Theorem 2lplnj 39621

Description: The join of two different lattice planes in a (3-dimensional) lattice volume equals the volume. (Contributed by NM, 12-Jul-2012.)

Hypotheses

Ref	Expression
2lplnj.l	⊢ ≤ = (le‘𝐾)
2lplnj.j	⊢ ∨ = (join‘𝐾)
2lplnj.p	⊢ 𝑃 = (LPlanes‘𝐾)
2lplnj.v	⊢ 𝑉 = (LVols‘𝐾)

Assertion

Ref	Expression
2lplnj	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∨ 𝑌) = 𝑊)

Proof of Theorem 2lplnj

Dummy variables 𝑟 𝑞 𝑠 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		2lplnj.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
3		2lplnj.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
4		eqid 2730	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		2lplnj.p	. . . . . . . 8 ⊢ 𝑃 = (LPlanes‘𝐾)
6	1, 2, 3, 4, 5	islpln2 39537	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)))))
7		simpr 484	. . . . . . 7 ⊢ ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)))
8	6, 7	biimtrdi 253	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))))
9	1, 2, 3, 4, 5	islpln2 39537	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)))))
10		simpr 484	. . . . . . 7 ⊢ ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)))
11	9, 10	biimtrdi 253	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑃 → ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))))
12	8, 11	anim12d 609	. . . . 5 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)))))
13	12	imp 406	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))))
14	13	3adantr3 1172	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))))
15	14	3adant3 1132	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))))
16		simpl33 1257	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))
17	16	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))
18		simp33 1212	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))
19	17, 18	oveq12d 7408	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (𝑋 ∨ 𝑌) = (((𝑞 ∨ 𝑟) ∨ 𝑠) ∨ ((𝑡 ∨ 𝑢) ∨ 𝑣)))
20		simp11 1204	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) → 𝐾 ∈ HL)
21		simp123 1308	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) → 𝑊 ∈ 𝑉)
22	20, 21	jca 511	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝑉))
23	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝑉))
24	23	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝑉))
25		simp2l 1200	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) → 𝑞 ∈ (Atoms‘𝐾))
26		simp2rl 1243	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) → 𝑟 ∈ (Atoms‘𝐾))
27		simp2rr 1244	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) → 𝑠 ∈ (Atoms‘𝐾))
28	25, 26, 27	3jca 1128	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) → (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)))
29	28	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)))
30	29	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)))
31		simpl31 1255	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → 𝑞 ≠ 𝑟)
32	31	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → 𝑞 ≠ 𝑟)
33		simpl32 1256	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → ¬ 𝑠 ≤ (𝑞 ∨ 𝑟))
34	33	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → ¬ 𝑠 ≤ (𝑞 ∨ 𝑟))
35	32, 34	jca 511	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟)))
36		simp1r 1199	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → 𝑡 ∈ (Atoms‘𝐾))
37		simp2l 1200	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → 𝑢 ∈ (Atoms‘𝐾))
38		simp2r 1201	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → 𝑣 ∈ (Atoms‘𝐾))
39	36, 37, 38	3jca 1128	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)))
40		simp31 1210	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → 𝑡 ≠ 𝑢)
41		simp32 1211	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → ¬ 𝑣 ≤ (𝑡 ∨ 𝑢))
42	40, 41	jca 511	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢)))
43		simpl13 1251	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌))
44	43	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌))
45		breq1 5113	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → (𝑋 ≤ 𝑊 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊))
46		neeq1 2988	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → (𝑋 ≠ 𝑌 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌))
47	45, 46	3anbi13d 1440	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ (((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌)))
48		breq1 5113	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣) → (𝑌 ≤ 𝑊 ↔ ((𝑡 ∨ 𝑢) ∨ 𝑣) ≤ 𝑊))
49		neeq2 2989	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣) → (((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ ((𝑡 ∨ 𝑢) ∨ 𝑣)))
50	48, 49	3anbi23d 1441	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣) → ((((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌) ↔ (((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊 ∧ ((𝑡 ∨ 𝑢) ∨ 𝑣) ≤ 𝑊 ∧ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ ((𝑡 ∨ 𝑢) ∨ 𝑣))))
51	47, 50	sylan9bb 509	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ (((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊 ∧ ((𝑡 ∨ 𝑢) ∨ 𝑣) ≤ 𝑊 ∧ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ ((𝑡 ∨ 𝑢) ∨ 𝑣))))
52	17, 18, 51	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ (((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊 ∧ ((𝑡 ∨ 𝑢) ∨ 𝑣) ≤ 𝑊 ∧ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ ((𝑡 ∨ 𝑢) ∨ 𝑣))))
53	44, 52	mpbid 232	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊 ∧ ((𝑡 ∨ 𝑢) ∨ 𝑣) ≤ 𝑊 ∧ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ ((𝑡 ∨ 𝑢) ∨ 𝑣)))
54		2lplnj.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (LVols‘𝐾)
55	2, 3, 4, 54	2lplnja 39620	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝑉) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢))) ∧ (((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊 ∧ ((𝑡 ∨ 𝑢) ∨ 𝑣) ≤ 𝑊 ∧ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (((𝑞 ∨ 𝑟) ∨ 𝑠) ∨ ((𝑡 ∨ 𝑢) ∨ 𝑣)) = 𝑊)
56	24, 30, 35, 39, 42, 53, 55	syl321anc 1394	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (((𝑞 ∨ 𝑟) ∨ 𝑠) ∨ ((𝑡 ∨ 𝑢) ∨ 𝑣)) = 𝑊)
57	19, 56	eqtrd 2765	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (𝑋 ∨ 𝑌) = 𝑊)
58	57	3exp 1119	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → ((𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) → ((𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)) → (𝑋 ∨ 𝑌) = 𝑊)))
59	58	rexlimdvv 3194	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)) → (𝑋 ∨ 𝑌) = 𝑊))
60	59	rexlimdva 3135	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠))) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)) → (𝑋 ∨ 𝑌) = 𝑊))
61	60	3exp 1119	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → ((𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) → ((𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)) → (𝑋 ∨ 𝑌) = 𝑊))))
62	61	expdimp 452	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) → ((𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)) → (𝑋 ∨ 𝑌) = 𝑊))))
63	62	rexlimdvv 3194	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)) → (𝑋 ∨ 𝑌) = 𝑊)))
64	63	rexlimdva 3135	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣)) → (𝑋 ∨ 𝑌) = 𝑊)))
65	64	impd 410	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → ((∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ ¬ 𝑠 ≤ (𝑞 ∨ 𝑟) ∧ 𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡 ∨ 𝑢) ∧ 𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣))) → (𝑋 ∨ 𝑌) = 𝑊))
66	15, 65	mpd 15	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑊 ∈ 𝑉) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∨ 𝑌) = 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  Atomscatm 39263  HLchlt 39350  LPlanesclpl 39493  LVolsclvol 39494

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-lat 18398  df-clat 18465  df-oposet 39176  df-ol 39178  df-oml 39179  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351  df-llines 39499  df-lplanes 39500  df-lvols 39501

This theorem is referenced by:  2lplnm2N  39622  dalem13  39677