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Theorem lncmp 35581
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 786 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → (𝑀𝑋) ∈ 𝑁)
2 simpll1 1262 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → 𝐾 ∈ HL)
3 simpll2 1264 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → 𝑋𝐵)
4 lncmp.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 eqid 2817 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
6 eqid 2817 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
7 lncmp.n . . . . . . 7 𝑁 = (Lines‘𝐾)
8 lncmp.m . . . . . . 7 𝑀 = (pmap‘𝐾)
94, 5, 6, 7, 8isline3 35574 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
102, 3, 9syl2anc 575 . . . . 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 223 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))
12 simp3rr 1321 . . . . . . . 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 1358 . . . . . . . . 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 1360 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑌𝐵)
15 simp1rr 1313 . . . . . . . . 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 1318 . . . . . . . . 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 1319 . . . . . . . . 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 1320 . . . . . . . . 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 35162 . . . . . . . . . 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 35087 . . . . . . . . . . 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 1359 . . . . . . . . . 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 35173 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑝 (𝑝(join‘𝐾)𝑞))
2513, 16, 17, 24syl3anc 1483 . . . . . . . . . . 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 4883 . . . . . . . . . 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 1160 . . . . . . . . . 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 17282 . . . . . . . . 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 35087 . . . . . . . . . . 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 35174 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → 𝑞 (𝑝(join‘𝐾)𝑞))
3213, 16, 17, 31syl3anc 1483 . . . . . . . . . . 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 4883 . . . . . . . . . 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 17282 . . . . . . . . 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 35576 . . . . . . . . 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 1513 . . . . . . . 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 2854 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌 ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))) → 𝑋 = 𝑌)
38373expia 1143 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → (((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))) → 𝑋 = 𝑌))
3938expd 402 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) ∧ 𝑋 𝑌) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) → 𝑋 = 𝑌)))
4039rexlimdvv 3236 . . . 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 399 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 𝑌𝑋 = 𝑌))
43 simpl1 1235 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → 𝐾 ∈ HL)
4443hllatd 35162 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → 𝐾 ∈ Lat)
45 simpl2 1237 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → 𝑋𝐵)
464, 19latref 17277 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → 𝑋 𝑋)
4744, 45, 46syl2anc 575 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → 𝑋 𝑋)
48 breq2 4859 . . 3 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
4947, 48syl5ibcom 236 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 = 𝑌𝑋 𝑌))
5042, 49impbid 203 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wne 2989  wrex 3108   class class class wbr 4855  cfv 6110  (class class class)co 6883  Basecbs 16087  lecple 16179  joincjn 17168  Latclat 17269  Atomscatm 35061  HLchlt 35148  Linesclines 35292  pmapcpmap 35295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-riota 6844  df-ov 6886  df-oprab 6887  df-proset 17152  df-poset 17170  df-plt 17182  df-lub 17198  df-glb 17199  df-join 17200  df-meet 17201  df-p0 17263  df-lat 17270  df-clat 17332  df-oposet 34974  df-ol 34976  df-oml 34977  df-covers 35064  df-ats 35065  df-atl 35096  df-cvlat 35120  df-hlat 35149  df-lines 35299  df-pmap 35302
This theorem is referenced by:  2lnat  35582
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