Theorem mrelatlub 18651

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 49922 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 2760	. 2 ⊢ (le‘𝐼) = (le‘𝐼)
2		mreclat.i	. . . 4 ⊢ 𝐼 = (toInc‘𝐶)
3	2	ipobas 18620	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼))
4	3	adantr 486	. 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 18625	. . 3 ⊢ 𝐼 ∈ Poset
8	7	a1i 11	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝐼 ∈ Poset)
9		simpr 490	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝑈 ⊆ 𝐶)
10		uniss 4875	. . . . 5 ⊢ (𝑈 ⊆ 𝐶 → ∪ 𝑈 ⊆ ∪ 𝐶)
11	10	adantl 487	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ ∪ 𝐶)
12		mreuni 17685	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
13	12	adantr 486	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝐶 = 𝑋)
14	11, 13	sseqtrd 3967	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ 𝑋)
15		mrelatlub.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
16	15	mrccl 17700	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘∪ 𝑈) ∈ 𝐶)
17	14, 16	syldan 603	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐹‘∪ 𝑈) ∈ 𝐶)
18		elssuni 4899	. . . 4 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ⊆ ∪ 𝑈)
19	15	mrcssid 17706	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
20	14, 19	syldan 603	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
21	18, 20	sylan9ssr 3945	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ (𝐹‘∪ 𝑈))
22		simpll 779	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
23	9	sselda 3931	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
24	17	adantr 486	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝐹‘∪ 𝑈) ∈ 𝐶)
25	2, 1	ipole 18623	. . . 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 787	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
30		simplr 781	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) → 𝑈 ⊆ 𝐶)
31	30	sselda 3931	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
32		simplr 781	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑦 ∈ 𝐶)
33	2, 1	ipole 18623	. . . . . . . . 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 3174	. . . . . 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 4901	. . . . 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 17709	. . . 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 18623	. . . 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 18501	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  ∪ cuni 4867   class class class wbr 5103  ‘cfv 6533  Basecbs 17302  lecple 17350  Moorecmre 17667  mrClscmrc 17668  Posetcpo 18396  lubclub 18398  toInccipo 18616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-er 8697  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-9 12335  df-n0 12530  df-z 12617  df-dec 12738  df-uz 12889  df-fz 13563  df-struct 17240  df-slot 17275  df-ndx 17287  df-base 17303  df-tset 17362  df-ple 17363  df-ocomp 17364  df-mre 17671  df-mrc 17672  df-proset 18383  df-poset 18402  df-lub 18433  df-ipo 18617

This theorem is used by:  mreclatBAD  18652