Theorem 4atexlemntlpq 41126

Description: Lemma for 4atexlem7 41133. (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 41125	. 2 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
10	1	4atexlemkc 41116	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ CvLat)
11	1, 2, 3, 4, 5, 6, 7	4atexlemu 41122	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
12	1, 2, 3, 4, 5, 6, 7, 8	4atexlemv 41123	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
13	1	4atexlemt 41111	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
14	1, 2, 3, 4, 5, 6, 7, 8	4atexlemunv 41124	. . . . . 6 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
15	1	4atexlemutvt 41112	. . . . . 6 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
16	5, 3	cvlsupr5 40404	. . . . . 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 1415	. . . . 5 ⊢ (𝜑 → 𝑇 ≠ 𝑈)
18	17	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≠ 𝑈)
19	1	4atexlemk 41105	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ HL)
20	1	4atexlemw 41106	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
21	19, 20	jca 521	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	21	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	1	4atexlempw 41107	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
24	23	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
25	1	4atexlemq 41109	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
26	25	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
27	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ∈ 𝐴)
28	1	4atexlempnq 41113	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
29	28	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑄)
30		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≤ (𝑃 ∨ 𝑄))
31	2, 3, 4, 5, 6, 7	lhpat3 41104	. . . . 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 1413	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (¬ 𝑇 ≤ 𝑊 ↔ 𝑇 ≠ 𝑈))
33	18, 32	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑇 ≤ 𝑊)
34	33	ex 418	. 2 ⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) → ¬ 𝑇 ≤ 𝑊))
35	9, 34	mt2d 137	1 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  lecple 17435  joincjn 18485  meetcmee 18486  Atomscatm 40321  CvLatclc 40323  HLchlt 40408  LHypclh 41042

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-oposet 40234  df-ol 40236  df-oml 40237  df-covers 40324  df-ats 40325  df-atl 40356  df-cvlat 40380  df-hlat 40409  df-lhyp 41046

This theorem is used by:  4atexlemc  41127  4atexlemex2  41129  4atexlemcnd  41130