Theorem 2lplnj 40112

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 2739	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		2lplnj.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
3		2lplnj.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
4		eqid 2739	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		2lplnj.p	. . . . . . . 8 ⊢ 𝑃 = (LPlanes‘𝐾)
6	1, 2, 3, 4, 5	islpln2 40028	. . . . . . 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 485	. . . . . . 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 254	. . . . . 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 40028	. . . . . . 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 485	. . . . . . 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 254	. . . . . 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 615	. . . . 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 407	. . . 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 1178	. . 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 1138	. 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 1263	. . . . . . . . . . . . 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 1139	. . . . . . . . . . . 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 1218	. . . . . . . . . . . 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 7374	. . . . . . . . . . 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 1210	. . . . . . . . . . . . . . 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 1314	. . . . . . . . . . . . . . 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 516	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 1139	. . . . . . . . . . . 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 1206	. . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . 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 1250	. . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 1139	. . . . . . . . . . . 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 1261	. . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . 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 1262	. . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . 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 516	. . . . . . . . . . . 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 1205	. . . . . . . . . . . . 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 1206	. . . . . . . . . . . . 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 1207	. . . . . . . . . . . . 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 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))) → (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)))
40		simp31 1216	. . . . . . . . . . . . 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 1217	. . . . . . . . . . . . 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 516	. . . . . . . . . . . 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 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)) → (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌))
44	43	3ad2ant1 1139	. . . . . . . . . . . . 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 5075	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → (𝑋 ≤ 𝑊 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊))
46		neeq1 2996	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → (𝑋 ≠ 𝑌 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌))
47	45, 46	3anbi13d 1446	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ (((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌)))
48		breq1 5075	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣) → (𝑌 ≤ 𝑊 ↔ ((𝑡 ∨ 𝑢) ∨ 𝑣) ≤ 𝑊))
49		neeq2 2997	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣) → (((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ 𝑌 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≠ ((𝑡 ∨ 𝑢) ∨ 𝑣)))
50	48, 49	3anbi23d 1447	. . . . . . . . . . . . . . 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 514	. . . . . . . . . . . . . 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 590	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 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 40111	. . . . . . . . . . . 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 1400	. . . . . . . . . . 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 2774	. . . . . . . . . 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 1125	. . . . . . . . 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 3195	. . . . . . . 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 3140	. . . . . . 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 1125	. . . . . 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 453	. . . . 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 3195	. . . 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 3140	. . 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 411	. 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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  lecple 17218  joincjn 18268  Atomscatm 39755  HLchlt 39842  LPlanesclpl 39984  LVolsclvol 39985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18389  df-clat 18456  df-oposet 39668  df-ol 39670  df-oml 39671  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843  df-llines 39990  df-lplanes 39991  df-lvols 39992

This theorem is referenced by:  2lplnm2N  40113  dalem13  40168