Theorem lncmp 35858

Description: If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)

Hypotheses

Ref	Expression
lncmp.b	⊢ 𝐵 = (Base‘𝐾)
lncmp.l	⊢ ≤ = (le‘𝐾)
lncmp.n	⊢ 𝑁 = (Lines‘𝐾)
lncmp.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
lncmp	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Proof of Theorem lncmp

Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplrl 797	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → (𝑀‘𝑋) ∈ 𝑁)
2		simpll1 1275	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ HL)
3		simpll2 1277	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
4		lncmp.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
5		eqid 2825	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
6		eqid 2825	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
7		lncmp.n	. . . . . . 7 ⊢ 𝑁 = (Lines‘𝐾)
8		lncmp.m	. . . . . . 7 ⊢ 𝑀 = (pmap‘𝐾)
9	4, 5, 6, 7, 8	isline3 35851	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞))))
10	2, 3, 9	syl2anc 581	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞))))
11	1, 10	mpbid 224	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))
12		simp3rr 1334	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑋 = (𝑝(join‘𝐾)𝑞))
13		simp1l1 1371	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝐾 ∈ HL)
14		simp1l3 1373	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑌 ∈ 𝐵)
15		simp1rr 1326	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → (𝑀‘𝑌) ∈ 𝑁)
16		simp3ll 1331	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 ∈ (Atoms‘𝐾))
17		simp3lr 1332	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞 ∈ (Atoms‘𝐾))
18		simp3rl 1333	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 ≠ 𝑞)
19		lncmp.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
20	13	hllatd 35439	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝐾 ∈ Lat)
21	4, 6	atbase 35364	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
22	16, 21	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 ∈ 𝐵)
23		simp1l2 1372	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑋 ∈ 𝐵)
24	19, 5, 6	hlatlej1 35450	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑝 ≤ (𝑝(join‘𝐾)𝑞))
25	13, 16, 17, 24	syl3anc 1496	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 ≤ (𝑝(join‘𝐾)𝑞))
26	25, 12	breqtrrd 4901	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 ≤ 𝑋)
27		simp2 1173	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑋 ≤ 𝑌)
28	4, 19, 20, 22, 23, 14, 26, 27	lattrd 17411	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 ≤ 𝑌)
29	4, 6	atbase 35364	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ 𝐵)
30	17, 29	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞 ∈ 𝐵)
31	19, 5, 6	hlatlej2 35451	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑞 ≤ (𝑝(join‘𝐾)𝑞))
32	13, 16, 17, 31	syl3anc 1496	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞 ≤ (𝑝(join‘𝐾)𝑞))
33	32, 12	breqtrrd 4901	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞 ≤ 𝑋)
34	4, 19, 20, 30, 23, 14, 33, 27	lattrd 17411	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞 ≤ 𝑌)
35	4, 19, 5, 6, 7, 8	lneq2at 35853	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ (𝑀‘𝑌) ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾) ∧ 𝑝 ≠ 𝑞) ∧ (𝑝 ≤ 𝑌 ∧ 𝑞 ≤ 𝑌)) → 𝑌 = (𝑝(join‘𝐾)𝑞))
36	13, 14, 15, 16, 17, 18, 28, 34, 35	syl332anc 1526	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑌 = (𝑝(join‘𝐾)𝑞))
37	12, 36	eqtr4d 2864	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑋 = 𝑌)
38	37	3expia 1156	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → (((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞))) → 𝑋 = 𝑌))
39	38	expd 406	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) → 𝑋 = 𝑌)))
40	39	rexlimdvv 3247	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) → 𝑋 = 𝑌))
41	11, 40	mpd 15	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → 𝑋 = 𝑌)
42	41	ex 403	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
43		simpl1 1248	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝐾 ∈ HL)
44	43	hllatd 35439	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝐾 ∈ Lat)
45		simpl2 1250	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝑋 ∈ 𝐵)
46	4, 19	latref 17406	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
47	44, 45, 46	syl2anc 581	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝑋 ≤ 𝑋)
48		breq2 4877	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
49	47, 48	syl5ibcom 237	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
50	42, 49	impbid 204	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166   ≠ wne 2999  ∃wrex 3118   class class class wbr 4873  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  joincjn 17297  Latclat 17398  Atomscatm 35338  HLchlt 35425  Linesclines 35569  pmapcpmap 35572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-lat 17399  df-clat 17461  df-oposet 35251  df-ol 35253  df-oml 35254  df-covers 35341  df-ats 35342  df-atl 35373  df-cvlat 35397  df-hlat 35426  df-lines 35576  df-pmap 35579

This theorem is referenced by:  2lnat  35859