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Theorem lncmp 37547
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 ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 𝑌𝑋 = 𝑌))

Proof of Theorem lncmp
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplrl 777 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → (𝑀𝑋) ∈ 𝑁)
2 simpll1 1214 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → 𝐾 ∈ HL)
3 simpll2 1215 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → 𝑋𝐵)
4 lncmp.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 eqid 2738 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
6 eqid 2738 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
7 lncmp.n . . . . . . 7 𝑁 = (Lines‘𝐾)
8 lncmp.m . . . . . . 7 𝑀 = (pmap‘𝐾)
94, 5, 6, 7, 8isline3 37540 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
102, 3, 9syl2anc 587 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
111, 10mpbid 235 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))
12 simp3rr 1249 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑋 = (𝑝(join‘𝐾)𝑞))
13 simp1l1 1268 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝐾 ∈ HL)
14 simp1l3 1270 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑌𝐵)
15 simp1rr 1241 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → (𝑀𝑌) ∈ 𝑁)
16 simp3ll 1246 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 ∈ (Atoms‘𝐾))
17 simp3lr 1247 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞 ∈ (Atoms‘𝐾))
18 simp3rl 1248 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝𝑞)
19 lncmp.l . . . . . . . . . 10 = (le‘𝐾)
2013hllatd 37128 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝐾 ∈ Lat)
214, 6atbase 37053 . . . . . . . . . . 11 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
2216, 21syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝𝐵)
23 simp1l2 1269 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑋𝐵)
2419, 5, 6hlatlej1 37139 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑝 (𝑝(join‘𝐾)𝑞))
2513, 16, 17, 24syl3anc 1373 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 (𝑝(join‘𝐾)𝑞))
2625, 12breqtrrd 5090 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 𝑋)
27 simp2 1139 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑋 𝑌)
284, 19, 20, 22, 23, 14, 26, 27lattrd 17965 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑝 𝑌)
294, 6atbase 37053 . . . . . . . . . . 11 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
3017, 29syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞𝐵)
3119, 5, 6hlatlej2 37140 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑞 (𝑝(join‘𝐾)𝑞))
3213, 16, 17, 31syl3anc 1373 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞 (𝑝(join‘𝐾)𝑞))
3332, 12breqtrrd 5090 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞 𝑋)
344, 19, 20, 30, 23, 14, 33, 27lattrd 17965 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑞 𝑌)
354, 19, 5, 6, 7, 8lneq2at 37542 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑌𝐵 ∧ (𝑀𝑌) ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾) ∧ 𝑝𝑞) ∧ (𝑝 𝑌𝑞 𝑌)) → 𝑌 = (𝑝(join‘𝐾)𝑞))
3613, 14, 15, 16, 17, 18, 28, 34, 35syl332anc 1403 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑌 = (𝑝(join‘𝐾)𝑞))
3712, 36eqtr4d 2781 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑋 = 𝑌)
38373expia 1123 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → (((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))) → 𝑋 = 𝑌))
3938expd 419 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) → 𝑋 = 𝑌)))
4039rexlimdvv 3219 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) → 𝑋 = 𝑌))
4111, 40mpd 15 . . 3 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → 𝑋 = 𝑌)
4241ex 416 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 𝑌𝑋 = 𝑌))
43 simpl1 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → 𝐾 ∈ HL)
4443hllatd 37128 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → 𝐾 ∈ Lat)
45 simpl2 1194 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → 𝑋𝐵)
464, 19latref 17960 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → 𝑋 𝑋)
4744, 45, 46syl2anc 587 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → 𝑋 𝑋)
48 breq2 5066 . . 3 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
4947, 48syl5ibcom 248 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 = 𝑌𝑋 𝑌))
5042, 49impbid 215 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2111  wne 2941  wrex 3063   class class class wbr 5062  cfv 6389  (class class class)co 7222  Basecbs 16773  lecple 16822  joincjn 17831  Latclat 17950  Atomscatm 37027  HLchlt 37114  Linesclines 37258  pmapcpmap 37261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5188  ax-sep 5201  ax-nul 5208  ax-pow 5267  ax-pr 5331  ax-un 7532
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-iun 4915  df-br 5063  df-opab 5125  df-mpt 5145  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-f1 6394  df-fo 6395  df-f1o 6396  df-fv 6397  df-riota 7179  df-ov 7225  df-oprab 7226  df-proset 17815  df-poset 17833  df-plt 17849  df-lub 17865  df-glb 17866  df-join 17867  df-meet 17868  df-p0 17944  df-lat 17951  df-clat 18018  df-oposet 36940  df-ol 36942  df-oml 36943  df-covers 37030  df-ats 37031  df-atl 37062  df-cvlat 37086  df-hlat 37115  df-lines 37265  df-pmap 37268
This theorem is referenced by:  2lnat  37548
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