Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlema1N Structured version   Visualization version   GIF version

Theorem cdlema1N 37036
Description: A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlema1.b 𝐵 = (Base‘𝐾)
cdlema1.l = (le‘𝐾)
cdlema1.j = (join‘𝐾)
cdlema1.m = (meet‘𝐾)
cdlema1.a 𝐴 = (Atoms‘𝐾)
cdlema1.n 𝑁 = (Lines‘𝐾)
cdlema1.f 𝐹 = (pmap‘𝐾)
Assertion
Ref Expression
cdlema1N (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) = (𝑋 𝑌))

Proof of Theorem cdlema1N
StepHypRef Expression
1 cdlema1.b . 2 𝐵 = (Base‘𝐾)
2 cdlema1.l . 2 = (le‘𝐾)
3 simp11 1200 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝐾 ∈ HL)
43hllatd 36609 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝐾 ∈ Lat)
5 simp12 1201 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑋𝐵)
6 simp23 1205 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅𝐴)
7 cdlema1.a . . . . 5 𝐴 = (Atoms‘𝐾)
81, 7atbase 36534 . . . 4 (𝑅𝐴𝑅𝐵)
96, 8syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅𝐵)
10 cdlema1.j . . . 4 = (join‘𝐾)
111, 10latjcl 17661 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑅𝐵) → (𝑋 𝑅) ∈ 𝐵)
124, 5, 9, 11syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) ∈ 𝐵)
13 simp13 1202 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑌𝐵)
141, 10latjcl 17661 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
154, 5, 13, 14syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) ∈ 𝐵)
161, 2, 10latlej1 17670 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))
174, 5, 13, 16syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑋 (𝑋 𝑌))
18 simp21 1203 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑃𝐴)
191, 7atbase 36534 . . . . . 6 (𝑃𝐴𝑃𝐵)
2018, 19syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑃𝐵)
21 simp22 1204 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄𝐴)
221, 7atbase 36534 . . . . . 6 (𝑄𝐴𝑄𝐵)
2321, 22syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄𝐵)
241, 10latjcl 17661 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
254, 20, 23, 24syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑄) ∈ 𝐵)
26 simp31r 1294 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅 (𝑃 𝑄))
27 simp32l 1295 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑃 𝑋)
28 simp32r 1296 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 𝑌)
291, 2, 10latjlej12 17677 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑋𝐵) ∧ (𝑄𝐵𝑌𝐵)) → ((𝑃 𝑋𝑄 𝑌) → (𝑃 𝑄) (𝑋 𝑌)))
304, 20, 5, 23, 13, 29syl122anc 1376 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → ((𝑃 𝑋𝑄 𝑌) → (𝑃 𝑄) (𝑋 𝑌)))
3127, 28, 30mp2and 698 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑄) (𝑋 𝑌))
321, 2, 4, 9, 25, 15, 26, 31lattrd 17668 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅 (𝑋 𝑌))
331, 2, 10latjle12 17672 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑅𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → ((𝑋 (𝑋 𝑌) ∧ 𝑅 (𝑋 𝑌)) ↔ (𝑋 𝑅) (𝑋 𝑌)))
344, 5, 9, 15, 33syl13anc 1369 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → ((𝑋 (𝑋 𝑌) ∧ 𝑅 (𝑋 𝑌)) ↔ (𝑋 𝑅) (𝑋 𝑌)))
3517, 32, 34mpbi2and 711 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) (𝑋 𝑌))
361, 2, 10latlej1 17670 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑅𝐵) → 𝑋 (𝑋 𝑅))
374, 5, 9, 36syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑋 (𝑋 𝑅))
38 simp331 1323 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝐹𝑌) ∈ 𝑁)
39 simp332 1324 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) ∈ 𝐴)
40 simp333 1325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → ¬ 𝑄 𝑋)
41 cdlema1.m . . . . . . . . . 10 = (meet‘𝐾)
421, 2, 41latmle1 17686 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑋)
