Theorem cdleme37m 40867

Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 ∨ 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 13-Mar-2013.)

Hypotheses

Ref	Expression
cdleme37.l	⊢ ≤ = (le‘𝐾)
cdleme37.j	⊢ ∨ = (join‘𝐾)
cdleme37.m	⊢ ∧ = (meet‘𝐾)
cdleme37.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme37.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme37.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme37.e	⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme37.d	⊢ 𝐷 = ((𝑢 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑢) ∧ 𝑊)))
cdleme37.v	⊢ 𝑉 = ((𝑡 ∨ 𝐸) ∧ 𝑊)
cdleme37.x	⊢ 𝑋 = ((𝑢 ∨ 𝐷) ∧ 𝑊)
cdleme37.c	⊢ 𝐶 = ((𝑆 ∨ 𝑉) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑆) ∧ 𝑊)))
cdleme37.g	⊢ 𝐺 = ((𝑆 ∨ 𝑋) ∧ (𝐷 ∨ ((𝑢 ∨ 𝑆) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme37m	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝐶 = 𝐺)

Proof of Theorem cdleme37m

Step	Hyp	Ref	Expression
1		simp1 1137	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
2		simp23 1210	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
3		simp32l 1300	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))
4		simp33l 1302	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊))
5		simp21 1208	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ≠ 𝑄)
6		simp32r 1301	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))
7		simp33r 1303	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑢 ≤ (𝑃 ∨ 𝑄))
8		simp31r 1299	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ≤ (𝑃 ∨ 𝑄))
9	6, 7, 8	3jca 1129	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (¬ 𝑡 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))
10		cdleme37.l	. . . 4 ⊢ ≤ = (le‘𝐾)
11		cdleme37.j	. . . 4 ⊢ ∨ = (join‘𝐾)
12		cdleme37.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
13		cdleme37.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
14		cdleme37.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
15		cdleme37.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
16		cdleme37.e	. . . 4 ⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
17		cdleme37.d	. . . 4 ⊢ 𝐷 = ((𝑢 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑢) ∧ 𝑊)))
18		eqid 2737	. . . 4 ⊢ ((𝑆 ∨ 𝑡) ∧ 𝑊) = ((𝑆 ∨ 𝑡) ∧ 𝑊)
19		eqid 2737	. . . 4 ⊢ ((𝑆 ∨ 𝑢) ∧ 𝑊) = ((𝑆 ∨ 𝑢) ∧ 𝑊)
20		eqid 2737	. . . 4 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑆 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑆 ∨ 𝑡) ∧ 𝑊)))
21		eqid 2737	. . . 4 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑆 ∨ 𝑢) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑆 ∨ 𝑢) ∧ 𝑊)))
22	10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21	cdleme21k 40743	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (¬ 𝑡 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑆 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑆 ∨ 𝑢) ∧ 𝑊))))
23	1, 2, 3, 4, 5, 9, 22	syl132anc 1391	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑆 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑆 ∨ 𝑢) ∧ 𝑊))))
24		cdleme37.c	. . 3 ⊢ 𝐶 = ((𝑆 ∨ 𝑉) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑆) ∧ 𝑊)))
25		simp11 1205	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26		simp12l 1288	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ∈ 𝐴)
27		simp13l 1290	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑄 ∈ 𝐴)
28	10, 11, 12, 13, 14, 15	cdleme4 40643	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑈))
29	25, 26, 27, 2, 8, 28	syl131anc 1386	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑈))
30		cdleme37.v	. . . . . . 7 ⊢ 𝑉 = ((𝑡 ∨ 𝐸) ∧ 𝑊)
31	10, 11, 12, 13, 14, 15, 16	cdleme2 40633	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → ((𝑡 ∨ 𝐸) ∧ 𝑊) = 𝑈)
32	25, 26, 27, 3, 31	syl13anc 1375	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ((𝑡 ∨ 𝐸) ∧ 𝑊) = 𝑈)
33	30, 32	eqtrid 2784	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑉 = 𝑈)
34	33	oveq2d 7386	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑉) = (𝑆 ∨ 𝑈))
35	29, 34	eqtr4d 2775	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑉))
36		simp11l 1286	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝐾 ∈ HL)
37		simp23l 1296	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ∈ 𝐴)
38	3	simpld 494	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑡 ∈ 𝐴)
39	11, 13	hlatjcom 39773	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) → (𝑆 ∨ 𝑡) = (𝑡 ∨ 𝑆))
40	36, 37, 38, 39	syl3anc 1374	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑡) = (𝑡 ∨ 𝑆))
41	40	oveq1d 7385	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ((𝑆 ∨ 𝑡) ∧ 𝑊) = ((𝑡 ∨ 𝑆) ∧ 𝑊))
42	41	oveq2d 7386	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝐸 ∨ ((𝑆 ∨ 𝑡) ∧ 𝑊)) = (𝐸 ∨ ((𝑡 ∨ 𝑆) ∧ 𝑊)))
43	35, 42	oveq12d 7388	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑆 ∨ 𝑡) ∧ 𝑊))) = ((𝑆 ∨ 𝑉) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑆) ∧ 𝑊))))
44	24, 43	eqtr4id 2791	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝐶 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑆 ∨ 𝑡) ∧ 𝑊))))
45		cdleme37.g	. . 3 ⊢ 𝐺 = ((𝑆 ∨ 𝑋) ∧ (𝐷 ∨ ((𝑢 ∨ 𝑆) ∧ 𝑊)))
46		cdleme37.x	. . . . . . 7 ⊢ 𝑋 = ((𝑢 ∨ 𝐷) ∧ 𝑊)
47	10, 11, 12, 13, 14, 15, 17	cdleme2 40633	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊))) → ((𝑢 ∨ 𝐷) ∧ 𝑊) = 𝑈)
48	25, 26, 27, 4, 47	syl13anc 1375	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ((𝑢 ∨ 𝐷) ∧ 𝑊) = 𝑈)
49	46, 48	eqtrid 2784	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑋 = 𝑈)
50	49	oveq2d 7386	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑋) = (𝑆 ∨ 𝑈))
51	29, 50	eqtr4d 2775	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑋))
52	4	simpld 494	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑢 ∈ 𝐴)
53	11, 13	hlatjcom 39773	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝑆 ∨ 𝑢) = (𝑢 ∨ 𝑆))
54	36, 37, 52, 53	syl3anc 1374	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑢) = (𝑢 ∨ 𝑆))
55	54	oveq1d 7385	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ((𝑆 ∨ 𝑢) ∧ 𝑊) = ((𝑢 ∨ 𝑆) ∧ 𝑊))
56	55	oveq2d 7386	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝐷 ∨ ((𝑆 ∨ 𝑢) ∧ 𝑊)) = (𝐷 ∨ ((𝑢 ∨ 𝑆) ∧ 𝑊)))
57	51, 56	oveq12d 7388	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑆 ∨ 𝑢) ∧ 𝑊))) = ((𝑆 ∨ 𝑋) ∧ (𝐷 ∨ ((𝑢 ∨ 𝑆) ∧ 𝑊))))
58	45, 57	eqtr4id 2791	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑆 ∨ 𝑢) ∧ 𝑊))))
59	23, 44, 58	3eqtr4d 2782	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝐶 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5100  ‘cfv 6502  (class class class)co 7370  lecple 17198  joincjn 18248  meetcmee 18249  Atomscatm 39668  HLchlt 39755  LHypclh 40389

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-proset 18231  df-poset 18250  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-p1 18361  df-lat 18369  df-clat 18436  df-oposet 39581  df-ol 39583  df-oml 39584  df-covers 39671  df-ats 39672  df-atl 39703  df-cvlat 39727  df-hlat 39756  df-llines 39903  df-lplanes 39904  df-lvols 39905  df-lines 39906  df-psubsp 39908  df-pmap 39909  df-padd 40201  df-lhyp 40393

This theorem is referenced by:  cdleme38m  40868