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Theorem 4atexlemntlpq 37186
Description: Lemma for 4atexlem7 37193. (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 37185 . 2 (𝜑𝑇 𝑊)
1014atexlemkc 37176 . . . . . 6 (𝜑𝐾 ∈ CvLat)
111, 2, 3, 4, 5, 6, 74atexlemu 37182 . . . . . 6 (𝜑𝑈𝐴)
121, 2, 3, 4, 5, 6, 7, 84atexlemv 37183 . . . . . 6 (𝜑𝑉𝐴)
1314atexlemt 37171 . . . . . 6 (𝜑𝑇𝐴)
141, 2, 3, 4, 5, 6, 7, 84atexlemunv 37184 . . . . . 6 (𝜑𝑈𝑉)
1514atexlemutvt 37172 . . . . . 6 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
165, 3cvlsupr5 36464 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇𝑈)
1710, 11, 12, 13, 14, 15, 16syl132anc 1382 . . . . 5 (𝜑𝑇𝑈)
1817adantr 483 . . . 4 ((𝜑𝑇 (𝑃 𝑄)) → 𝑇𝑈)
1914atexlemk 37165 . . . . . . 7 (𝜑𝐾 ∈ HL)
2014atexlemw 37166 . . . . . . 7 (𝜑𝑊𝐻)
2119, 20jca 514 . . . . . 6 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
2221adantr 483 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2314atexlempw 37167 . . . . . 6 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2423adantr 483 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2514atexlemq 37169 . . . . . 6 (𝜑𝑄𝐴)
2625adantr 483 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑄𝐴)
2713adantr 483 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑇𝐴)
2814atexlempnq 37173 . . . . . 6 (𝜑𝑃𝑄)
2928adantr 483 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑃𝑄)
30 simpr 487 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑇 (𝑃 𝑄))
312, 3, 4, 5, 6, 7lhpat3 37164 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑄𝐴𝑇𝐴) ∧ (𝑃𝑄𝑇 (𝑃 𝑄))) → (¬ 𝑇 𝑊𝑇𝑈))
3222, 24, 26, 27, 29, 30, 31syl222anc 1380 . . . 4 ((𝜑𝑇 (𝑃 𝑄)) → (¬ 𝑇 𝑊𝑇𝑈))
3318, 32mpbird 259 . . 3 ((𝜑𝑇 (𝑃 𝑄)) → ¬ 𝑇 𝑊)
3433ex 415 . 2 (𝜑 → (𝑇 (𝑃 𝑄) → ¬ 𝑇 𝑊))
359, 34mt2d 138 1 (𝜑 → ¬ 𝑇 (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wne 3014   class class class wbr 5057  cfv 6348  (class class class)co 7148  lecple 16564  joincjn 17546  meetcmee 17547  Atomscatm 36381  CvLatclc 36383  HLchlt 36468  LHypclh 37102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36294  df-ol 36296  df-oml 36297  df-covers 36384  df-ats 36385  df-atl 36416  df-cvlat 36440  df-hlat 36469  df-lhyp 37106
This theorem is referenced by:  4atexlemc  37187  4atexlemex2  37189  4atexlemcnd  37190
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