Theorem mreclatBAD 17187

Description: A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6611 update): Reprove using isclat 17109 instead of the isclatBAD. hypothesis. See commented-out mreclat above.

Hypotheses

Ref	Expression
mreclat.i	|- I = (toInc ` C )
isclatBAD.	|- ( I e. CLat <-> ( I e. Poset /\ A. x ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) ) )

Assertion

Ref	Expression
mreclatBAD	|- ( C e. (Moore ` X ) -> I e. CLat )

Distinct variable groups:   x, I   x, C   x, X

Proof of Theorem mreclatBAD

Step	Hyp	Ref	Expression
1		mreclat.i	. . . 4 |- I = (toInc ` C )
2	1	ipopos 17160	. . 3 |- I e. Poset
3	2	a1i 11	. 2 |- ( C e. (Moore ` X ) -> I e. Poset )
4		eqid 2622	. . . . . . . 8 |- (mrCls ` C ) = (mrCls ` C )
5		eqid 2622	. . . . . . . 8 |- ( lub ` I ) = ( lub ` I )
6	1, 4, 5	mrelatlub 17186	. . . . . . 7 |- ( ( C e. (Moore ` X ) /\ x C_ C ) -> ( ( lub ` I ) ` x ) = ( (mrCls ` C ) ` U. x ) )
7		uniss 4458	. . . . . . . . . 10 |- ( x C_ C -> U. x C_ U. C )
8	7	adantl 482	. . . . . . . . 9 |- ( ( C e. (Moore ` X ) /\ x C_ C ) -> U. x C_ U. C )
9		mreuni 16260	. . . . . . . . . 10 |- ( C e. (Moore ` X ) -> U. C = X )
10	9	adantr 481	. . . . . . . . 9 |- ( ( C e. (Moore ` X ) /\ x C_ C ) -> U. C = X )
11	8, 10	sseqtrd 3641	. . . . . . . 8 |- ( ( C e. (Moore ` X ) /\ x C_ C ) -> U. x C_ X )
12	4	mrccl 16271	. . . . . . . 8 |- ( ( C e. (Moore ` X ) /\ U. x C_ X ) -> ( (mrCls ` C ) ` U. x ) e. C )
13	11, 12	syldan 487	. . . . . . 7 |- ( ( C e. (Moore ` X ) /\ x C_ C ) -> ( (mrCls ` C ) ` U. x ) e. C )
14	6, 13	eqeltrd 2701	. . . . . 6 |- ( ( C e. (Moore ` X ) /\ x C_ C ) -> ( ( lub ` I ) ` x ) e. C )
15		fveq2 6191	. . . . . . . . . 10 |- ( x = (/) -> ( ( glb ` I ) ` x ) = ( ( glb ` I ) ` (/) ) )
16	15	adantl 482	. . . . . . . . 9 |- ( ( ( C e. (Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> ( ( glb ` I ) ` x ) = ( ( glb ` I ) ` (/) ) )
17		eqid 2622	. . . . . . . . . . 11 |- ( glb ` I ) = ( glb ` I )
18	1, 17	mrelatglb0 17185	. . . . . . . . . 10 |- ( C e. (Moore ` X ) -> ( ( glb ` I ) ` (/) ) = X )
19	18	ad2antrr 762	. . . . . . . . 9 |- ( ( ( C e. (Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> ( ( glb ` I ) ` (/) ) = X )
20	16, 19	eqtrd 2656	. . . . . . . 8 |- ( ( ( C e. (Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> ( ( glb ` I ) ` x ) = X )
21		mre1cl 16254	. . . . . . . . 9 |- ( C e. (Moore ` X ) -> X e. C )
22	21	ad2antrr 762	. . . . . . . 8 |- ( ( ( C e. (Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> X e. C )
23	20, 22	eqeltrd 2701	. . . . . . 7 |- ( ( ( C e. (Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> ( ( glb ` I ) ` x ) e. C )
24	1, 17	mrelatglb 17184	. . . . . . . . 9 |- ( ( C e. (Moore ` X ) /\ x C_ C /\ x =/= (/) ) -> ( ( glb ` I ) ` x ) = |^| x )
25		mreintcl 16255	. . . . . . . . 9 |- ( ( C e. (Moore ` X ) /\ x C_ C /\ x =/= (/) ) -> |^| x e. C )
26	24, 25	eqeltrd 2701	. . . . . . . 8 |- ( ( C e. (Moore ` X ) /\ x C_ C /\ x =/= (/) ) -> ( ( glb ` I ) ` x ) e. C )
27	26	3expa 1265	. . . . . . 7 |- ( ( ( C e. (Moore ` X ) /\ x C_ C ) /\ x =/= (/) ) -> ( ( glb ` I ) ` x ) e. C )
28	23, 27	pm2.61dane 2881	. . . . . 6 |- ( ( C e. (Moore ` X ) /\ x C_ C ) -> ( ( glb ` I ) ` x ) e. C )
29	14, 28	jca 554	. . . . 5 |- ( ( C e. (Moore ` X ) /\ x C_ C ) -> ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) )
30	29	ex 450	. . . 4 |- ( C e. (Moore ` X ) -> ( x C_ C -> ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) ) )
31	1	ipobas 17155	. . . . 5 |- ( C e. (Moore ` X ) -> C = ( Base ` I ) )
32		sseq2 3627	. . . . . 6 |- ( C = ( Base ` I ) -> ( x C_ C <-> x C_ ( Base ` I ) ) )
33		eleq2 2690	. . . . . . 7 |- ( C = ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. C <-> ( ( lub ` I ) ` x ) e. ( Base ` I ) ) )
34		eleq2 2690	. . . . . . 7 |- ( C = ( Base ` I ) -> ( ( ( glb ` I ) ` x ) e. C <-> ( ( glb ` I ) ` x ) e. ( Base ` I ) ) )
35	33, 34	anbi12d 747	. . . . . 6 |- ( C = ( Base ` I ) -> ( ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) <-> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) )
36	32, 35	imbi12d 334	. . . . 5 |- ( C = ( Base ` I ) -> ( ( x C_ C -> ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) ) <-> ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) ) )
37	31, 36	syl 17	. . . 4 |- ( C e. (Moore ` X ) -> ( ( x C_ C -> ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) ) <-> ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) ) )
38	30, 37	mpbid 222	. . 3 |- ( C e. (Moore ` X ) -> ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) )
39	38	alrimiv 1855	. 2 |- ( C e. (Moore ` X ) -> A. x ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) )
40		isclatBAD.	. 2 |- ( I e. CLat <-> ( I e. Poset /\ A. x ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) ) )
41	3, 39, 40	sylanbrc 698	1 |- ( C e. (Moore ` X ) -> I e. CLat )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   U.cuni 4436   |^|cint 4475   ` cfv 5888   Basecbs 15857  Moorecmre 16242  mrClscmrc 16243   Posetcpo 16940   lubclub 16942   glbcglb 16943   CLatccla 17107  toInccipo 17151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-tset 15960  df-ple 15961  df-ocomp 15963  df-mre 16246  df-mrc 16247  df-preset 16928  df-poset 16946  df-lub 16974  df-glb 16975  df-odu 17129  df-ipo 17152

This theorem is referenced by: (None)