Theorem 2llnjN 33871

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 2610	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		2llnj.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
3		eqid 2610	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		2llnj.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
5	1, 2, 3, 4	islln2 33815	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)))))
6		simpr 476	. . . . . . 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	syl6bi 242	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))))
8	1, 2, 3, 4	islln2 33815	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))))
9		simpr 476	. . . . . . 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	syl6bi 242	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))))
11	7, 10	anim12d 584	. . . . 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 444	. . . 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 1215	. . 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 1074	. 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 1124	. . . . . . . . . . 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 1128	. . . . . . . . . . 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 6567	. . . . . . . . . 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 1086	. . . . . . . . . . . 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 4586	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → (𝑋 ≤ 𝑊 ↔ (𝑞 ∨ 𝑟) ≤ 𝑊))
20		neeq1 2844	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → (𝑋 ≠ 𝑌 ↔ (𝑞 ∨ 𝑟) ≠ 𝑌))
21	19, 20	3anbi13d 1393	. . . . . . . . . . . . . 14 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ 𝑌)))
22		breq1 4586	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → (𝑌 ≤ 𝑊 ↔ (𝑠 ∨ 𝑡) ≤ 𝑊))
23		neeq2 2845	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → ((𝑞 ∨ 𝑟) ≠ 𝑌 ↔ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡)))
24	22, 23	3anbi23d 1394	. . . . . . . . . . . . . 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 732	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 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 221	. . . . . . . . . . 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 1084	. . . . . . . . . . . 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 1188	. . . . . . . . . . . 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 1121	. . . . . . . . . . . 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 1122	. . . . . . . . . . . 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 1123	. . . . . . . . . . . 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 1125	. . . . . . . . . . . 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 1126	. . . . . . . . . . . 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 1127	. . . . . . . . . . . 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 33870	. . . . . . . . . . . . 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 449	. . . . . . . . . . . 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 1347	. . . . . . . . . . 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 2644	. . . . . . . . 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 1256	. . . . . . . 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 1254	. . . . . . 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 451	. . . . . 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 3019	. . . . 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 1256	. . . 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 3019	. . 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 446	. 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Atomscatm 33568  HLchlt 33655  LLinesclln 33795  LPlanesclpl 33796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803

This theorem is referenced by:  2llnm2N  33872