Theorem mrelatlub 18613

Description: Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) See mrelatlubALT 49793 for an alternate proof.

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 2763	. 2 ⊢ (le‘𝐼) = (le‘𝐼)
2		mreclat.i	. . . 4 ⊢ 𝐼 = (toInc‘𝐶)
3	2	ipobas 18582	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼))
4	3	adantr 485	. 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 18587	. . 3 ⊢ 𝐼 ∈ Poset
8	7	a1i 11	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝐼 ∈ Poset)
9		simpr 489	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝑈 ⊆ 𝐶)
10		uniss 4880	. . . . 5 ⊢ (𝑈 ⊆ 𝐶 → ∪ 𝑈 ⊆ ∪ 𝐶)
11	10	adantl 486	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ ∪ 𝐶)
12		mreuni 17647	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
13	12	adantr 485	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝐶 = 𝑋)
14	11, 13	sseqtrd 3973	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ 𝑋)
15		mrelatlub.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
16	15	mrccl 17662	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘∪ 𝑈) ∈ 𝐶)
17	14, 16	syldan 602	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐹‘∪ 𝑈) ∈ 𝐶)
18		elssuni 4904	. . . 4 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ⊆ ∪ 𝑈)
19	15	mrcssid 17668	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
20	14, 19	syldan 602	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
21	18, 20	sylan9ssr 3951	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ (𝐹‘∪ 𝑈))
22		simpll 778	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
23	9	sselda 3937	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
24	17	adantr 485	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝐹‘∪ 𝑈) ∈ 𝐶)
25	2, 1	ipole 18585	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝐶 ∧ (𝐹‘∪ 𝑈) ∈ 𝐶) → (𝑥(le‘𝐼)(𝐹‘∪ 𝑈) ↔ 𝑥 ⊆ (𝐹‘∪ 𝑈)))
26	22, 23, 24, 25	syl3anc 1398	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)(𝐹‘∪ 𝑈) ↔ 𝑥 ⊆ (𝐹‘∪ 𝑈)))
27	21, 26	mpbird 260	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥(le‘𝐼)(𝐹‘∪ 𝑈))
28		simp1l 1216	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → 𝐶 ∈ (Moore‘𝑋))
29		simplll 786	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
30		simplr 780	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) → 𝑈 ⊆ 𝐶)
31	30	sselda 3937	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
32		simplr 780	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑦 ∈ 𝐶)
33	2, 1	ipole 18585	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥(le‘𝐼)𝑦 ↔ 𝑥 ⊆ 𝑦))
34	29, 31, 32, 33	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)𝑦 ↔ 𝑥 ⊆ 𝑦))
35	34	biimpd 232	. . . . . . 7 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)𝑦 → 𝑥 ⊆ 𝑦))
36	35	ralimdva 3177	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦 → ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦))
37	36	3impia 1135	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦)
38		unissb 4906	. . . . 5 ⊢ (∪ 𝑈 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦)
39	37, 38	sylibr 237	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → ∪ 𝑈 ⊆ 𝑦)
40		simp2 1155	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → 𝑦 ∈ 𝐶)
41	15	mrcsscl 17671	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐶) → (𝐹‘∪ 𝑈) ⊆ 𝑦)
42	28, 39, 40, 41	syl3anc 1398	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → (𝐹‘∪ 𝑈) ⊆ 𝑦)
43	17	3ad2ant1 1151	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → (𝐹‘∪ 𝑈) ∈ 𝐶)
44	2, 1	ipole 18585	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘∪ 𝑈) ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘∪ 𝑈)(le‘𝐼)𝑦 ↔ (𝐹‘∪ 𝑈) ⊆ 𝑦))
45	28, 43, 40, 44	syl3anc 1398	. . 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 260	. 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 18463	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905  ∪ cuni 4872   class class class wbr 5109  ‘cfv 6536  Basecbs 17264  lecple 17312  Moorecmre 17629  mrClscmrc 17630  Posetcpo 18358  lubclub 18360  toInccipo 18578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-struct 17202  df-slot 17237  df-ndx 17249  df-base 17265  df-tset 17324  df-ple 17325  df-ocomp 17326  df-mre 17633  df-mrc 17634  df-proset 18345  df-poset 18364  df-lub 18395  df-ipo 18579

This theorem is referenced by:  mreclatBAD  18614