Theorem cdleme11g 36883

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 36888. (Contributed by NM, 14-Jun-2012.)

Hypotheses

Ref	Expression
cdleme11.l	⊢ ≤ = (le‘𝐾)
cdleme11.j	⊢ ∨ = (join‘𝐾)
cdleme11.m	⊢ ∧ = (meet‘𝐾)
cdleme11.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme11.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme11.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme11.c	⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
cdleme11.d	⊢ 𝐷 = ((𝑃 ∨ 𝑇) ∧ 𝑊)
cdleme11.f	⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme11g	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ 𝐹) = (𝑄 ∨ 𝐶))

Proof of Theorem cdleme11g

Step	Hyp	Ref	Expression
1		cdleme11.f	. . . 4 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
2	1	oveq2i 6985	. . 3 ⊢ (𝑄 ∨ 𝐹) = (𝑄 ∨ ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))))
3		simp1l 1178	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ HL)
4		simp22l 1273	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴)
5	3	hllatd 35982	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ Lat)
6		simp23 1189	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑆 ∈ 𝐴)
7		eqid 2771	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		cdleme11.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
9	7, 8	atbase 35907	. . . . . 6 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
10	6, 9	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑆 ∈ (Base‘𝐾))
11		simp1 1117	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simp21 1187	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴)
13		cdleme11.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
14		cdleme11.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
15		cdleme11.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
16		cdleme11.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
17		cdleme11.u	. . . . . . 7 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
18	13, 14, 15, 8, 16, 17, 7	cdleme0aa 36828	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑈 ∈ (Base‘𝐾))
19	11, 12, 4, 18	syl3anc 1352	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑈 ∈ (Base‘𝐾))
20	7, 14	latjcl 17531	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑆 ∨ 𝑈) ∈ (Base‘𝐾))
21	5, 10, 19, 20	syl3anc 1352	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑆 ∨ 𝑈) ∈ (Base‘𝐾))
22	7, 8	atbase 35907	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
23	4, 22	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ (Base‘𝐾))
24	7, 8	atbase 35907	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
25	12, 24	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ (Base‘𝐾))
26	7, 14	latjcl 17531	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
27	5, 25, 10, 26	syl3anc 1352	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
28		simp1r 1179	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑊 ∈ 𝐻)
29	7, 16	lhpbase 36616	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
30	28, 29	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑊 ∈ (Base‘𝐾))
31	7, 15	latmcl 17532	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾))
32	5, 27, 30, 31	syl3anc 1352	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾))
33	7, 14	latjcl 17531	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) ∈ (Base‘𝐾))
34	5, 23, 32, 33	syl3anc 1352	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) ∈ (Base‘𝐾))
35	7, 13, 14	latlej1 17540	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾)) → 𝑄 ≤ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
36	5, 23, 32, 35	syl3anc 1352	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑄 ≤ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
37	7, 13, 14, 15, 8	atmod1i1 36475	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ (𝑆 ∨ 𝑈) ∈ (Base‘𝐾) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) ∈ (Base‘𝐾)) ∧ 𝑄 ≤ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) → (𝑄 ∨ ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))) = ((𝑄 ∨ (𝑆 ∨ 𝑈)) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))))
38	3, 4, 21, 34, 36, 37	syl131anc 1364	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))) = ((𝑄 ∨ (𝑆 ∨ 𝑈)) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))))
39	2, 38	syl5eq 2819	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ 𝐹) = ((𝑄 ∨ (𝑆 ∨ 𝑈)) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))))
40		simp22 1188	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
41	13, 14, 15, 8, 16, 17	cdleme0cq 36833	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝑄 ∨ 𝑈) = (𝑃 ∨ 𝑄))
42	11, 12, 40, 41	syl12anc 825	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ 𝑈) = (𝑃 ∨ 𝑄))
43	42	oveq2d 6990	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑆 ∨ (𝑄 ∨ 𝑈)) = (𝑆 ∨ (𝑃 ∨ 𝑄)))
44	7, 14	latj12 17576	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → (𝑄 ∨ (𝑆 ∨ 𝑈)) = (𝑆 ∨ (𝑄 ∨ 𝑈)))
45	5, 23, 10, 19, 44	syl13anc 1353	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ (𝑆 ∨ 𝑈)) = (𝑆 ∨ (𝑄 ∨ 𝑈)))
46	7, 14	latj13 17578	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑄 ∨ (𝑃 ∨ 𝑆)) = (𝑆 ∨ (𝑃 ∨ 𝑄)))
47	5, 23, 25, 10, 46	syl13anc 1353	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ (𝑃 ∨ 𝑆)) = (𝑆 ∨ (𝑃 ∨ 𝑄)))
48	43, 45, 47	3eqtr4d 2817	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ (𝑆 ∨ 𝑈)) = (𝑄 ∨ (𝑃 ∨ 𝑆)))
49	48	oveq1d 6989	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑄 ∨ (𝑆 ∨ 𝑈)) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) = ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))))
50	7, 13, 15	latmle1 17556	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
51	5, 27, 30, 50	syl3anc 1352	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
52	7, 13, 14	latjlej2 17546	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆) → (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) ≤ (𝑄 ∨ (𝑃 ∨ 𝑆))))
53	5, 32, 27, 23, 52	syl13anc 1353	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆) → (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) ≤ (𝑄 ∨ (𝑃 ∨ 𝑆))))
54	51, 53	mpd 15	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) ≤ (𝑄 ∨ (𝑃 ∨ 𝑆)))
55	7, 14	latjcl 17531	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → (𝑄 ∨ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
56	5, 23, 27, 55	syl3anc 1352	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
57	7, 13, 15	latleeqm2 17560	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) ∈ (Base‘𝐾) ∧ (𝑄 ∨ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾)) → ((𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) ≤ (𝑄 ∨ (𝑃 ∨ 𝑆)) ↔ ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) = (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))))
58	5, 34, 56, 57	syl3anc 1352	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) ≤ (𝑄 ∨ (𝑃 ∨ 𝑆)) ↔ ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) = (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))))
59	54, 58	mpbid 224	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) = (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
60		cdleme11.c	. . . 4 ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
61	60	oveq2i 6985	. . 3 ⊢ (𝑄 ∨ 𝐶) = (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))
62	59, 61	syl6eqr 2825	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) = (𝑄 ∨ 𝐶))
63	39, 49, 62	3eqtrd 2811	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑄 ∨ 𝐹) = (𝑄 ∨ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   ≠ wne 2960   class class class wbr 4925  ‘cfv 6185  (class class class)co 6974  Basecbs 16337  lecple 16426  joincjn 17424  meetcmee 17425  Latclat 17525  Atomscatm 35881  HLchlt 35968  LHypclh 36602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-proset 17408  df-poset 17426  df-plt 17438  df-lub 17454  df-glb 17455  df-join 17456  df-meet 17457  df-p0 17519  df-p1 17520  df-lat 17526  df-clat 17588  df-oposet 35794  df-ol 35796  df-oml 35797  df-covers 35884  df-ats 35885  df-atl 35916  df-cvlat 35940  df-hlat 35969  df-psubsp 36121  df-pmap 36122  df-padd 36414  df-lhyp 36606

This theorem is referenced by:  cdleme11h  36884  cdleme11j  36885  cdleme15a  36892