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Theorem 4atexlemunv 37307
Description: Lemma for 4atexlem7 37316. (Contributed by NM, 21-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))
4thatlem0.l = (le‘𝐾)
4thatlem0.j = (join‘𝐾)
4thatlem0.m = (meet‘𝐾)
4thatlem0.a 𝐴 = (Atoms‘𝐾)
4thatlem0.h 𝐻 = (LHyp‘𝐾)
4thatlem0.u 𝑈 = ((𝑃 𝑄) 𝑊)
4thatlem0.v 𝑉 = ((𝑃 𝑆) 𝑊)
Assertion
Ref Expression
4atexlemunv (𝜑𝑈𝑉)

Proof of Theorem 4atexlemunv
StepHypRef Expression
1 4thatlem.ph . . 3 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑇𝐴 ∧ (𝑈 𝑇) = (𝑉 𝑇))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))
214atexlemnslpq 37297 . 2 (𝜑 → ¬ 𝑆 (𝑃 𝑄))
314atexlemk 37288 . . . . . . 7 (𝜑𝐾 ∈ HL)
414atexlemp 37291 . . . . . . 7 (𝜑𝑃𝐴)
514atexlems 37293 . . . . . . 7 (𝜑𝑆𝐴)
6 4thatlem0.l . . . . . . . 8 = (le‘𝐾)
7 4thatlem0.j . . . . . . . 8 = (join‘𝐾)
8 4thatlem0.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatlej2 36617 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑆 (𝑃 𝑆))
103, 4, 5, 9syl3anc 1368 . . . . . 6 (𝜑𝑆 (𝑃 𝑆))
1110adantr 484 . . . . 5 ((𝜑𝑈 = 𝑉) → 𝑆 (𝑃 𝑆))
12 4thatlem0.v . . . . . . . . 9 𝑉 = ((𝑃 𝑆) 𝑊)
1314atexlemkl 37298 . . . . . . . . . 10 (𝜑𝐾 ∈ Lat)
141, 7, 84atexlempsb 37301 . . . . . . . . . 10 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
15 4thatlem0.h . . . . . . . . . . 11 𝐻 = (LHyp‘𝐾)
161, 154atexlemwb 37300 . . . . . . . . . 10 (𝜑𝑊 ∈ (Base‘𝐾))
17 eqid 2824 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
18 4thatlem0.m . . . . . . . . . . 11 = (meet‘𝐾)
1917, 6, 18latmle1 17686 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
2013, 14, 16, 19syl3anc 1368 . . . . . . . . 9 (𝜑 → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
2112, 20eqbrtrid 5087 . . . . . . . 8 (𝜑𝑉 (𝑃 𝑆))
2214atexlemkc 37299 . . . . . . . . 9 (𝜑𝐾 ∈ CvLat)
23 4thatlem0.u . . . . . . . . . 10 𝑈 = ((𝑃 𝑄) 𝑊)
241, 6, 7, 18, 8, 15, 23, 124atexlemv 37306 . . . . . . . . 9 (𝜑𝑉𝐴)
2517, 6, 18latmle2 17687 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) 𝑊)
2613, 14, 16, 25syl3anc 1368 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑆) 𝑊) 𝑊)
2712, 26eqbrtrid 5087 . . . . . . . . . 10 (𝜑𝑉 𝑊)
2814atexlempw 37290 . . . . . . . . . . 11 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2928simprd 499 . . . . . . . . . 10 (𝜑 → ¬ 𝑃 𝑊)
30 nbrne2 5072 . . . . . . . . . 10 ((𝑉 𝑊 ∧ ¬ 𝑃 𝑊) → 𝑉𝑃)
