110  df-base 15111  df-sets 15112  df-plusg 15188  df-0g 15325  df-mre 15477  df-mgm 16473  df-sgrp 16512  df-mnd 16522  df-grp 16658  df-minusg 16659  df-sbg 16660  df-mgp 17709  df-ur 17721  df-ring 17767  df-lmod 18078  df-lss 18141  df-top 19905  df-topon 19907  df-topsp 19908  df-cld 20018

This theorem is referenced by: (None)