Theorem 2llnjN 39172

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 2725	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		2llnj.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
3		eqid 2725	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		2llnj.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
5	1, 2, 3, 4	islln2 39116	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟)))))
6		simpr 483	. . . . . . 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 252	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞 ∨ 𝑟))))
8	1, 2, 3, 4	islln2 39116	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡)))))
9		simpr 483	. . . . . . 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 252	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ 𝑌 = (𝑠 ∨ 𝑡))))
11	7, 10	anim12d 607	. . . . 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 405	. . . 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 1168	. . 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 1129	. 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 1240	. . . . . . . . . . 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 1244	. . . . . . . . . . 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 7437	. . . . . . . . . 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 1202	. . . . . . . . . . . 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 5152	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → (𝑋 ≤ 𝑊 ↔ (𝑞 ∨ 𝑟) ≤ 𝑊))
20		neeq1 2992	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → (𝑋 ≠ 𝑌 ↔ (𝑞 ∨ 𝑟) ≠ 𝑌))
21	19, 20	3anbi13d 1434	. . . . . . . . . . . . . 14 ⊢ (𝑋 = (𝑞 ∨ 𝑟) → ((𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌) ↔ ((𝑞 ∨ 𝑟) ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ (𝑞 ∨ 𝑟) ≠ 𝑌)))
22		breq1 5152	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → (𝑌 ≤ 𝑊 ↔ (𝑠 ∨ 𝑡) ≤ 𝑊))
23		neeq2 2993	. . . . . . . . . . . . . . 15 ⊢ (𝑌 = (𝑠 ∨ 𝑡) → ((𝑞 ∨ 𝑟) ≠ 𝑌 ↔ (𝑞 ∨ 𝑟) ≠ (𝑠 ∨ 𝑡)))
24	22, 23	3anbi23d 1435	. . . . . . . . . . . . . 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 508	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 1200	. . . . . . . . . . . 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 1304	. . . . . . . . . . . 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 1237	. . . . . . . . . . . 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 1238	. . . . . . . . . . . 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 1239	. . . . . . . . . . . 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 1241	. . . . . . . . . . . 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 1242	. . . . . . . . . . . 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 1243	. . . . . . . . . . . 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 39171	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 1396	. . . . . . . . . . 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 2765	. . . . . . . . 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 1116	. . . . . . . 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 1113	. . . . . . 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 414	. . . . . 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 3200	. . . . 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 1116	. . . 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 3200	. . 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 409	. 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 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∃wrex 3059   class class class wbr 5149  ‘cfv 6549  (class class class)co 7419  Basecbs 17188  lecple 17248  joincjn 18311  Atomscatm 38867  HLchlt 38954  LLinesclln 39096  LPlanesclpl 39097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-proset 18295  df-poset 18313  df-plt 18330  df-lub 18346  df-glb 18347  df-join 18348  df-meet 18349  df-p0 18425  df-lat 18432  df-clat 18499  df-oposet 38780  df-ol 38782  df-oml 38783  df-covers 38870  df-ats 38871  df-atl 38902  df-cvlat 38926  df-hlat 38955  df-llines 39103  df-lplanes 39104

This theorem is referenced by:  2llnm2N  39173