4atexlemntlpq - Mathbox for Norm Megill

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Theorem 4atexlemntlpq 39545

Description: Lemma for 4atexlem7 39552. (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 39544	. 2 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
10	1	4atexlemkc 39535	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ CvLat)
11	1, 2, 3, 4, 5, 6, 7	4atexlemu 39541	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
12	1, 2, 3, 4, 5, 6, 7, 8	4atexlemv 39542	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
13	1	4atexlemt 39530	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
14	1, 2, 3, 4, 5, 6, 7, 8	4atexlemunv 39543	. . . . . 6 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
15	1	4atexlemutvt 39531	. . . . . 6 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
16	5, 3	cvlsupr5 38822	. . . . . 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 1385	. . . . 5 ⊢ (𝜑 → 𝑇 ≠ 𝑈)
18	17	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≠ 𝑈)
19	1	4atexlemk 39524	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ HL)
20	1	4atexlemw 39525	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
21	19, 20	jca 510	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	21	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	1	4atexlempw 39526	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
24	23	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
25	1	4atexlemq 39528	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
26	25	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
27	13	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ∈ 𝐴)
28	1	4atexlempnq 39532	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
29	28	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑄)
30		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≤ (𝑃 ∨ 𝑄))
31	2, 3, 4, 5, 6, 7	lhpat3 39523	. . . . 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 1383	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (¬ 𝑇 ≤ 𝑊 ↔ 𝑇 ≠ 𝑈))
33	18, 32	mpbird 256	. . 3 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑇 ≤ 𝑊)
34	33	ex 411	. 2 ⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) → ¬ 𝑇 ≤ 𝑊))
35	9, 34	mt2d 136	1 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2936 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 lecple 17245 joincjn 18308 meetcmee 18309 Atomscatm 38739 CvLatclc 38741 HLchlt 38826 LHypclh 39461

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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-proset 18292 df-poset 18310 df-plt 18327 df-lub 18343 df-glb 18344 df-join 18345 df-meet 18346 df-p0 18422 df-p1 18423 df-lat 18429 df-clat 18496 df-oposet 38652 df-ol 38654 df-oml 38655 df-covers 38742 df-ats 38743 df-atl 38774 df-cvlat 38798 df-hlat 38827 df-lhyp 39465

This theorem is referenced by: 4atexlemc 39546 4atexlemex2 39548 4atexlemcnd 39549

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