Theorem cdleme7e 40241

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 40242 and cdleme7 40243. (Contributed by NM, 8-Jun-2012.)

Hypotheses

Ref	Expression
cdleme4.l	⊢ ≤ = (le‘𝐾)
cdleme4.j	⊢ ∨ = (join‘𝐾)
cdleme4.m	⊢ ∧ = (meet‘𝐾)
cdleme4.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme4.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme4.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme4.f	⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme4.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
cdleme7.v	⊢ 𝑉 = ((𝑅 ∨ 𝑆) ∧ 𝑊)

Assertion

Ref	Expression
cdleme7e	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐺 ≠ (0.‘𝐾))

Proof of Theorem cdleme7e

Step	Hyp	Ref	Expression
1		simp11l 1285	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
2	1	hllatd 39357	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ Lat)
3		simp2ll 1241	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴)
4		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		cdleme4.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 39282	. . . . . 6 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
7	3, 6	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ (Base‘𝐾))
8		hlop 39355	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
9		eqid 2729	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
10	4, 9	op0cl 39177	. . . . . 6 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
11	1, 8, 10	3syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (0.‘𝐾) ∈ (Base‘𝐾))
12		cdleme4.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
13	4, 12	latjcl 18398	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (0.‘𝐾) ∈ (Base‘𝐾)) → (𝑅 ∨ (0.‘𝐾)) ∈ (Base‘𝐾))
14	2, 7, 11, 13	syl3anc 1373	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ (0.‘𝐾)) ∈ (Base‘𝐾))
15		cdleme4.g	. . . . . 6 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
16		simp12l 1287	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
17		simp13l 1289	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
18	4, 12, 5	hlatjcl 39360	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
19	1, 16, 17, 18	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
20		simp11 1204	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		simp2rl 1243	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ∈ 𝐴)
22		cdleme4.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
23		cdleme4.m	. . . . . . . . . 10 ⊢ ∧ = (meet‘𝐾)
24		cdleme4.h	. . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾)
25		cdleme4.u	. . . . . . . . . 10 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
26		cdleme4.f	. . . . . . . . . 10 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
27	22, 12, 23, 5, 24, 25, 26, 4	cdleme1b 40220	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐹 ∈ (Base‘𝐾))
28	20, 16, 17, 21, 27	syl13anc 1374	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐹 ∈ (Base‘𝐾))
29	4, 12, 5	hlatjcl 39360	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
30	1, 3, 21, 29	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
31		simp11r 1286	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ 𝐻)
32	4, 24	lhpbase 39992	. . . . . . . . . 10 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
33	31, 32	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ (Base‘𝐾))
34	4, 23	latmcl 18399	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾))
35	2, 30, 33, 34	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾))
36	4, 12	latjcl 18398	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) ∈ (Base‘𝐾))
37	2, 28, 35, 36	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) ∈ (Base‘𝐾))
38	4, 23	latmcl 18399	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) ∈ (Base‘𝐾))
39	2, 19, 37, 38	syl3anc 1373	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) ∈ (Base‘𝐾))
40	15, 39	eqeltrid 2832	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐺 ∈ (Base‘𝐾))
41	4, 12	latjcl 18398	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝐺 ∈ (Base‘𝐾)) → (𝑅 ∨ 𝐺) ∈ (Base‘𝐾))
42	2, 7, 40, 41	syl3anc 1373	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝐺) ∈ (Base‘𝐾))
43	4, 22, 23	latmle2 18424	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
44	2, 19, 33, 43	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
45	25, 44	eqbrtrid 5142	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑈 ≤ 𝑊)
46		simp2lr 1242	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑅 ≤ 𝑊)
47		nbrne2 5127	. . . . . . . 8 ⊢ ((𝑈 ≤ 𝑊 ∧ ¬ 𝑅 ≤ 𝑊) → 𝑈 ≠ 𝑅)
48	47	necomd 2980	. . . . . . 7 ⊢ ((𝑈 ≤ 𝑊 ∧ ¬ 𝑅 ≤ 𝑊) → 𝑅 ≠ 𝑈)
49	45, 46, 48	syl2anc 584	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≠ 𝑈)
50		simp12 1205	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
51		simp31 1210	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≠ 𝑄)
52	22, 12, 23, 5, 24, 25	lhpat2 40039	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴)
53	20, 50, 17, 51, 52	syl112anc 1376	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑈 ∈ 𝐴)
54		eqid 2729	. . . . . . . 8 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
55	12, 54, 5	atcvr1 39411	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑅 ≠ 𝑈 ↔ 𝑅( ⋖ ‘𝐾)(𝑅 ∨ 𝑈)))
56	1, 3, 53, 55	syl3anc 1373	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≠ 𝑈 ↔ 𝑅( ⋖ ‘𝐾)(𝑅 ∨ 𝑈)))
57	49, 56	mpbid 232	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅( ⋖ ‘𝐾)(𝑅 ∨ 𝑈))
58		hlol 39354	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
59	1, 58	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ OL)
60	4, 12, 9	olj01 39218	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑅 ∨ (0.‘𝐾)) = 𝑅)
61	59, 7, 60	syl2anc 584	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ (0.‘𝐾)) = 𝑅)
62		simp2l 1200	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
63		simp2r 1201	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
64		simp32 1211	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
65	22, 12, 23, 5, 24, 25, 26, 15	cdleme5 40234	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝐺) = (𝑃 ∨ 𝑄))
66	20, 16, 17, 62, 63, 64, 65	syl132anc 1390	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝐺) = (𝑃 ∨ 𝑄))
67	22, 12, 23, 5, 24, 25	cdleme4 40232	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑈))
68	20, 16, 17, 62, 64, 67	syl131anc 1385	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑈))
69	66, 68	eqtrd 2764	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝐺) = (𝑅 ∨ 𝑈))
70	57, 61, 69	3brtr4d 5139	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ (0.‘𝐾))( ⋖ ‘𝐾)(𝑅 ∨ 𝐺))
71	4, 54	cvrne 39274	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑅 ∨ (0.‘𝐾)) ∈ (Base‘𝐾) ∧ (𝑅 ∨ 𝐺) ∈ (Base‘𝐾)) ∧ (𝑅 ∨ (0.‘𝐾))( ⋖ ‘𝐾)(𝑅 ∨ 𝐺)) → (𝑅 ∨ (0.‘𝐾)) ≠ (𝑅 ∨ 𝐺))
72	1, 14, 42, 70, 71	syl31anc 1375	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ (0.‘𝐾)) ≠ (𝑅 ∨ 𝐺))
73		oveq2 7395	. . . 4 ⊢ ((0.‘𝐾) = 𝐺 → (𝑅 ∨ (0.‘𝐾)) = (𝑅 ∨ 𝐺))
74	73	necon3i 2957	. . 3 ⊢ ((𝑅 ∨ (0.‘𝐾)) ≠ (𝑅 ∨ 𝐺) → (0.‘𝐾) ≠ 𝐺)
75	72, 74	syl 17	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (0.‘𝐾) ≠ 𝐺)
76	75	necomd 2980	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐺 ≠ (0.‘𝐾))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  meetcmee 18273  0.cp0 18382  Latclat 18390  OPcops 39165  OLcol 39167   ⋖ ccvr 39255  Atomscatm 39256  HLchlt 39343  LHypclh 39978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  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 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-psubsp 39497  df-pmap 39498  df-padd 39790  df-lhyp 39982

This theorem is referenced by:  cdleme7ga  40242