Theorem mrelatlub 17186

Description: Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypotheses

Ref	Expression
mreclat.i	|- I = (toInc ` C )
mrelatlub.f	|- F = (mrCls ` C )
mrelatlub.l	|- L = ( lub ` I )

Assertion

Ref	Expression
mrelatlub	|- ( ( C e. (Moore ` X ) /\ U C_ C ) -> ( L ` U ) = ( F ` U. U ) )

Proof of Theorem mrelatlub

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( le ` I ) = ( le ` I )
2		mreclat.i	. . . 4 |- I = (toInc ` C )
3	2	ipobas 17155	. . 3 |- ( C e. (Moore ` X ) -> C = ( Base ` I ) )
4	3	adantr 481	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> C = ( Base ` I ) )
5		mrelatlub.l	. . 3 |- L = ( lub ` I )
6	5	a1i 11	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> L = ( lub ` I ) )
7	2	ipopos 17160	. . 3 |- I e. Poset
8	7	a1i 11	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> I e. Poset )
9		simpr 477	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> U C_ C )
10		uniss 4458	. . . . 5 |- ( U C_ C -> U. U C_ U. C )
11	10	adantl 482	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> U. U C_ U. C )
12		mreuni 16260	. . . . 5 |- ( C e. (Moore ` X ) -> U. C = X )
13	12	adantr 481	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> U. C = X )
14	11, 13	sseqtrd 3641	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> U. U C_ X )
15		mrelatlub.f	. . . 4 |- F = (mrCls ` C )
16	15	mrccl 16271	. . 3 |- ( ( C e. (Moore ` X ) /\ U. U C_ X ) -> ( F ` U. U ) e. C )
17	14, 16	syldan 487	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> ( F ` U. U ) e. C )
18		elssuni 4467	. . . 4 |- ( x e. U -> x C_ U. U )
19	15	mrcssid 16277	. . . . 5 |- ( ( C e. (Moore ` X ) /\ U. U C_ X ) -> U. U C_ ( F ` U. U ) )
20	14, 19	syldan 487	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> U. U C_ ( F ` U. U ) )
21	18, 20	sylan9ssr 3617	. . 3 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ x e. U ) -> x C_ ( F ` U. U ) )
22		simpll 790	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ x e. U ) -> C e. (Moore ` X ) )
23	9	sselda 3603	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ x e. U ) -> x e. C )
24	17	adantr 481	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ x e. U ) -> ( F ` U. U ) e. C )
25	2, 1	ipole 17158	. . . 4 |- ( ( C e. (Moore ` X ) /\ x e. C /\ ( F ` U. U ) e. C ) -> ( x ( le ` I ) ( F ` U. U ) <-> x C_ ( F ` U. U ) ) )
26	22, 23, 24, 25	syl3anc 1326	. . 3 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ x e. U ) -> ( x ( le ` I ) ( F ` U. U ) <-> x C_ ( F ` U. U ) ) )
27	21, 26	mpbird 247	. 2 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ x e. U ) -> x ( le ` I ) ( F ` U. U ) )
28		simp1l 1085	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C /\ A. x e. U x ( le ` I ) y ) -> C e. (Moore ` X ) )
29		simplll 798	. . . . . . . . 9 |- ( ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C ) /\ x e. U ) -> C e. (Moore ` X ) )
30		simplr 792	. . . . . . . . . 10 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C ) -> U C_ C )
31	30	sselda 3603	. . . . . . . . 9 |- ( ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C ) /\ x e. U ) -> x e. C )
32		simplr 792	. . . . . . . . 9 |- ( ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C ) /\ x e. U ) -> y e. C )
33	2, 1	ipole 17158	. . . . . . . . 9 |- ( ( C e. (Moore ` X ) /\ x e. C /\ y e. C ) -> ( x ( le ` I ) y <-> x C_ y ) )
34	29, 31, 32, 33	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C ) /\ x e. U ) -> ( x ( le ` I ) y <-> x C_ y ) )
35	34	biimpd 219	. . . . . . 7 |- ( ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C ) /\ x e. U ) -> ( x ( le ` I ) y -> x C_ y ) )
36	35	ralimdva 2962	. . . . . 6 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C ) -> ( A. x e. U x ( le ` I ) y -> A. x e. U x C_ y ) )
37	36	3impia 1261	. . . . 5 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C /\ A. x e. U x ( le ` I ) y ) -> A. x e. U x C_ y )
38		unissb 4469	. . . . 5 |- ( U. U C_ y <-> A. x e. U x C_ y )
39	37, 38	sylibr 224	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C /\ A. x e. U x ( le ` I ) y ) -> U. U C_ y )
40		simp2 1062	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C /\ A. x e. U x ( le ` I ) y ) -> y e. C )
41	15	mrcsscl 16280	. . . 4 |- ( ( C e. (Moore ` X ) /\ U. U C_ y /\ y e. C ) -> ( F ` U. U ) C_ y )
42	28, 39, 40, 41	syl3anc 1326	. . 3 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C /\ A. x e. U x ( le ` I ) y ) -> ( F ` U. U ) C_ y )
43	17	3ad2ant1 1082	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C /\ A. x e. U x ( le ` I ) y ) -> ( F ` U. U ) e. C )
44	2, 1	ipole 17158	. . . 4 |- ( ( C e. (Moore ` X ) /\ ( F ` U. U ) e. C /\ y e. C ) -> ( ( F ` U. U ) ( le ` I ) y <-> ( F ` U. U ) C_ y ) )
45	28, 43, 40, 44	syl3anc 1326	. . 3 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C /\ A. x e. U x ( le ` I ) y ) -> ( ( F ` U. U ) ( le ` I ) y <-> ( F ` U. U ) C_ y ) )
46	42, 45	mpbird 247	. 2 |- ( ( ( C e. (Moore ` X ) /\ U C_ C ) /\ y e. C /\ A. x e. U x ( le ` I ) y ) -> ( F ` U. U ) ( le ` I ) y )
47	1, 4, 6, 8, 9, 17, 27, 46	poslubdg 17149	1 |- ( ( C e. (Moore ` X ) /\ U C_ C ) -> ( L ` U ) = ( F ` U. U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   U.cuni 4436   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948  Moorecmre 16242  mrClscmrc 16243   Posetcpo 16940   lubclub 16942  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-tset 15960  df-ple 15961  df-ocomp 15963  df-mre 16246  df-mrc 16247  df-preset 16928  df-poset 16946  df-lub 16974  df-ipo 17152

This theorem is referenced by:  mreclatBAD  17187