Theorem 2lplnj 40066

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 2736	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		2lplnj.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
3		2lplnj.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
4		eqid 2736	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		2lplnj.p	. . . . . . . 8 ⊢ 𝑃 = (LPlanes‘𝐾)
6	1, 2, 3, 4, 5	islpln2 39982	. . . . . . 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 39982	. . . . . . 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 610	. . . . 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 1173	. . 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 1133	. 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 1258	. . . . . . . . . . . . 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 1134	. . . . . . . . . . . 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 1213	. . . . . . . . . . . 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 7385	. . . . . . . . . . 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 1205	. . . . . . . . . . . . . . 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 1309	. . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . 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 1201	. . . . . . . . . . . . . . 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 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))) → 𝑟 ∈ (Atoms‘𝐾))
27		simp2rr 1245	. . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . 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 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)) → 𝑞 ≠ 𝑟)
32	31	3ad2ant1 1134	. . . . . . . . . . . . 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 1257	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . 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 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))) → 𝑡 ∈ (Atoms‘𝐾))
37		simp2l 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))) → 𝑢 ∈ (Atoms‘𝐾))
38		simp2r 1202	. . . . . . . . . . . . 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 1129	. . . . . . . . . . . 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 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))) → 𝑡 ≠ 𝑢)
41		simp32 1212	. . . . . . . . . . . . 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 1252	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . 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 5088	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → (𝑋 ≤ 𝑊 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊))
46		neeq1 2994	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → (𝑋 ≠ 𝑌 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌))
47	45, 46	3anbi13d 1441	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ (((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌)))
48		breq1 5088	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣) → (𝑌 ≤ 𝑊 ↔ ((𝑡 ∨ 𝑢) ∨ 𝑣) ≤ 𝑊))
49		neeq2 2995	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣) → (((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ ((𝑡 ∨ 𝑢) ∨ 𝑣)))
50	48, 49	3anbi23d 1442	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 40065	. . . . . . . . . . . 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 1395	. . . . . . . . . . 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 2771	. . . . . . . . . 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 1120	. . . . . . . . 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 3193	. . . . . . . 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 3138	. . . . . . 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 1120	. . . . . 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 3193	. . . 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 3138	. . 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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  lecple 17227  joincjn 18277  Atomscatm 39709  HLchlt 39796  LPlanesclpl 39938  LVolsclvol 39939

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-proset 18260  df-poset 18279  df-plt 18294  df-lub 18310  df-glb 18311  df-join 18312  df-meet 18313  df-p0 18389  df-lat 18398  df-clat 18465  df-oposet 39622  df-ol 39624  df-oml 39625  df-covers 39712  df-ats 39713  df-atl 39744  df-cvlat 39768  df-hlat 39797  df-llines 39944  df-lplanes 39945  df-lvols 39946

This theorem is referenced by:  2lplnm2N  40067  dalem13  40122