Theorem topdlat 50112

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 50106	. . 3 ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat)
3		clatl 18682	. . 3 ⊢ (𝐼 ∈ CLat → 𝐼 ∈ Lat)
4	2, 3	syl 18	. 2 ⊢ (𝐽 ∈ Top → 𝐼 ∈ Lat)
5		simpl 488	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝐽 ∈ Top)
6		simpr2 1214	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑦 ∈ (Base‘𝐼))
7	1	ipobas 18705	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 = (Base‘𝐼))
8	5, 7	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝐽 = (Base‘𝐼))
9	6, 8	eleqtrrd 2863	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑦 ∈ 𝐽)
10		simpr3 1215	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑧 ∈ (Base‘𝐼))
11	10, 8	eleqtrrd 2863	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑧 ∈ 𝐽)
12		eqid 2760	. . . . . 6 ⊢ (join‘𝐼) = (join‘𝐼)
13	1, 5, 9, 11, 12	toplatjoin 50110	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑦(join‘𝐼)𝑧) = (𝑦 ∪ 𝑧))
14	13	oveq2d 7436	. . . 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 1213	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑥 ∈ (Base‘𝐼))
16	15, 8	eleqtrrd 2863	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → 𝑥 ∈ 𝐽)
17		unopn 23221	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑦 ∪ 𝑧) ∈ 𝐽)
18	5, 9, 11, 17	syl3anc 1398	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑦 ∪ 𝑧) ∈ 𝐽)
19		eqid 2760	. . . . 5 ⊢ (meet‘𝐼) = (meet‘𝐼)
20	1, 5, 16, 18, 19	toplatmeet 50111	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)(𝑦 ∪ 𝑧)) = (𝑥 ∩ (𝑦 ∪ 𝑧)))
21		inopn 23217	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∩ 𝑦) ∈ 𝐽)
22	5, 16, 9, 21	syl3anc 1398	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ 𝑦) ∈ 𝐽)
23		inopn 23217	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑧 ∈ 𝐽) → (𝑥 ∩ 𝑧) ∈ 𝐽)
24	5, 16, 11, 23	syl3anc 1398	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥 ∩ 𝑧) ∈ 𝐽)
25	1, 5, 22, 24, 12	toplatjoin 50110	. . . . 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 50111	. . . . . 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 50111	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ (Base‘𝐼) ∧ 𝑦 ∈ (Base‘𝐼) ∧ 𝑧 ∈ (Base‘𝐼))) → (𝑥(meet‘𝐼)𝑧) = (𝑥 ∩ 𝑧))
28	26, 27	oveq12d 7438	. . . . 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 4229	. . . . . 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 2806	. . . 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 2799	. . 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 3208	. 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 2760	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
35	34, 12, 19	isdlat 18696	. 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 595	1 ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∩ cin 4089   ∪ cun 4149  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  joincjn 18485  meetcmee 18486  Latclat 18605  CLatccla 18672  DLatcdlat 18694  toInccipo 18701  Topctop 23211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-tset 17447  df-ple 17448  df-ocomp 17449  df-odu 18461  df-proset 18468  df-poset 18487  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-lat 18606  df-clat 18673  df-dlat 18695  df-ipo 18702  df-top 23212

This theorem is used by: (None)