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Theorem 4atexlemntlpq 35672
Description: Lemma for 4atexlem7 35679. (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 35671 . 2 (𝜑𝑇 𝑊)
1014atexlemkc 35662 . . . . . 6 (𝜑𝐾 ∈ CvLat)
111, 2, 3, 4, 5, 6, 74atexlemu 35668 . . . . . 6 (𝜑𝑈𝐴)
121, 2, 3, 4, 5, 6, 7, 84atexlemv 35669 . . . . . 6 (𝜑𝑉𝐴)
1314atexlemt 35657 . . . . . 6 (𝜑𝑇𝐴)
141, 2, 3, 4, 5, 6, 7, 84atexlemunv 35670 . . . . . 6 (𝜑𝑈𝑉)
1514atexlemutvt 35658 . . . . . 6 (𝜑 → (𝑈 𝑇) = (𝑉 𝑇))
165, 3cvlsupr5 34951 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑉𝐴𝑇𝐴) ∧ (𝑈𝑉 ∧ (𝑈 𝑇) = (𝑉 𝑇))) → 𝑇𝑈)
1710, 11, 12, 13, 14, 15, 16syl132anc 1384 . . . . 5 (𝜑𝑇𝑈)
1817adantr 480 . . . 4 ((𝜑𝑇 (𝑃 𝑄)) → 𝑇𝑈)
1914atexlemk 35651 . . . . . . 7 (𝜑𝐾 ∈ HL)
2014atexlemw 35652 . . . . . . 7 (𝜑𝑊𝐻)
2119, 20jca 553 . . . . . 6 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
2221adantr 480 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2314atexlempw 35653 . . . . . 6 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2423adantr 480 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2514atexlemq 35655 . . . . . 6 (𝜑𝑄𝐴)
2625adantr 480 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑄𝐴)
2713adantr 480 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑇𝐴)
2814atexlempnq 35659 . . . . . 6 (𝜑𝑃𝑄)
2928adantr 480 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑃𝑄)
30 simpr 476 . . . . 5 ((𝜑𝑇 (𝑃 𝑄)) → 𝑇 (𝑃 𝑄))
312, 3, 4, 5, 6, 7lhpat3 35650 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑄𝐴𝑇𝐴) ∧ (𝑃𝑄𝑇 (𝑃 𝑄))) → (¬ 𝑇 𝑊𝑇𝑈))
3222, 24, 26, 27, 29, 30, 31syl222anc 1382 . . . 4 ((𝜑𝑇 (𝑃 𝑄)) → (¬ 𝑇 𝑊𝑇𝑈))
3318, 32mpbird 247 . . 3 ((𝜑𝑇 (𝑃 𝑄)) → ¬ 𝑇 𝑊)
3433ex 449 . 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  meetcmee 16992  Atomscatm 34868  CvLatclc 34870  HLchlt 34955  LHypclh 35588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-lhyp 35592
This theorem is referenced by:  4atexlemc  35673  4atexlemex2  35675  4atexlemcnd  35676
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