Theorem mrelatlub 18468

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 49105 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 2731	. 2 ⊢ (le‘𝐼) = (le‘𝐼)
2		mreclat.i	. . . 4 ⊢ 𝐼 = (toInc‘𝐶)
3	2	ipobas 18437	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼))
4	3	adantr 480	. 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 18442	. . 3 ⊢ 𝐼 ∈ Poset
8	7	a1i 11	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝐼 ∈ Poset)
9		simpr 484	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝑈 ⊆ 𝐶)
10		uniss 4864	. . . . 5 ⊢ (𝑈 ⊆ 𝐶 → ∪ 𝑈 ⊆ ∪ 𝐶)
11	10	adantl 481	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ ∪ 𝐶)
12		mreuni 17502	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
13	12	adantr 480	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝐶 = 𝑋)
14	11, 13	sseqtrd 3966	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ 𝑋)
15		mrelatlub.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
16	15	mrccl 17517	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘∪ 𝑈) ∈ 𝐶)
17	14, 16	syldan 591	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐹‘∪ 𝑈) ∈ 𝐶)
18		elssuni 4887	. . . 4 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ⊆ ∪ 𝑈)
19	15	mrcssid 17523	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
20	14, 19	syldan 591	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
21	18, 20	sylan9ssr 3944	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ (𝐹‘∪ 𝑈))
22		simpll 766	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
23	9	sselda 3929	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
24	17	adantr 480	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝐹‘∪ 𝑈) ∈ 𝐶)
25	2, 1	ipole 18440	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝐶 ∧ (𝐹‘∪ 𝑈) ∈ 𝐶) → (𝑥(le‘𝐼)(𝐹‘∪ 𝑈) ↔ 𝑥 ⊆ (𝐹‘∪ 𝑈)))
26	22, 23, 24, 25	syl3anc 1373	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)(𝐹‘∪ 𝑈) ↔ 𝑥 ⊆ (𝐹‘∪ 𝑈)))
27	21, 26	mpbird 257	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥(le‘𝐼)(𝐹‘∪ 𝑈))
28		simp1l 1198	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → 𝐶 ∈ (Moore‘𝑋))
29		simplll 774	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
30		simplr 768	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) → 𝑈 ⊆ 𝐶)
31	30	sselda 3929	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
32		simplr 768	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑦 ∈ 𝐶)
33	2, 1	ipole 18440	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥(le‘𝐼)𝑦 ↔ 𝑥 ⊆ 𝑦))
34	29, 31, 32, 33	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)𝑦 ↔ 𝑥 ⊆ 𝑦))
35	34	biimpd 229	. . . . . . 7 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)𝑦 → 𝑥 ⊆ 𝑦))
36	35	ralimdva 3144	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦 → ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦))
37	36	3impia 1117	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦)
38		unissb 4889	. . . . 5 ⊢ (∪ 𝑈 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦)
39	37, 38	sylibr 234	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → ∪ 𝑈 ⊆ 𝑦)
40		simp2 1137	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → 𝑦 ∈ 𝐶)
41	15	mrcsscl 17526	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐶) → (𝐹‘∪ 𝑈) ⊆ 𝑦)
42	28, 39, 40, 41	syl3anc 1373	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → (𝐹‘∪ 𝑈) ⊆ 𝑦)
43	17	3ad2ant1 1133	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → (𝐹‘∪ 𝑈) ∈ 𝐶)
44	2, 1	ipole 18440	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘∪ 𝑈) ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘∪ 𝑈)(le‘𝐼)𝑦 ↔ (𝐹‘∪ 𝑈) ⊆ 𝑦))
45	28, 43, 40, 44	syl3anc 1373	. . 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 257	. 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 18318	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897  ∪ cuni 4856   class class class wbr 5089  ‘cfv 6481  Basecbs 17120  lecple 17168  Moorecmre 17484  mrClscmrc 17485  Posetcpo 18213  lubclub 18215  toInccipo 18433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-tset 17180  df-ple 17181  df-ocomp 17182  df-mre 17488  df-mrc 17489  df-proset 18200  df-poset 18219  df-lub 18250  df-ipo 18434

This theorem is referenced by:  mreclatBAD  18469