mreclatBAD - Metamath Proof Explorer

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Theorem mreclatBAD 18512

Description: A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7361 update): Reprove using isclat 18449 instead of the isclatBAD. hypothesis. See commented-out mreclat above. See mreclat 47575 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 18485	. . 3 ⊢ 𝐼 ∈ Poset
3	2	a1i 11	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ Poset)
4		eqid 2732	. . . . . . . 8 ⊢ (mrCls‘𝐶) = (mrCls‘𝐶)
5		eqid 2732	. . . . . . . 8 ⊢ (lub‘𝐼) = (lub‘𝐼)
6	1, 4, 5	mrelatlub 18511	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((lub‘𝐼)‘𝑥) = ((mrCls‘𝐶)‘∪ 𝑥))
7		uniss 4915	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐶 → ∪ 𝑥 ⊆ ∪ 𝐶)
8	7	adantl 482	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ∪ 𝑥 ⊆ ∪ 𝐶)
9		mreuni 17540	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
10	9	adantr 481	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ∪ 𝐶 = 𝑋)
11	8, 10	sseqtrd 4021	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ∪ 𝑥 ⊆ 𝑋)
12	4	mrccl 17551	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑥 ⊆ 𝑋) → ((mrCls‘𝐶)‘∪ 𝑥) ∈ 𝐶)
13	11, 12	syldan 591	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((mrCls‘𝐶)‘∪ 𝑥) ∈ 𝐶)
14	6, 13	eqeltrd 2833	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((lub‘𝐼)‘𝑥) ∈ 𝐶)
15		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((glb‘𝐼)‘𝑥) = ((glb‘𝐼)‘∅))
16	15	adantl 482	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘𝑥) = ((glb‘𝐼)‘∅))
17		eqid 2732	. . . . . . . . . . 11 ⊢ (glb‘𝐼) = (glb‘𝐼)
18	1, 17	mrelatglb0 18510	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((glb‘𝐼)‘∅) = 𝑋)
19	18	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘∅) = 𝑋)
20	16, 19	eqtrd 2772	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘𝑥) = 𝑋)
21		mre1cl 17534	. . . . . . . . 9 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
22	21	ad2antrr 724	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → 𝑋 ∈ 𝐶)
23	20, 22	eqeltrd 2833	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘𝑥) ∈ 𝐶)
24	1, 17	mrelatglb 18509	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ((glb‘𝐼)‘𝑥) = ∩ 𝑥)
25		mreintcl 17535	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝐶)
26	24, 25	eqeltrd 2833	. . . . . . . 8 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ((glb‘𝐼)‘𝑥) ∈ 𝐶)
27	26	3expa 1118	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 ≠ ∅) → ((glb‘𝐼)‘𝑥) ∈ 𝐶)
28	23, 27	pm2.61dane 3029	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((glb‘𝐼)‘𝑥) ∈ 𝐶)
29	14, 28	jca 512	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶))
30	29	ex 413	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ⊆ 𝐶 → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶)))
31	1	ipobas 18480	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼))
32		sseq2 4007	. . . . . 6 ⊢ (𝐶 = (Base‘𝐼) → (𝑥 ⊆ 𝐶 ↔ 𝑥 ⊆ (Base‘𝐼)))
33		eleq2 2822	. . . . . . 7 ⊢ (𝐶 = (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ↔ ((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼)))
34		eleq2 2822	. . . . . . 7 ⊢ (𝐶 = (Base‘𝐼) → (((glb‘𝐼)‘𝑥) ∈ 𝐶 ↔ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))
35	33, 34	anbi12d 631	. . . . . 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 344	. . . . 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 1930	. 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 583	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3947 ∅c0 4321 ∪ cuni 4907 ∩ cint 4949 ‘cfv 6540 Basecbs 17140 Moorecmre 17522 mrClscmrc 17523 Posetcpo 18256 lubclub 18258 glbcglb 18259 CLatccla 18447 toInccipo 18476

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-tset 17212 df-ple 17213 df-ocomp 17214 df-mre 17526 df-mrc 17527 df-odu 18236 df-proset 18244 df-poset 18262 df-lub 18295 df-glb 18296 df-ipo 18477

This theorem is referenced by: (None)

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