Theorem mreclatdemoBAD 18722

Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 15378. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6073 update): This proof uses the old df-clat 15299 and references the required instance of mreclatBAD 15378 as a hypothesis. When mreclatBAD 15378 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 5722	. . . . 5 |- ( TopOpen ` W ) e. _V
2	1	uniex 6397	. . . 4 |- U. ( TopOpen ` W ) e. _V
3		mremre 14563	. . . 4 |- ( U. ( TopOpen ` W ) e. _V -> (Moore ` U. ( TopOpen ` W ) ) e. (Moore ` ~P U. ( TopOpen ` W ) ) )
4	2, 3	mp1i 12	. . 3 |- ( W e. ( TopSp i^i LMod ) -> (Moore ` U. ( TopOpen ` W ) ) e. (Moore ` ~P U. ( TopOpen ` W ) ) )
5		inss2 3592	. . . . . 6 |- ( TopSp i^i LMod ) C_ LMod
6	5	sseli 3373	. . . . 5 |- ( W e. ( TopSp i^i LMod ) -> W e. LMod )
7		eqid 2443	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
8		eqid 2443	. . . . . 6 |- ( LSubSp ` W ) = ( LSubSp ` W )
9	7, 8	lssmre 17069	. . . . 5 |- ( W e. LMod -> ( LSubSp ` W ) e. (Moore ` ( Base ` W ) ) )
10	6, 9	syl 16	. . . 4 |- ( W e. ( TopSp i^i LMod ) -> ( LSubSp ` W ) e. (Moore ` ( Base ` W ) ) )
11		inss1 3591	. . . . . 6 |- ( TopSp i^i LMod ) C_ TopSp
12	11	sseli 3373	. . . . 5 |- ( W e. ( TopSp i^i LMod ) -> W e. TopSp )
13		eqid 2443	. . . . . . 7 |- ( TopOpen ` W ) = ( TopOpen ` W )
14	7, 13	tpsuni 18565	. . . . . 6 |- ( W e. TopSp -> ( Base ` W ) = U. ( TopOpen ` W ) )
15	14	fveq2d 5716	. . . . 5 |- ( W e. TopSp -> (Moore ` ( Base ` W ) ) = (Moore ` U. ( TopOpen ` W ) ) )
16	12, 15	syl 16	. . . 4 |- ( W e. ( TopSp i^i LMod ) -> (Moore ` ( Base ` W ) ) = (Moore ` U. ( TopOpen ` W ) ) )
17	10, 16	eleqtrd 2519	. . 3 |- ( W e. ( TopSp i^i LMod ) -> ( LSubSp ` W ) e. (Moore ` U. ( TopOpen ` W ) ) )
18	13	tpstop 18566	. . . 4 |- ( W e. TopSp -> ( TopOpen ` W ) e. Top )
19		eqid 2443	. . . . 5 |- U. ( TopOpen ` W ) = U. ( TopOpen ` W )
20	19	cldmre 18704	. . . 4 |- ( ( TopOpen ` W ) e. Top -> ( Clsd ` ( TopOpen ` W ) ) e. (Moore ` U. ( TopOpen ` W ) ) )
21	12, 18, 20	3syl 20	. . 3 |- ( W e. ( TopSp i^i LMod ) -> ( Clsd ` ( TopOpen ` W ) ) e. (Moore ` U. ( TopOpen ` W ) ) )
22		mreincl 14558	. . 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 1218	. 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 16	1 |- ( W e. ( TopSp i^i LMod ) -> (toInc ` ( ( LSubSp ` W ) i^i ( Clsd ` ( TopOpen ` W ) ) ) ) e. CLat )

Syntax hints:   -> wi 4   = wceq 1369   e. wcel 1756   _Vcvv 2993   i^i cin 3348   ~Pcpw 3881   U.cuni 4112   ` cfv 5439   Basecbs 14195   TopOpenctopn 14381  Moorecmre 14541   CLatccla 15298  toInccipo 15342   LModclmod 16970   LSubSpclss 17035   Topctop 18520   TopSpctps 18523   Clsdccld 18642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-plusg 14272  df-0g 14401  df-mre 14545  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mgp 16614  df-ur 16626  df-rng 16669  df-lmod 16972  df-lss 17036  df-top 18525  df-topon 18528  df-topsp 18529  df-cld 18645

This theorem is referenced by: (None)