Theorem cdleme26f2ALTN 41422

Description: Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 41420 with s and t swapped (this case is not mentioned by them). If s ≤ t ∨ v, then f(s) ≤ f_s(t) ∨ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.)

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)
cdleme26f2.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme26f2.f	⊢ 𝐺 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme26f2.n	⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑠) ∧ 𝑊)))
cdleme26f2.e	⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))

Assertion

Ref	Expression
cdleme26f2ALTN	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐺 ≤ (𝐸 ∨ 𝑉))

Distinct variable groups:   𝑢,𝑠,𝐴   𝐵,𝑠,𝑢   𝐻,𝑠   ∨ ,𝑠,𝑢   𝐾,𝑠   ≤ ,𝑠,𝑢   ∧ ,𝑠,𝑢   𝑢,𝑂   𝑃,𝑠,𝑢   𝑄,𝑠,𝑢   𝑇,𝑠,𝑢   𝑈,𝑠,𝑢   𝑊,𝑠,𝑢

Allowed substitution hints:   𝐸(𝑢, 𝑠)   𝐺(𝑢, 𝑠)   𝐻(𝑢)   𝐾(𝑢)   𝑂(𝑠)   𝑉(𝑢, 𝑠)

Proof of Theorem cdleme26f2ALTN

Step	Hyp	Ref	Expression
1		simp11 1222	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp23 1227	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊))
3		simp31r 1316	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → ¬ 𝑠 ≤ (𝑃 ∨ 𝑄))
4		simp12r 1306	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑇 ≤ (𝑃 ∨ 𝑄))
5		simp12l 1305	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑃 ≠ 𝑄)
6	3, 4, 5	3jca 1146	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (¬ 𝑠 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄))
7		simp21 1225	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
8		simp22 1226	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
9		simp13 1224	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊))
10		simp32 1229	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)))
11		simp33 1230	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
12		cdleme26.l	. . . 4 ⊢ ≤ = (le‘𝐾)
13		cdleme26.j	. . . 4 ⊢ ∨ = (join‘𝐾)
14		cdleme26.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
15		cdleme26.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
16		cdleme26.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
17		cdleme26f2.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
18		cdleme26f2.f	. . . 4 ⊢ 𝐺 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
19		cdleme26f2.n	. . . 4 ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑠) ∧ 𝑊)))
20	12, 13, 14, 15, 16, 17, 18, 19	cdleme22f2 41405	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (¬ 𝑠 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐺 ≤ (𝑂 ∨ 𝑉))
21	1, 2, 6, 7, 8, 9, 10, 11, 20	syl323anc 1427	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐺 ≤ (𝑂 ∨ 𝑉))
22		simp23l 1313	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑇 ∈ 𝐴)
23		simp23r 1314	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → ¬ 𝑇 ≤ 𝑊)
24		cdleme26.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
25		cdleme26f2.e	. . . . . 6 ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))
26	24, 12, 13, 14, 15, 16, 17, 18, 19, 25	cdleme25cl 41415	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) → 𝐸 ∈ 𝐵)
27	1, 7, 8, 22, 23, 5, 4, 26	syl322anc 1425	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐸 ∈ 𝐵)
28		simp13l 1307	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑠 ∈ 𝐴)
29		simp31 1228	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))
30	24	fvexi 6899	. . . . 5 ⊢ 𝐵 ∈ V
31	30, 25	riotasv 40017	. . . 4 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄))) → 𝐸 = 𝑂)
32	27, 28, 29, 31	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐸 = 𝑂)
33	32	oveq1d 7435	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝐸 ∨ 𝑉) = (𝑂 ∨ 𝑉))
34	21, 33	breqtrrd 5132	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑠 ≠ 𝑇 ∧ 𝑠 ≤ (𝑇 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐺 ≤ (𝐸 ∨ 𝑉))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076   class class class wbr 5102  ‘cfv 6538  ℩crio 7376  (class class class)co 7420  Basecbs 17387  lecple 17435  joincjn 18485  meetcmee 18486  Atomscatm 40321  HLchlt 40408  LHypclh 41042

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-riotaBAD 40011

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-undef 8290  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-oposet 40234  df-ol 40236  df-oml 40237  df-covers 40324  df-ats 40325  df-atl 40356  df-cvlat 40380  df-hlat 40409  df-llines 40556  df-lplanes 40557  df-lvols 40558  df-lines 40559  df-psubsp 40561  df-pmap 40562  df-padd 40854  df-lhyp 41046

This theorem is used by: (None)