Theorem topdlat 49191

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 49185	. . 3 ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat)
3		clatl 18429	. . 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 18452	. . . . . . . 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 2734	. . . . . 6 ⊢ (join‘𝐼) = (join‘𝐼)
13	1, 5, 9, 11, 12	toplatjoin 49189	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑦(join‘𝐼)𝑧) = (𝑦 ∪ 𝑧))
14	13	oveq2d 7372	. . . 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 22845	. . . . . 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 2734	. . . . 5 ⊢ (meet‘𝐼) = (meet‘𝐼)
20	1, 5, 16, 18, 19	toplatmeet 49190	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)(𝑦 ∪ 𝑧)) = (𝑥 ∩ (𝑦 ∪ 𝑧)))
21		inopn 22841	. . . . . . 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 22841	. . . . . . 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 49189	. . . . 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 49190	. . . . . 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 49190	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)𝑧) = (𝑥 ∩ 𝑧))
28	26, 27	oveq12d 7374	. . . . 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 4234	. . . . . 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 2780	. . . 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 2773	. . 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 3180	. 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 2734	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
35	34, 12, 19	isdlat 18443	. 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 1541   ∈ wcel 2113  ∀wral 3049   ∪ cun 3897   ∩ cin 3898  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  joincjn 18232  meetcmee 18233  Latclat 18352  CLatccla 18419  DLatcdlat 18441  toInccipo 18448  Topctop 22835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-tset 17194  df-ple 17195  df-ocomp 17196  df-odu 18208  df-proset 18215  df-poset 18234  df-lub 18265  df-glb 18266  df-join 18267  df-meet 18268  df-lat 18353  df-clat 18420  df-dlat 18442  df-ipo 18449  df-top 22836

This theorem is referenced by: (None)