Theorem 4atexlemntlpq 40534

Description: Lemma for 4atexlem7 40541. (Contributed by NM, 24-Nov-2012.)

Hypotheses

Ref	Expression
4thatlem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 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

Step	Hyp	Ref	Expression
1		4thatlem.ph	. . 3 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 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)
9	1, 2, 3, 4, 5, 6, 7, 8	4atexlemtlw 40533	. 2 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
10	1	4atexlemkc 40524	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ CvLat)
11	1, 2, 3, 4, 5, 6, 7	4atexlemu 40530	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
12	1, 2, 3, 4, 5, 6, 7, 8	4atexlemv 40531	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
13	1	4atexlemt 40519	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
14	1, 2, 3, 4, 5, 6, 7, 8	4atexlemunv 40532	. . . . . 6 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
15	1	4atexlemutvt 40520	. . . . . 6 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
16	5, 3	cvlsupr5 39812	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ≠ 𝑉 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) → 𝑇 ≠ 𝑈)
17	10, 11, 12, 13, 14, 15, 16	syl132anc 1391	. . . . 5 ⊢ (𝜑 → 𝑇 ≠ 𝑈)
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≠ 𝑈)
19	1	4atexlemk 40513	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ HL)
20	1	4atexlemw 40514	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
21	19, 20	jca 511	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	1	4atexlempw 40515	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
25	1	4atexlemq 40517	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
26	25	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
27	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ∈ 𝐴)
28	1	4atexlempnq 40521	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
29	28	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑄)
30		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≤ (𝑃 ∨ 𝑄))
31	2, 3, 4, 5, 6, 7	lhpat3 40512	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) → (¬ 𝑇 ≤ 𝑊 ↔ 𝑇 ≠ 𝑈))
32	22, 24, 26, 27, 29, 30, 31	syl222anc 1389	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (¬ 𝑇 ≤ 𝑊 ↔ 𝑇 ≠ 𝑈))
33	18, 32	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑇 ≤ 𝑊)
34	33	ex 412	. 2 ⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) → ¬ 𝑇 ≤ 𝑊))
35	9, 34	mt2d 136	1 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5086  ‘cfv 6494  (class class class)co 7362  lecple 17222  joincjn 18272  meetcmee 18273  Atomscatm 39729  CvLatclc 39731  HLchlt 39816  LHypclh 40450

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-p1 18385  df-lat 18393  df-clat 18460  df-oposet 39642  df-ol 39644  df-oml 39645  df-covers 39732  df-ats 39733  df-atl 39764  df-cvlat 39788  df-hlat 39817  df-lhyp 40454

This theorem is referenced by:  4atexlemc  40535  4atexlemex2  40537  4atexlemcnd  40538