Theorem 2llnjN 40442

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 2762	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		2llnj.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
3		eqid 2762	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		2llnj.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
5	1, 2, 3, 4	islln2 40386	. . . . . . 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 40386	. . . . . . 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 7434	. . . . . . . . . 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 5110	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → (𝑋 ≤ 𝑊 ↔ (𝑞 ∨ 𝑟) ≤ 𝑊))
20		neeq1 3019	. . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → (𝑌 ≤ 𝑊 ↔ (𝑠 ∨ 𝑡) ≤ 𝑊))
23		neeq2 3020	. . . . . . . . . . . . . . 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 40441	. . . . . . . . . . . . 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 2797	. . . . . . . . 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 3220	. . . . 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 3220	. . 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 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  lecple 17353  joincjn 18403  Atomscatm 40138  HLchlt 40225  LLinesclln 40366  LPlanesclpl 40367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-proset 18386  df-poset 18405  df-plt 18420  df-lub 18436  df-glb 18437  df-join 18438  df-meet 18439  df-p0 18515  df-lat 18524  df-clat 18591  df-oposet 40051  df-ol 40053  df-oml 40054  df-covers 40141  df-ats 40142  df-atl 40173  df-cvlat 40197  df-hlat 40226  df-llines 40373  df-lplanes 40374

This theorem is used by:  2llnm2N  40443