mreclatBAD - Metamath Proof Explorer

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Theorem mreclatBAD 18516

Description: A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7365 update): Reprove using isclat 18453 instead of the isclatBAD. hypothesis. See commented-out mreclat above. See mreclat 47622 for a good version.

**Hypotheses**
Ref	Expression
mreclat.i	⊢ 𝐼 = (toInc‘𝐶)
isclatBAD.	⊢ (𝐼 ∈ CLat ↔ (𝐼 ∈ Poset ∧ ∀𝑥(𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))))

**Assertion**
Ref	Expression
mreclatBAD	⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)

Distinct variable groups: 𝑥,𝐼 𝑥,𝐶 𝑥,𝑋

**Proof of Theorem mreclatBAD**
Step	Hyp	Ref	Expression
1		mreclat.i	. . . 4 ⊢ 𝐼 = (toInc‘𝐶)
2	1	ipopos 18489	. . 3 ⊢ 𝐼 ∈ Poset
3	2	a1i 11	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ Poset)
4		eqid 2733	. . . . . . . 8 ⊢ (mrCls‘𝐶) = (mrCls‘𝐶)
5		eqid 2733	. . . . . . . 8 ⊢ (lub‘𝐼) = (lub‘𝐼)
6	1, 4, 5	mrelatlub 18515	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((lub‘𝐼)‘𝑥) = ((mrCls‘𝐶)‘∪ 𝑥))
7		uniss 4917	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐶 → ∪ 𝑥 ⊆ ∪ 𝐶)
8	7	adantl 483	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ∪ 𝑥 ⊆ ∪ 𝐶)
9		mreuni 17544	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
10	9	adantr 482	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ∪ 𝐶 = 𝑋)
11	8, 10	sseqtrd 4023	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ∪ 𝑥 ⊆ 𝑋)
12	4	mrccl 17555	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑥 ⊆ 𝑋) → ((mrCls‘𝐶)‘∪ 𝑥) ∈ 𝐶)
13	11, 12	syldan 592	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((mrCls‘𝐶)‘∪ 𝑥) ∈ 𝐶)
14	6, 13	eqeltrd 2834	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((lub‘𝐼)‘𝑥) ∈ 𝐶)
15		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((glb‘𝐼)‘𝑥) = ((glb‘𝐼)‘∅))
16	15	adantl 483	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘𝑥) = ((glb‘𝐼)‘∅))
17		eqid 2733	. . . . . . . . . . 11 ⊢ (glb‘𝐼) = (glb‘𝐼)
18	1, 17	mrelatglb0 18514	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((glb‘𝐼)‘∅) = 𝑋)
19	18	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘∅) = 𝑋)
20	16, 19	eqtrd 2773	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘𝑥) = 𝑋)
21		mre1cl 17538	. . . . . . . . 9 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
22	21	ad2antrr 725	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → 𝑋 ∈ 𝐶)
23	20, 22	eqeltrd 2834	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘𝑥) ∈ 𝐶)
24	1, 17	mrelatglb 18513	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ((glb‘𝐼)‘𝑥) = ∩ 𝑥)
25		mreintcl 17539	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝐶)
26	24, 25	eqeltrd 2834	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ((glb‘𝐼)‘𝑥) ∈ 𝐶)
27	26	3expa 1119	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 ≠ ∅) → ((glb‘𝐼)‘𝑥) ∈ 𝐶)
28	23, 27	pm2.61dane 3030	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((glb‘𝐼)‘𝑥) ∈ 𝐶)
29	14, 28	jca 513	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶))
30	29	ex 414	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ⊆ 𝐶 → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶)))
31	1	ipobas 18484	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼))
32		sseq2 4009	. . . . . 6 ⊢ (𝐶 = (Base‘𝐼) → (𝑥 ⊆ 𝐶 ↔ 𝑥 ⊆ (Base‘𝐼)))
33		eleq2 2823	. . . . . . 7 ⊢ (𝐶 = (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ↔ ((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼)))
34		eleq2 2823	. . . . . . 7 ⊢ (𝐶 = (Base‘𝐼) → (((glb‘𝐼)‘𝑥) ∈ 𝐶 ↔ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))
35	33, 34	anbi12d 632	. . . . . 6 ⊢ (𝐶 = (Base‘𝐼) → ((((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶) ↔ (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼))))
36	32, 35	imbi12d 345	. . . . 5 ⊢ (𝐶 = (Base‘𝐼) → ((𝑥 ⊆ 𝐶 → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶)) ↔ (𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))))
37	31, 36	syl 17	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((𝑥 ⊆ 𝐶 → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶)) ↔ (𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))))
38	30, 37	mpbid 231	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼))))
39	38	alrimiv 1931	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∀𝑥(𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼))))
40		isclatBAD.	. 2 ⊢ (𝐼 ∈ CLat ↔ (𝐼 ∈ Poset ∧ ∀𝑥(𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))))
41	3, 39, 40	sylanbrc 584	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ⊆ wss 3949 ∅c0 4323 ∪ cuni 4909 ∩ cint 4951 ‘cfv 6544 Basecbs 17144 Moorecmre 17526 mrClscmrc 17527 Posetcpo 18260 lubclub 18262 glbcglb 18263 CLatccla 18451 toInccipo 18480

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-tset 17216 df-ple 17217 df-ocomp 17218 df-mre 17530 df-mrc 17531 df-odu 18240 df-proset 18248 df-poset 18266 df-lub 18299 df-glb 18300 df-ipo 18481

This theorem is referenced by: (None)

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