Theorem lncmp 39750

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 776	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → (𝑀‘𝑋) ∈ 𝑁)
2		simpll1 1213	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ HL)
3		simpll2 1214	. . . . . 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 2729	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
6		eqid 2729	. . . . . . 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 39743	. . . . . 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 584	. . . . 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 232	. . . 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 1248	. . . . . . . 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 1267	. . . . . . . . 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 1269	. . . . . . . . 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 1240	. . . . . . . . 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 1245	. . . . . . . . 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 1246	. . . . . . . . 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 1247	. . . . . . . . 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 39330	. . . . . . . . . 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 39255	. . . . . . . . . . 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 1268	. . . . . . . . . 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 39341	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑝 ≤ (𝑝(join‘𝐾)𝑞))
25	13, 16, 17, 24	syl3anc 1373	. . . . . . . . . . 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 5130	. . . . . . . . . 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 1137	. . . . . . . . . 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 18381	. . . . . . . . 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 39255	. . . . . . . . . . 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 39342	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑞 ≤ (𝑝(join‘𝐾)𝑞))
32	13, 16, 17, 31	syl3anc 1373	. . . . . . . . . . 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 5130	. . . . . . . . . 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 18381	. . . . . . . . 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 39745	. . . . . . . . 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 1403	. . . . . . . 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 2767	. . . . . . 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 1121	. . . . . 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 415	. . . . 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 3191	. . . 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 412	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
43		simpl1 1192	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝐾 ∈ HL)
44	43	hllatd 39330	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝐾 ∈ Lat)
45		simpl2 1193	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝑋 ∈ 𝐵)
46	4, 19	latref 18376	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
47	44, 45, 46	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝑋 ≤ 𝑋)
48		breq2 5106	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
49	47, 48	syl5ibcom 245	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
50	42, 49	impbid 212	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  lecple 17203  joincjn 18248  Latclat 18366  Atomscatm 39229  HLchlt 39316  Linesclines 39461  pmapcpmap 39464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-proset 18231  df-poset 18250  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-lat 18367  df-clat 18434  df-oposet 39142  df-ol 39144  df-oml 39145  df-covers 39232  df-ats 39233  df-atl 39264  df-cvlat 39288  df-hlat 39317  df-lines 39468  df-pmap 39471

This theorem is referenced by:  2lnat  39751