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Theorem 4atexlemntlpq 35180
Description: Lemma for 4atexlem7 35187. (Contributed by NM, 24-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
4atexlemntlpq (𝜑 → ¬ 𝑇 (𝑃 𝑄))

Proof of Theorem 4atexlemntlpq
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))))
2 4thatlem0.l . . 3 = (le‘𝐾)
3 4thatlem0.j . . 3 = (join‘𝐾)
4 4thatlem0.m . . 3 = (meet‘𝐾)
5 4thatlem0.a . . 3 𝐴 = (Atoms‘𝐾)
6 4thatlem0.h . . 3 𝐻 = (LHyp‘𝐾)
7 4thatlem0.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
8 4thatlem0.v . . 3 𝑉 = ((𝑃 𝑆) 𝑊)
91, 2, 3, 4, 5, 6, 7, 84atexlemtlw 35179 . 2 (𝜑𝑇 𝑊)
1014atexlemkc 35170 . . . . . 6 (𝜑𝐾 ∈ CvLat)
111, 2, 3, 4, 5, 6, 74atexlemu 35176 . . . . . 6 (𝜑𝑈𝐴)
121, 2, 3, 4, 5, 6, 7, 84atexlemv 35177 . . . . . 6 (𝜑𝑉𝐴)
1314atexlemt 35165 . . . . . 6 (𝜑𝑇𝐴)
141, 2, 3, 4, 5, 6, 7, 84atexlemunv 35178 . . . . . 6 (𝜑𝑈𝑉)
1514atexlemutvt 35166 . . . . . 6 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
165, 3cvlsupr5 34459 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇𝑈)
1710, 11, 12, 13, 14, 15, 16syl132anc 1343 . . . . 5 (𝜑𝑇𝑈)
1817adantr 481 . . . 4 ((𝜑𝑇 (𝑃 𝑄)) → 𝑇𝑈)
1914atexlemk 35159 . . . . . . 7 (𝜑𝐾 ∈ HL)
2014atexlemw 35160 . . . . . . 7 (𝜑𝑊𝐻)
2119, 20jca 554 . . . . . 6 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
2221adantr 481 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2314atexlempw 35161 . . . . . 6 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2423adantr 481 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2514atexlemq 35163 . . . . . 6 (𝜑𝑄𝐴)
2625adantr 481 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑄𝐴)
2713adantr 481 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑇𝐴)
2814atexlempnq 35167 . . . . . 6 (𝜑𝑃𝑄)
2928adantr 481 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑃𝑄)
30 simpr 477 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑇 (𝑃 𝑄))
312, 3, 4, 5, 6, 7lhpat3 35158 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑄𝐴𝑇𝐴) ∧ (𝑃𝑄𝑇 (𝑃 𝑄))) → (¬ 𝑇 𝑊𝑇𝑈))
3222, 24, 26, 27, 29, 30, 31syl222anc 1341 . . . 4 ((𝜑𝑇 (𝑃 𝑄)) → (¬ 𝑇 𝑊𝑇𝑈))
3318, 32mpbird 247 . . 3 ((𝜑𝑇 (𝑃 𝑄)) → ¬ 𝑇 𝑊)
3433ex 450 . 2 (𝜑 → (𝑇 (𝑃 𝑄) → ¬ 𝑇 𝑊))
359, 34mt2d 131 1 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  wne 2793   class class class wbr 4651  cfv 5886  (class class class)co 6647  lecple 15942  joincjn 16938  meetcmee 16939  Atomscatm 34376  CvLatclc 34378  HLchlt 34463  LHypclh 35096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-preset 16922  df-poset 16940  df-plt 16952  df-lub 16968  df-glb 16969  df-join 16970  df-meet 16971  df-p0 17033  df-p1 17034  df-lat 17040  df-clat 17102  df-oposet 34289  df-ol 34291  df-oml 34292  df-covers 34379  df-ats 34380  df-atl 34411  df-cvlat 34435  df-hlat 34464  df-lhyp 35100
This theorem is referenced by:  4atexlemc  35181  4atexlemex2  35183  4atexlemcnd  35184
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