Theorem mrelatlub 18594

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 49616 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 2762	. 2 ⊢ (le‘𝐼) = (le‘𝐼)
2		mreclat.i	. . . 4 ⊢ 𝐼 = (toInc‘𝐶)
3	2	ipobas 18563	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼))
4	3	adantr 484	. 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 18568	. . 3 ⊢ 𝐼 ∈ Poset
8	7	a1i 11	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝐼 ∈ Poset)
9		simpr 488	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → 𝑈 ⊆ 𝐶)
10		uniss 4873	. . . . 5 ⊢ (𝑈 ⊆ 𝐶 → ∪ 𝑈 ⊆ ∪ 𝐶)
11	10	adantl 485	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ ∪ 𝐶)
12		mreuni 17628	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
13	12	adantr 484	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝐶 = 𝑋)
14	11, 13	sseqtrd 3972	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ 𝑋)
15		mrelatlub.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
16	15	mrccl 17643	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → (𝐹‘∪ 𝑈) ∈ 𝐶)
17	14, 16	syldan 600	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐹‘∪ 𝑈) ∈ 𝐶)
18		elssuni 4897	. . . 4 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ⊆ ∪ 𝑈)
19	15	mrcssid 17649	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑋) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
20	14, 19	syldan 600	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → ∪ 𝑈 ⊆ (𝐹‘∪ 𝑈))
21	18, 20	sylan9ssr 3950	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ (𝐹‘∪ 𝑈))
22		simpll 776	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
23	9	sselda 3936	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
24	17	adantr 484	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝐹‘∪ 𝑈) ∈ 𝐶)
25	2, 1	ipole 18566	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝐶 ∧ (𝐹‘∪ 𝑈) ∈ 𝐶) → (𝑥(le‘𝐼)(𝐹‘∪ 𝑈) ↔ 𝑥 ⊆ (𝐹‘∪ 𝑈)))
26	22, 23, 24, 25	syl3anc 1390	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)(𝐹‘∪ 𝑈) ↔ 𝑥 ⊆ (𝐹‘∪ 𝑈)))
27	21, 26	mpbird 259	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥(le‘𝐼)(𝐹‘∪ 𝑈))
28		simp1l 1211	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → 𝐶 ∈ (Moore‘𝑋))
29		simplll 784	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (Moore‘𝑋))
30		simplr 778	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) → 𝑈 ⊆ 𝐶)
31	30	sselda 3936	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝐶)
32		simplr 778	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → 𝑦 ∈ 𝐶)
33	2, 1	ipole 18566	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥(le‘𝐼)𝑦 ↔ 𝑥 ⊆ 𝑦))
34	29, 31, 32, 33	syl3anc 1390	. . . . . . . 8 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝑈) → (𝑥(le‘𝐼)𝑦 ↔ 𝑥 ⊆ 𝑦))
35	34	biimpd 231	. . . . . . 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 1130	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦)
38		unissb 4899	. . . . 5 ⊢ (∪ 𝑈 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝑈 𝑥 ⊆ 𝑦)
39	37, 38	sylibr 236	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → ∪ 𝑈 ⊆ 𝑦)
40		simp2 1150	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → 𝑦 ∈ 𝐶)
41	15	mrcsscl 17652	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑈 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐶) → (𝐹‘∪ 𝑈) ⊆ 𝑦)
42	28, 39, 40, 41	syl3anc 1390	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → (𝐹‘∪ 𝑈) ⊆ 𝑦)
43	17	3ad2ant1 1146	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) ∧ 𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝑈 𝑥(le‘𝐼)𝑦) → (𝐹‘∪ 𝑈) ∈ 𝐶)
44	2, 1	ipole 18566	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘∪ 𝑈) ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘∪ 𝑈)(le‘𝐼)𝑦 ↔ (𝐹‘∪ 𝑈) ⊆ 𝑦))
45	28, 43, 40, 44	syl3anc 1390	. . 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 259	. 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 18444	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ⊆ wss 3904  ∪ cuni 4865   class class class wbr 5100  ‘cfv 6521  Basecbs 17245  lecple 17293  Moorecmre 17610  mrClscmrc 17611  Posetcpo 18339  lubclub 18341  toInccipo 18559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-fz 13513  df-struct 17183  df-slot 17218  df-ndx 17230  df-base 17246  df-tset 17305  df-ple 17306  df-ocomp 17307  df-mre 17614  df-mrc 17615  df-proset 18326  df-poset 18345  df-lub 18376  df-ipo 18560

This theorem is referenced by:  mreclatBAD  18595