Theorem lncmp 39383

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 775	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → (𝑀‘𝑋) ∈ 𝑁)
2		simpll1 1209	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ HL)
3		simpll2 1210	. . . . . 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 2725	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
6		eqid 2725	. . . . . . 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 39376	. . . . . 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 582	. . . . 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 231	. . . 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 1244	. . . . . . . 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 1263	. . . . . . . . 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 1265	. . . . . . . . 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 1236	. . . . . . . . 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 1241	. . . . . . . . 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 1242	. . . . . . . . 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 1243	. . . . . . . . 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 38963	. . . . . . . . . 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 38888	. . . . . . . . . . 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 1264	. . . . . . . . . 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 38974	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑝 ≤ (𝑝(join‘𝐾)𝑞))
25	13, 16, 17, 24	syl3anc 1368	. . . . . . . . . . 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 5177	. . . . . . . . . 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 1134	. . . . . . . . . 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 18441	. . . . . . . . 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 38888	. . . . . . . . . . 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 38975	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑞 ≤ (𝑝(join‘𝐾)𝑞))
32	13, 16, 17, 31	syl3anc 1368	. . . . . . . . . . 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 5177	. . . . . . . . . 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 18441	. . . . . . . . 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 39378	. . . . . . . . 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 1398	. . . . . . . 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 2768	. . . . . . 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 1118	. . . . . 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 414	. . . . 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 3200	. . . 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 411	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
43		simpl1 1188	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝐾 ∈ HL)
44	43	hllatd 38963	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝐾 ∈ Lat)
45		simpl2 1189	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝑋 ∈ 𝐵)
46	4, 19	latref 18436	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
47	44, 45, 46	syl2anc 582	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → 𝑋 ≤ 𝑋)
48		breq2 5153	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
49	47, 48	syl5ibcom 244	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
50	42, 49	impbid 211	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

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 17183  lecple 17243  joincjn 18306  Latclat 18426  Atomscatm 38862  HLchlt 38949  Linesclines 39094  pmapcpmap 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 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-lat 18427  df-clat 18494  df-oposet 38775  df-ol 38777  df-oml 38778  df-covers 38865  df-ats 38866  df-atl 38897  df-cvlat 38921  df-hlat 38950  df-lines 39101  df-pmap 39104

This theorem is referenced by:  2lnat  39384