434, 5, 13, 42syl3anc 1368 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) 𝑋)
44 breq1 5055 . . . . . . . 8 (𝑄 = (𝑋 𝑌) → (𝑄 𝑋 ↔ (𝑋 𝑌) 𝑋))
4543, 44syl5ibrcom 250 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑄 = (𝑋 𝑌) → 𝑄 𝑋))
4645necon3bd 3028 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (¬ 𝑄 𝑋𝑄 ≠ (𝑋 𝑌)))
4740, 46mpd 15 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 ≠ (𝑋 𝑌))
481, 2, 41latmle2 17687 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
494, 5, 13, 48syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) 𝑌)
50 cdlema1.n . . . . . 6 𝑁 = (Lines‘𝐾)
51 cdlema1.f . . . . . 6 𝐹 = (pmap‘𝐾)
521, 2, 10, 7, 50, 51lneq2at 37023 . . . . 5 (((𝐾 ∈ HL ∧ 𝑌𝐵 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑄𝐴 ∧ (𝑋 𝑌) ∈ 𝐴𝑄 ≠ (𝑋 𝑌)) ∧ (𝑄 𝑌 ∧ (𝑋 𝑌) 𝑌)) → 𝑌 = (𝑄 (𝑋 𝑌)))
533, 13, 38, 21, 39, 47, 28, 49, 52syl332anc 1398 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑌 = (𝑄 (𝑋 𝑌)))
541, 10latjcl 17661 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑅𝐵) → (𝑃 𝑅) ∈ 𝐵)
554, 20, 9, 54syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑅) ∈ 𝐵)
566, 21, 183jca 1125 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑅𝐴𝑄𝐴𝑃𝐴))
57 simp31l 1293 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅𝑃)
583, 56, 573jca 1125 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃))
592, 10, 7hlatexch1 36640 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃) → (𝑅 (𝑃 𝑄) → 𝑄 (𝑃 𝑅)))
6058, 26, 59sylc 65 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 (𝑃 𝑅))
6120, 5, 93jca 1125 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃𝐵𝑋𝐵𝑅𝐵))
624, 61jca 515 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝐾 ∈ Lat ∧ (𝑃𝐵𝑋𝐵𝑅𝐵)))
631, 2, 10latjlej1 17675 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑋𝐵𝑅𝐵)) → (𝑃 𝑋 → (𝑃 𝑅) (𝑋 𝑅)))
6462, 27, 63sylc 65 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑅) (𝑋 𝑅))
651, 2, 4, 23, 55, 12, 60, 64lattrd 17668 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 (𝑋 𝑅))
661, 2, 10, 41latmlej11 17700 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑅𝐵)) → (𝑋 𝑌) (𝑋 𝑅))
674, 5, 13, 9, 66syl13anc 1369 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) (𝑋 𝑅))
681, 41latmcl 17662 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
694, 5, 13, 68syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) ∈ 𝐵)
701, 2, 10latjle12 17672 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄𝐵 ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑅) ∈ 𝐵)) → ((𝑄 (𝑋 𝑅) ∧ (𝑋 𝑌) (𝑋 𝑅)) ↔ (𝑄 (𝑋 𝑌)) (𝑋 𝑅)))
714, 23, 69, 12, 70syl13anc 1369 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → ((𝑄 (𝑋 𝑅) ∧ (𝑋 𝑌) (𝑋 𝑅)) ↔ (𝑄 (𝑋 𝑌)) (𝑋 𝑅)))
7265, 67, 71mpbi2and 711 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑄 (𝑋 𝑌)) (𝑋 𝑅))
7353, 72eqbrtrd 5074 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑌 (𝑋 𝑅))
741, 2, 10latjle12 17672 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 𝑅) ∈ 𝐵)) → ((𝑋 (𝑋 𝑅) ∧ 𝑌 (𝑋 𝑅)) ↔ (𝑋 𝑌) (𝑋 𝑅)))
754, 5, 13, 12, 74syl13anc 1369 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → ((𝑋 (𝑋 𝑅) ∧ 𝑌 (𝑋 𝑅)) ↔ (𝑋 𝑌) (𝑋 𝑅)))
7637, 73, 75mpbi2and 711 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) (𝑋 𝑅))
771, 2, 4, 12, 15, 35, 76latasymd 17667 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) = (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  Latclat 17655  Atomscatm 36508  HLchlt 36595  Linesclines 36739  pmapcpmap 36742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-clat 17718  df-oposet 36421  df-ol 36423  df-oml 36424  df-covers 36511  df-ats 36512  df-atl 36543  df-cvlat 36567  df-hlat 36596  df-lines 36746  df-pmap 36749
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator