Theorem topdlat 49495

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 49489	. . 3 ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat)
3		clatl 18469	. . 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 1197	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑦 ∈ (Base‘𝐼))
7	1	ipobas 18492	. . . . . . . 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 2840	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑦 ∈ 𝐽)
10		simpr3 1198	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑧 ∈ (Base‘𝐼))
11	10, 8	eleqtrrd 2840	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑧 ∈ 𝐽)
12		eqid 2737	. . . . . 6 ⊢ (join‘𝐼) = (join‘𝐼)
13	1, 5, 9, 11, 12	toplatjoin 49493	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑦(join‘𝐼)𝑧) = (𝑦 ∪ 𝑧))
14	13	oveq2d 7378	. . . 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 1196	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑥 ∈ (Base‘𝐼))
16	15, 8	eleqtrrd 2840	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑥 ∈ 𝐽)
17		unopn 22882	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑦 ∪ 𝑧) ∈ 𝐽)
18	5, 9, 11, 17	syl3anc 1374	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑦 ∪ 𝑧) ∈ 𝐽)
19		eqid 2737	. . . . 5 ⊢ (meet‘𝐼) = (meet‘𝐼)
20	1, 5, 16, 18, 19	toplatmeet 49494	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)(𝑦 ∪ 𝑧)) = (𝑥 ∩ (𝑦 ∪ 𝑧)))
21		inopn 22878	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∩ 𝑦) ∈ 𝐽)
22	5, 16, 9, 21	syl3anc 1374	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ 𝑦) ∈ 𝐽)
23		inopn 22878	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑥 ∩ 𝑧) ∈ 𝐽)
24	5, 16, 11, 23	syl3anc 1374	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ 𝑧) ∈ 𝐽)
25	1, 5, 22, 24, 12	toplatjoin 49493	. . . . 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 49494	. . . . . 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 49494	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)𝑧) = (𝑥 ∩ 𝑧))
28	26, 27	oveq12d 7380	. . . . 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 4225	. . . . . 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 2783	. . . 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 2776	. . 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 3184	. 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 2737	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
35	34, 12, 19	isdlat 18483	. 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 584	1 ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∪ cun 3888   ∩ cin 3889  ‘cfv 6494  (class class class)co 7362  Basecbs 17174  joincjn 18272  meetcmee 18273  Latclat 18392  CLatccla 18459  DLatcdlat 18481  toInccipo 18488  Topctop 22872

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-tset 17234  df-ple 17235  df-ocomp 17236  df-odu 18248  df-proset 18255  df-poset 18274  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-lat 18393  df-clat 18460  df-dlat 18482  df-ipo 18489  df-top 22873

This theorem is referenced by: (None)