3127, 29, 30syl2anc 587 . . . . . . . . 9 (𝜑𝑉𝑃)
326, 7, 8cvlatexchb1 36575 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑉𝐴𝑆𝐴𝑃𝐴) ∧ 𝑉𝑃) → (𝑉 (𝑃 𝑆) ↔ (𝑃 𝑉) = (𝑃 𝑆)))
3322, 24, 5, 4, 31, 32syl131anc 1380 . . . . . . . 8 (𝜑 → (𝑉 (𝑃 𝑆) ↔ (𝑃 𝑉) = (𝑃 𝑆)))
3421, 33mpbid 235 . . . . . . 7 (𝜑 → (𝑃 𝑉) = (𝑃 𝑆))
3534adantr 484 . . . . . 6 ((𝜑𝑈 = 𝑉) → (𝑃 𝑉) = (𝑃 𝑆))
36 oveq2 7157 . . . . . . . 8 (𝑈 = 𝑉 → (𝑃 𝑈) = (𝑃 𝑉))
3736eqcomd 2830 . . . . . . 7 (𝑈 = 𝑉 → (𝑃 𝑉) = (𝑃 𝑈))
3814atexlemq 37292 . . . . . . . . . . 11 (𝜑𝑄𝐴)
3917, 7, 8hlatjcl 36608 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
403, 4, 38, 39syl3anc 1368 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
4117, 6, 18latmle1 17686 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
4213, 40, 16, 41syl3anc 1368 . . . . . . . . 9 (𝜑 → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
4323, 42eqbrtrid 5087 . . . . . . . 8 (𝜑𝑈 (𝑃 𝑄))
441, 6, 7, 18, 8, 15, 234atexlemu 37305 . . . . . . . . 9 (𝜑𝑈𝐴)
4517, 6, 18latmle2 17687 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
4613, 40, 16, 45syl3anc 1368 . . . . . . . . . . 11 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
4723, 46eqbrtrid 5087 . . . . . . . . . 10 (𝜑𝑈 𝑊)
48 nbrne2 5072 . . . . . . . . . 10 ((𝑈 𝑊 ∧ ¬ 𝑃 𝑊) → 𝑈𝑃)
4947, 29, 48syl2anc 587 . . . . . . . . 9 (𝜑𝑈𝑃)
506, 7, 8cvlatexchb1 36575 . . . . . . . . 9 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑄𝐴𝑃𝐴) ∧ 𝑈𝑃) → (𝑈 (𝑃 𝑄) ↔ (𝑃 𝑈) = (𝑃 𝑄)))
5122, 44, 38, 4, 49, 50syl131anc 1380 . . . . . . . 8 (𝜑 → (𝑈 (𝑃 𝑄) ↔ (𝑃 𝑈) = (𝑃 𝑄)))
5243, 51mpbid 235 . . . . . . 7 (𝜑 → (𝑃 𝑈) = (𝑃 𝑄))
5337, 52sylan9eqr 2881 . . . . . 6 ((𝜑𝑈 = 𝑉) → (𝑃 𝑉) = (𝑃 𝑄))
5435, 53eqtr3d 2861 . . . . 5 ((𝜑𝑈 = 𝑉) → (𝑃 𝑆) = (𝑃 𝑄))
5511, 54breqtrd 5078 . . . 4 ((𝜑𝑈 = 𝑉) → 𝑆 (𝑃 𝑄))
5655ex 416 . . 3 (𝜑 → (𝑈 = 𝑉𝑆 (𝑃 𝑄)))
5756necon3bd 3028 . 2 (𝜑 → (¬ 𝑆 (𝑃 𝑄) → 𝑈𝑉))
582, 57mpd 15 1 (𝜑𝑈𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  Latclat 17655  Atomscatm 36504  CvLatclc 36506  HLchlt 36591  LHypclh 37225
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-p1 17650  df-lat 17656  df-clat 17718  df-oposet 36417  df-ol 36419  df-oml 36420  df-covers 36507  df-ats 36508  df-atl 36539  df-cvlat 36563  df-hlat 36592  df-lhyp 37229
This theorem is referenced by:  4atexlemtlw  37308  4atexlemntlpq  37309  4atexlemc  37310  4atexlemnclw  37311
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