Theorem 2llnjN 40700

Description: The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
2llnj.l	⊢ ≤ = (le‘𝐾)
2llnj.j	⊢ ∨ = (join‘𝐾)
2llnj.n	⊢ 𝑁 = (LLines‘𝐾)
2llnj.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

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

Proof of Theorem 2llnjN

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

Step	Hyp	Ref	Expression
1		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		2llnj.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
3		eqid 2760	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		2llnj.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
5	1, 2, 3, 4	islln2 40644	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)))))
6		simpr 490	. . . . . . 7 ⊢ ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)))
7	5, 6	biimtrdi 256	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))))
8	1, 2, 3, 4	islln2 40644	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))))
9		simpr 490	. . . . . . 7 ⊢ ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))) → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))
10	8, 9	biimtrdi 256	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))))
11	7, 10	anim12d 621	. . . . 5 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))))
12	11	imp 412	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))))
13	12	3adantr3 1190	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))))
14	13	3adant3 1150	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))))
15		simp2rr 1262	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝑋 = (𝑞 ∨ 𝑟))
16		simp3rr 1266	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝑌 = (𝑠 ∨ 𝑡))
17	15, 16	oveq12d 7438	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → (𝑋 ∨ 𝑌) = ((𝑞 ∨ 𝑟) ∨ (𝑠 ∨ 𝑡)))
18		simp13 1224	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌))
19		breq1 5105	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → (𝑋 ≤ 𝑊 ↔ (𝑞 ∨ 𝑟) ≤ 𝑊))
20		neeq1 3017	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → (𝑋 ≠ 𝑌 ↔ (𝑞 ∨ 𝑟) ≠ 𝑌))
21	19, 20	3anbi13d 1466	. . . . . . . . . . . . . 14 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ 𝑌)))
22		breq1 5105	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → (𝑌 ≤ 𝑊 ↔ (𝑠 ∨ 𝑡) ≤ 𝑊))
23		neeq2 3018	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → ((𝑞 ∨ 𝑟) ≠ 𝑌 ↔ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡)))
24	22, 23	3anbi23d 1467	. . . . . . . . . . . . . 14 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → (((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ 𝑌) ↔ ((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ (𝑠 ∨ 𝑡) ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡))))
25	21, 24	sylan9bb 519	. . . . . . . . . . . . 13 ⊢ ((𝑋 = (𝑞 ∨ 𝑟) ∧ 𝑌 = (𝑠 ∨ 𝑡)) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ (𝑠 ∨ 𝑡) ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡))))
26	15, 16, 25	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ (𝑠 ∨ 𝑡) ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡))))
27	18, 26	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → ((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ (𝑠 ∨ 𝑡) ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡)))
28		simp11 1222	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝐾 ∈ HL)
29		simp123 1326	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝑊 ∈ 𝑃)
30		simp2ll 1259	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝑞 ∈ (Atoms‘𝐾))
31		simp2lr 1260	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝑟 ∈ (Atoms‘𝐾))
32		simp2rl 1261	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝑞 ≠ 𝑟)
33		simp3ll 1263	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝑠 ∈ (Atoms‘𝐾))
34		simp3lr 1264	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝑡 ∈ (Atoms‘𝐾))
35		simp3rl 1265	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → 𝑠 ≠ 𝑡)
36		2llnj.l	. . . . . . . . . . . . . 14 ⊢ ≤ = (le‘𝐾)
37		2llnj.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (LPlanes‘𝐾)
38	36, 2, 3, 4, 37	2llnjaN 40699	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝑃) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑞 ≠ 𝑟) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾) ∧ 𝑠 ≠ 𝑡)) ∧ ((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ (𝑠 ∨ 𝑡) ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡))) → ((𝑞 ∨ 𝑟) ∨ (𝑠 ∨ 𝑡)) = 𝑊)
39	38	ex 418	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝑃) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑞 ≠ 𝑟) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾) ∧ 𝑠 ≠ 𝑡)) → (((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ (𝑠 ∨ 𝑡) ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡)) → ((𝑞 ∨ 𝑟) ∨ (𝑠 ∨ 𝑡)) = 𝑊))
40	28, 29, 30, 31, 32, 33, 34, 35, 39	syl233anc 1426	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → (((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ (𝑠 ∨ 𝑡) ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡)) → ((𝑞 ∨ 𝑟) ∨ (𝑠 ∨ 𝑡)) = 𝑊))
41	27, 40	mpd 16	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → ((𝑞 ∨ 𝑟) ∨ (𝑠 ∨ 𝑡)) = 𝑊)
42	17, 41	eqtrd 2795	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))) → (𝑋 ∨ 𝑌) = 𝑊)
43	42	3exp 1137	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) → (((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))) → (𝑋 ∨ 𝑌) = 𝑊)))
44	43	3impib 1134	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) → (((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))) → (𝑋 ∨ 𝑌) = 𝑊))
45	44	expd 421	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → ((𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)) → (𝑋 ∨ 𝑌) = 𝑊)))
46	45	rexlimdvv 3218	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))) → (∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)) → (𝑋 ∨ 𝑌) = 𝑊))
47	46	3exp 1137	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)) → (∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)) → (𝑋 ∨ 𝑌) = 𝑊))))
48	47	rexlimdvv 3218	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)) → (∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)) → (𝑋 ∨ 𝑌) = 𝑊)))
49	48	impd 416	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → ((∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))) → (𝑋 ∨ 𝑌) = 𝑊))
50	14, 49	mpd 16	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∨ 𝑌) = 𝑊)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17388  lecple 17436  joincjn 18486  Atomscatm 40396  HLchlt 40483  LLinesclln 40624  LPlanesclpl 40625

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-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-proset 18469  df-poset 18488  df-plt 18503  df-lub 18519  df-glb 18520  df-join 18521  df-meet 18522  df-p0 18598  df-lat 18607  df-clat 18674  df-oposet 40309  df-ol 40311  df-oml 40312  df-covers 40399  df-ats 40400  df-atl 40431  df-cvlat 40455  df-hlat 40484  df-llines 40631  df-lplanes 40632

This theorem is used by:  2llnm2N  40701