Theorem 2llnjN 40139

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 2756	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		2llnj.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
3		eqid 2756	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		2llnj.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
5	1, 2, 3, 4	islln2 40083	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)))))
6		simpr 487	. . . . . . 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 255	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))))
8	1, 2, 3, 4	islln2 40083	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))))
9		simpr 487	. . . . . . 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 255	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))))
11	7, 10	anim12d 617	. . . . 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 409	. . . 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 1181	. . 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 1141	. 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 1253	. . . . . . . . . . 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 1257	. . . . . . . . . . 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 7403	. . . . . . . . . 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 1215	. . . . . . . . . . . 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 5097	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → (𝑋 ≤ 𝑊 ↔ (𝑞 ∨ 𝑟) ≤ 𝑊))
20		neeq1 3013	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → (𝑋 ≠ 𝑌 ↔ (𝑞 ∨ 𝑟) ≠ 𝑌))
21	19, 20	3anbi13d 1453	. . . . . . . . . . . . . 14 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ 𝑌)))
22		breq1 5097	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → (𝑌 ≤ 𝑊 ↔ (𝑠 ∨ 𝑡) ≤ 𝑊))
23		neeq2 3014	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → ((𝑞 ∨ 𝑟) ≠ 𝑌 ↔ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡)))
24	22, 23	3anbi23d 1454	. . . . . . . . . . . . . 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 516	. . . . . . . . . . . . 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 592	. . . . . . . . . . . 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 234	. . . . . . . . . . 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 1213	. . . . . . . . . . . 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 1317	. . . . . . . . . . . 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 1250	. . . . . . . . . . . 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 1251	. . . . . . . . . . . 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 1252	. . . . . . . . . . . 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 1254	. . . . . . . . . . . 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 1255	. . . . . . . . . . . 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 1256	. . . . . . . . . . . 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 40138	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 1414	. . . . . . . . . . 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 15	. . . . . . . . . 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 2791	. . . . . . . . 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 1128	. . . . . . . 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 1125	. . . . . . 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 418	. . . . . 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 3212	. . . . 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 1128	. . . 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 3212	. . 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 413	. 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 15	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∨ 𝑌) = 𝑊)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ≠ wne 2951  ∃wrex 3080   class class class wbr 5094  ‘cfv 6510  (class class class)co 7385  Basecbs 17221  lecple 17269  joincjn 18319  Atomscatm 39835  HLchlt 39922  LLinesclln 40063  LPlanesclpl 40064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-proset 18302  df-poset 18321  df-plt 18336  df-lub 18352  df-glb 18353  df-join 18354  df-meet 18355  df-p0 18431  df-lat 18440  df-clat 18507  df-oposet 39748  df-ol 39750  df-oml 39751  df-covers 39838  df-ats 39839  df-atl 39870  df-cvlat 39894  df-hlat 39923  df-llines 40070  df-lplanes 40071

This theorem is referenced by:  2llnm2N  40140