Theorem 2lplnj 40083

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 2737	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		2lplnj.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
3		2lplnj.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
4		eqid 2737	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		2lplnj.p	. . . . . . . 8 ⊢ 𝑃 = (LPlanes‘𝐾)
6	1, 2, 3, 4, 5	islpln2 39999	. . . . . . 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 39999	. . . . . . 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 7379	. . . . . . . . . . 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 5089	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ((𝑞 ∨ 𝑟) ∨ 𝑠) → (𝑋 ≤ 𝑊 ↔ ((𝑞 ∨ 𝑟) ∨ 𝑠) ≤ 𝑊))
46		neeq1 2995	. . . . . . . . . . . . . . . 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 5089	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 = ((𝑡 ∨ 𝑢) ∨ 𝑣) → (𝑌 ≤ 𝑊 ↔ ((𝑡 ∨ 𝑢) ∨ 𝑣) ≤ 𝑊))
49		neeq2 2996	. . . . . . . . . . . . . . . 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 40082	. . . . . . . . . . . 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 2772	. . . . . . . . . 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 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 3139	. . . . . . 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 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 3139	. . 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 2933  ∃wrex 3062   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  lecple 17221  joincjn 18271  Atomscatm 39726  HLchlt 39813  LPlanesclpl 39955  LVolsclvol 39956

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-proset 18254  df-poset 18273  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-lat 18392  df-clat 18459  df-oposet 39639  df-ol 39641  df-oml 39642  df-covers 39729  df-ats 39730  df-atl 39761  df-cvlat 39785  df-hlat 39814  df-llines 39961  df-lplanes 39962  df-lvols 39963

This theorem is referenced by:  2lplnm2N  40084  dalem13  40139