Theorem mrelatlub 17546

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 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))

Proof of Theorem mrelatlub

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

Step	Hyp	Ref	Expression
1		eqid 2825	. 2 ⊢ (le‘𝐼) = (le‘𝐼)
2		mreclat.i	. . . 4 ⊢ 𝐼 = (toInc‘𝐶)
3	2	ipobas 17515	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼))
4	3	adantr 474	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝐶 = (Base‘𝐼))
5		mrelatlub.l	. . 3 ⊢ 𝐿 = (lub‘𝐼)
6	5	a1i 11	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝐿 = (lub‘𝐼))
7	2	ipopos 17520	. . 3 ⊢ 𝐼 ∈ Poset
8	7	a1i 11	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝐼 ∈ Poset)
9		simpr 479	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝑈 ⊆ 𝐶)
10		uniss 4683	. . . . 5 ⊢ (𝑈 ⊆ 𝐶 → ∪ 𝑈 ⊆ ∪ 𝐶)
11	10	adantl 475	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ ∪ 𝐶)
12		mreuni 16620	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
13	12	adantr 474	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝐶 = 𝑋)
14	11, 13	sseqtrd 3866	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ 𝑋)
15		mrelatlub.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
16	15	mrccl 16631	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘∪ 𝑈) ∈ 𝐶)
17	14, 16	syldan 585	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐹‘∪ 𝑈) ∈ 𝐶)
18		elssuni 4691	. . . 4 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ⊆ ∪ 𝑈)
19	15	mrcssid 16637	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
20	14, 19	syldan 585	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
21	18, 20	sylan9ssr 3841	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ (𝐹‘∪ 𝑈))
22		simpll 783	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
23	9	sselda 3827	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
24	17	adantr 474	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝐹‘∪ 𝑈) ∈ 𝐶)
25	2, 1	ipole 17518	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝐶 ∧ (𝐹‘∪ 𝑈) ∈ 𝐶) → (𝑥(le‘𝐼)(𝐹‘∪ 𝑈) ↔ 𝑥 ⊆ (𝐹‘∪ 𝑈)))
26	22, 23, 24, 25	syl3anc 1494	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)(𝐹‘∪ 𝑈) ↔ 𝑥 ⊆ (𝐹‘∪ 𝑈)))
27	21, 26	mpbird 249	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥(le‘𝐼)(𝐹‘∪ 𝑈))
28		simp1l 1258	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → 𝐶 ∈ (Moore‘𝑋))
29		simplll 791	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
30		simplr 785	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) → 𝑈 ⊆ 𝐶)
31	30	sselda 3827	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
32		simplr 785	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑦 ∈ 𝐶)
33	2, 1	ipole 17518	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥(le‘𝐼)𝑦 ↔ 𝑥 ⊆ 𝑦))
34	29, 31, 32, 33	syl3anc 1494	. . . . . . . 8 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)𝑦 ↔ 𝑥 ⊆ 𝑦))
35	34	biimpd 221	. . . . . . 7 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)𝑦 → 𝑥 ⊆ 𝑦))
36	35	ralimdva 3171	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦 → ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦))
37	36	3impia 1149	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦)
38		unissb 4693	. . . . 5 ⊢ (∪ 𝑈 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦)
39	37, 38	sylibr 226	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → ∪ 𝑈 ⊆ 𝑦)
40		simp2 1171	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → 𝑦 ∈ 𝐶)
41	15	mrcsscl 16640	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐶) → (𝐹‘∪ 𝑈) ⊆ 𝑦)
42	28, 39, 40, 41	syl3anc 1494	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → (𝐹‘∪ 𝑈) ⊆ 𝑦)
43	17	3ad2ant1 1167	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → (𝐹‘∪ 𝑈) ∈ 𝐶)
44	2, 1	ipole 17518	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘∪ 𝑈) ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘∪ 𝑈)(le‘𝐼)𝑦 ↔ (𝐹‘∪ 𝑈) ⊆ 𝑦))
45	28, 43, 40, 44	syl3anc 1494	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → ((𝐹‘∪ 𝑈)(le‘𝐼)𝑦 ↔ (𝐹‘∪ 𝑈) ⊆ 𝑦))
46	42, 45	mpbird 249	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → (𝐹‘∪ 𝑈)(le‘𝐼)𝑦)
47	1, 4, 6, 8, 9, 17, 27, 46	poslubdg 17509	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117   ⊆ wss 3798  ∪ cuni 4660   class class class wbr 4875  ‘cfv 6127  Basecbs 16229  lecple 16319  Moorecmre 16602  mrClscmrc 16603  Posetcpo 17300  lubclub 17302  toInccipo 17511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-5 11424  df-6 11425  df-7 11426  df-8 11427  df-9 11428  df-n0 11626  df-z 11712  df-dec 11829  df-uz 11976  df-fz 12627  df-struct 16231  df-ndx 16232  df-slot 16233  df-base 16235  df-tset 16331  df-ple 16332  df-ocomp 16333  df-mre 16606  df-mrc 16607  df-proset 17288  df-poset 17306  df-lub 17334  df-ipo 17512

This theorem is referenced by:  mreclatBAD  17547