Theorem topdlat 48978

Description: A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)

Hypothesis

Ref	Expression
topdlat.i	⊢ 𝐼 = (toInc‘𝐽)

Assertion

Ref	Expression
topdlat	⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat)

Proof of Theorem topdlat

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topdlat.i	. . . 4 ⊢ 𝐼 = (toInc‘𝐽)
2	1	topclat 48972	. . 3 ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat)
3		clatl 18518	. . 3 ⊢ (𝐼 ∈ CLat → 𝐼 ∈ Lat)
4	2, 3	syl 17	. 2 ⊢ (𝐽 ∈ Top → 𝐼 ∈ Lat)
5		simpl 482	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝐽 ∈ Top)
6		simpr2 1196	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑦 ∈ (Base‘𝐼))
7	1	ipobas 18541	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 = (Base‘𝐼))
8	5, 7	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝐽 = (Base‘𝐼))
9	6, 8	eleqtrrd 2837	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑦 ∈ 𝐽)
10		simpr3 1197	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑧 ∈ (Base‘𝐼))
11	10, 8	eleqtrrd 2837	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑧 ∈ 𝐽)
12		eqid 2735	. . . . . 6 ⊢ (join‘𝐼) = (join‘𝐼)
13	1, 5, 9, 11, 12	toplatjoin 48976	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑦(join‘𝐼)𝑧) = (𝑦 ∪ 𝑧))
14	13	oveq2d 7421	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)(𝑦(join‘𝐼)𝑧)) = (𝑥(meet‘𝐼)(𝑦 ∪ 𝑧)))
15		simpr1 1195	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑥 ∈ (Base‘𝐼))
16	15, 8	eleqtrrd 2837	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑥 ∈ 𝐽)
17		unopn 22841	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑦 ∪ 𝑧) ∈ 𝐽)
18	5, 9, 11, 17	syl3anc 1373	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑦 ∪ 𝑧) ∈ 𝐽)
19		eqid 2735	. . . . 5 ⊢ (meet‘𝐼) = (meet‘𝐼)
20	1, 5, 16, 18, 19	toplatmeet 48977	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)(𝑦 ∪ 𝑧)) = (𝑥 ∩ (𝑦 ∪ 𝑧)))
21		inopn 22837	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∩ 𝑦) ∈ 𝐽)
22	5, 16, 9, 21	syl3anc 1373	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ 𝑦) ∈ 𝐽)
23		inopn 22837	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑥 ∩ 𝑧) ∈ 𝐽)
24	5, 16, 11, 23	syl3anc 1373	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ 𝑧) ∈ 𝐽)
25	1, 5, 22, 24, 12	toplatjoin 48976	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → ((𝑥 ∩ 𝑦)(join‘𝐼)(𝑥 ∩ 𝑧)) = ((𝑥 ∩ 𝑦) ∪ (𝑥 ∩ 𝑧)))
26	1, 5, 16, 9, 19	toplatmeet 48977	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)𝑦) = (𝑥 ∩ 𝑦))
27	1, 5, 16, 11, 19	toplatmeet 48977	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)𝑧) = (𝑥 ∩ 𝑧))
28	26, 27	oveq12d 7423	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → ((𝑥(meet‘𝐼)𝑦)(join‘𝐼)(𝑥(meet‘𝐼)𝑧)) = ((𝑥 ∩ 𝑦)(join‘𝐼)(𝑥 ∩ 𝑧)))
29		indi 4259	. . . . . 6 ⊢ (𝑥 ∩ (𝑦 ∪ 𝑧)) = ((𝑥 ∩ 𝑦) ∪ (𝑥 ∩ 𝑧))
30	29	a1i 11	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ (𝑦 ∪ 𝑧)) = ((𝑥 ∩ 𝑦) ∪ (𝑥 ∩ 𝑧)))
31	25, 28, 30	3eqtr4rd 2781	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ (𝑦 ∪ 𝑧)) = ((𝑥(meet‘𝐼)𝑦)(join‘𝐼)(𝑥(meet‘𝐼)𝑧)))
32	14, 20, 31	3eqtrd 2774	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)(𝑦(join‘𝐼)𝑧)) = ((𝑥(meet‘𝐼)𝑦)(join‘𝐼)(𝑥(meet‘𝐼)𝑧)))
33	32	ralrimivvva 3190	. 2 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)∀𝑧 ∈ (Base‘𝐼)(𝑥(meet‘𝐼)(𝑦(join‘𝐼)𝑧)) = ((𝑥(meet‘𝐼)𝑦)(join‘𝐼)(𝑥(meet‘𝐼)𝑧)))
34		eqid 2735	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
35	34, 12, 19	isdlat 18532	. 2 ⊢ (𝐼 ∈ DLat ↔ (𝐼 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐼)∀𝑦 ∈ (Base‘𝐼)∀𝑧 ∈ (Base‘𝐼)(𝑥(meet‘𝐼)(𝑦(join‘𝐼)𝑧)) = ((𝑥(meet‘𝐼)𝑦)(join‘𝐼)(𝑥(meet‘𝐼)𝑧))))
36	4, 33, 35	sylanbrc 583	1 ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ∪ cun 3924   ∩ cin 3925  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  joincjn 18323  meetcmee 18324  Latclat 18441  CLatccla 18508  DLatcdlat 18530  toInccipo 18537  Topctop 22831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-tset 17290  df-ple 17291  df-ocomp 17292  df-odu 18299  df-proset 18306  df-poset 18325  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-lat 18442  df-clat 18509  df-dlat 18531  df-ipo 18538  df-top 22832

This theorem is referenced by: (None)