Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  topdlat Structured version   Visualization version   GIF version

Theorem topdlat 46251
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.
StepHypRef Expression
1 topdlat.i . . . 4 𝐼 = (toInc‘𝐽)
21topclat 46245 . . 3 (𝐽 ∈ Top → 𝐼 ∈ CLat)
3 clatl 18216 . . 3 (𝐼 ∈ CLat → 𝐼 ∈ Lat)
42, 3syl 17 . 2 (𝐽 ∈ Top → 𝐼 ∈ Lat)
5 simpl 483 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝐽 ∈ Top)
6 simpr2 1194 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑦 ∈ (Base‘𝐼))
71ipobas 18239 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 = (Base‘𝐼))
85, 7syl 17 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝐽 = (Base‘𝐼))
96, 8eleqtrrd 2844 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑦𝐽)
10 simpr3 1195 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑧 ∈ (Base‘𝐼))
1110, 8eleqtrrd 2844 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑧𝐽)
12 eqid 2740 . . . . . 6 (join‘𝐼) = (join‘𝐼)
131, 5, 9, 11, 12toplatjoin 46249 . . . . 5 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑦(join‘𝐼)𝑧) = (𝑦𝑧))
1413oveq2d 7285 . . . 4 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)(𝑦(join‘𝐼)𝑧)) = (𝑥(meet‘𝐼)(𝑦𝑧)))
15 simpr1 1193 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑥 ∈ (Base‘𝐼))
1615, 8eleqtrrd 2844 . . . . 5 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑥𝐽)
17 unopn 22042 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑦𝐽𝑧𝐽) → (𝑦𝑧) ∈ 𝐽)
185, 9, 11, 17syl3anc 1370 . . . . 5 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑦𝑧) ∈ 𝐽)
19 eqid 2740 . . . . 5 (meet‘𝐼) = (meet‘𝐼)
201, 5, 16, 18, 19toplatmeet 46250 . . . 4 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)(𝑦𝑧)) = (𝑥 ∩ (𝑦𝑧)))
21 inopn 22038 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑦𝐽) → (𝑥𝑦) ∈ 𝐽)
225, 16, 9, 21syl3anc 1370 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥𝑦) ∈ 𝐽)
23 inopn 22038 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑧𝐽) → (𝑥𝑧) ∈ 𝐽)
245, 16, 11, 23syl3anc 1370 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥𝑧) ∈ 𝐽)
251, 5, 22, 24, 12toplatjoin 46249 . . . . 5 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → ((𝑥𝑦)(join‘𝐼)(𝑥𝑧)) = ((𝑥𝑦) ∪ (𝑥𝑧)))
261, 5, 16, 9, 19toplatmeet 46250 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)𝑦) = (𝑥𝑦))
271, 5, 16, 11, 19toplatmeet 46250 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)𝑧) = (𝑥𝑧))
2826, 27oveq12d 7287 . . . . 5 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → ((𝑥(meet‘𝐼)𝑦)(join‘𝐼)(𝑥(meet‘𝐼)𝑧)) = ((𝑥𝑦)(join‘𝐼)(𝑥𝑧)))
29 indi 4213 . . . . . 6 (𝑥 ∩ (𝑦𝑧)) = ((𝑥𝑦) ∪ (𝑥𝑧))
3029a1i 11 . . . . 5 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ (𝑦𝑧)) = ((𝑥𝑦) ∪ (𝑥𝑧)))
3125, 28, 303eqtr4rd 2791 . . . 4 ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ (𝑦𝑧)) = ((𝑥(meet‘𝐼)𝑦)(join‘𝐼)(𝑥(meet‘𝐼)𝑧)))
3214, 20, 313eqtrd 2784 . . 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)))
3332ralrimivvva 3118 . 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 2740 . . 3 (Base‘𝐼) = (Base‘𝐼)
3534, 12, 19isdlat 18230 . 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))))
364, 33, 35sylanbrc 583 1 (𝐽 ∈ Top → 𝐼 ∈ DLat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1542  wcel 2110  wral 3066  cun 3890  cin 3891  cfv 6431  (class class class)co 7269  Basecbs 16902  joincjn 18019  meetcmee 18020  Latclat 18139  CLatccla 18206  DLatcdlat 18228  toInccipo 18235  Topctop 22032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-cnex 10920  ax-resscn 10921  ax-1cn 10922  ax-icn 10923  ax-addcl 10924  ax-addrcl 10925  ax-mulcl 10926  ax-mulrcl 10927  ax-mulcom 10928  ax-addass 10929  ax-mulass 10930  ax-distr 10931  ax-i2m1 10932  ax-1ne0 10933  ax-1rid 10934  ax-rnegex 10935  ax-rrecex 10936  ax-cnre 10937  ax-pre-lttri 10938  ax-pre-lttrn 10939  ax-pre-ltadd 10940  ax-pre-mulgt0 10941
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7702  df-1st 7818  df-2nd 7819  df-frecs 8082  df-wrecs 8113  df-recs 8187  df-rdg 8226  df-1o 8282  df-er 8473  df-en 8709  df-dom 8710  df-sdom 8711  df-fin 8712  df-pnf 11004  df-mnf 11005  df-xr 11006  df-ltxr 11007  df-le 11008  df-sub 11199  df-neg 11200  df-nn 11966  df-2 12028  df-3 12029  df-4 12030  df-5 12031  df-6 12032  df-7 12033  df-8 12034  df-9 12035  df-n0 12226  df-z 12312  df-dec 12429  df-uz 12574  df-fz 13231  df-struct 16838  df-sets 16855  df-slot 16873  df-ndx 16885  df-base 16903  df-tset 16971  df-ple 16972  df-ocomp 16973  df-odu 17995  df-proset 18003  df-poset 18021  df-lub 18054  df-glb 18055  df-join 18056  df-meet 18057  df-lat 18140  df-clat 18207  df-dlat 18229  df-ipo 18236  df-top 22033
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator