Theorem cdleme26ee 35648

Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), f_z(s), f_z(t) respectively. When t ∨ v = p ∨ q, f_z(s) ≤ f_z(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 2-Feb-2013.)

Hypotheses

Ref	Expression
cdleme26.b	⊢ 𝐵 = (Base‘𝐾)
cdleme26.l	⊢ ≤ = (le‘𝐾)
cdleme26.j	⊢ ∨ = (join‘𝐾)
cdleme26.m	⊢ ∧ = (meet‘𝐾)
cdleme26.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme26.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme26e.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme26e.f	⊢ 𝐹 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
cdleme26e.n	⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊)))
cdleme26e.o	⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
cdleme26e.i	⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
cdleme26e.e	⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))

Assertion

Ref	Expression
cdleme26ee	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝐼 ≤ (𝐸 ∨ 𝑉))

Distinct variable groups:   𝑧,𝑢,𝐴   𝑧,𝐵,𝑢   𝑧,𝐻   𝑧, ∨ ,𝑢   𝑧,𝐾   𝑧, ≤ ,𝑢   𝑧, ∧ ,𝑢   𝑢,𝑁   𝑢,𝑂   𝑧,𝑃,𝑢   𝑧,𝑄,𝑢   𝑧,𝑆,𝑢   𝑧,𝑇,𝑢   𝑧,𝑈,𝑢   𝑧,𝑊,𝑢   𝑧,𝑉

Allowed substitution hints:   𝐸(𝑧,𝑢)   𝐹(𝑧,𝑢)   𝐻(𝑢)   𝐼(𝑧,𝑢)   𝐾(𝑢)   𝑁(𝑧)   𝑂(𝑧)   𝑉(𝑢)

Proof of Theorem cdleme26ee

Step	Hyp	Ref	Expression
1		simp11l 1172	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
2		simp11r 1173	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝑊 ∈ 𝐻)
3		simp12 1092	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		simp13 1093	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
5		simp3l1 1166	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝑃 ≠ 𝑄)
6		cdleme26.l	. . . 4 ⊢ ≤ = (le‘𝐾)
7		cdleme26.j	. . . 4 ⊢ ∨ = (join‘𝐾)
8		cdleme26.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdleme26.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
10	6, 7, 8, 9	cdlemb2 35327	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))
11	1, 2, 3, 4, 5, 10	syl221anc 1337	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))
12		nfv 1843	. . 3 ⊢ Ⅎ𝑧(((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)))
13		cdleme26e.i	. . . . 5 ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
14		nfra1 2941	. . . . . 6 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)
15		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑧𝐵
16	14, 15	nfriota 6620	. . . . 5 ⊢ Ⅎ𝑧(℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
17	13, 16	nfcxfr 2762	. . . 4 ⊢ Ⅎ𝑧𝐼
18		nfcv 2764	. . . 4 ⊢ Ⅎ𝑧 ≤
19		cdleme26e.e	. . . . . 6 ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))
20		nfra1 2941	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)
21	20, 15	nfriota 6620	. . . . . 6 ⊢ Ⅎ𝑧(℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))
22	19, 21	nfcxfr 2762	. . . . 5 ⊢ Ⅎ𝑧𝐸
23		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑧 ∨
24		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑧𝑉
25	22, 23, 24	nfov 6676	. . . 4 ⊢ Ⅎ𝑧(𝐸 ∨ 𝑉)
26	17, 18, 25	nfbr 4699	. . 3 ⊢ Ⅎ𝑧 𝐼 ≤ (𝐸 ∨ 𝑉)
27		simp111 1190	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28		simp112 1191	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
29		simp113 1192	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
30		simp121 1193	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
31		simp122 1194	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊))
32		simp123 1195	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
33		simp13l 1176	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)))
34		simp13r 1177	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))
35		simp3r 1090	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))
36	34, 35	jca 554	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → ((𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄) ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)))
37		simp2 1062	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → 𝑧 ∈ 𝐴)
38		simp3l 1089	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑧 ≤ 𝑊)
39	37, 38	jca 554	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))
40		cdleme26.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
41		cdleme26.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
42		cdleme26e.u	. . . . . 6 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
43		cdleme26e.f	. . . . . 6 ⊢ 𝐹 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
44		cdleme26e.n	. . . . . 6 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊)))
45		cdleme26e.o	. . . . . 6 ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
46	40, 6, 7, 41, 8, 9, 42, 43, 44, 45, 13, 19	cdleme26e 35647	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄) ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝐼 ≤ (𝐸 ∨ 𝑉))
47	27, 28, 29, 30, 31, 32, 33, 36, 39, 46	syl333anc 1358	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) ∧ 𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))) → 𝐼 ≤ (𝐸 ∨ 𝑉))
48	47	3exp 1264	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → (𝑧 ∈ 𝐴 → ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝐼 ≤ (𝐸 ∨ 𝑉))))
49	12, 26, 48	rexlimd 3026	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → (∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝐼 ≤ (𝐸 ∨ 𝑉)))
50	11, 49	mpd 15	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝐼 ≤ (𝐸 ∨ 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   class class class wbr 4653  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  Basecbs 15857  lecple 15948  joincjn 16944  meetcmee 16945  Atomscatm 34550  HLchlt 34637  LHypclh 35270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274

This theorem is referenced by:  cdleme27a  35655