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Theorem cdlema1N 35580
 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 1246 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝐾 ∈ HL)
4 hllat 35153 . . 3 (𝐾 ∈ HL → 𝐾 ∈ Lat)
53, 4syl 17 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝐾 ∈ Lat)
6 simp12 1247 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑋𝐵)
7 simp23 1251 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅𝐴)
8 cdlema1.a . . . . 5 𝐴 = (Atoms‘𝐾)
91, 8atbase 35079 . . . 4 (𝑅𝐴𝑅𝐵)
107, 9syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅𝐵)
11 cdlema1.j . . . 4 = (join‘𝐾)
121, 11latjcl 17252 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑅𝐵) → (𝑋 𝑅) ∈ 𝐵)
135, 6, 10, 12syl3anc 1477 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) ∈ 𝐵)
14 simp13 1248 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑌𝐵)
151, 11latjcl 17252 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
165, 6, 14, 15syl3anc 1477 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) ∈ 𝐵)
171, 2, 11latlej1 17261 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))
185, 6, 14, 17syl3anc 1477 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑋 (𝑋 𝑌))
19 simp21 1249 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑃𝐴)
201, 8atbase 35079 . . . . . 6 (𝑃𝐴𝑃𝐵)
2119, 20syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑃𝐵)
22 simp22 1250 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄𝐴)
231, 8atbase 35079 . . . . . 6 (𝑄𝐴𝑄𝐵)
2422, 23syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄𝐵)
251, 11latjcl 17252 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
265, 21, 24, 25syl3anc 1477 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑄) ∈ 𝐵)
27 simp31r 1382 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅 (𝑃 𝑄))
28 simp32l 1383 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑃 𝑋)
29 simp32r 1384 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 𝑌)
301, 2, 11latjlej12 17268 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑋𝐵) ∧ (𝑄𝐵𝑌𝐵)) → ((𝑃 𝑋𝑄 𝑌) → (𝑃 𝑄) (𝑋 𝑌)))
315, 21, 6, 24, 14, 30syl122anc 1486 . . . . 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)))
3228, 29, 31mp2and 717 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑄) (𝑋 𝑌))
331, 2, 5, 10, 26, 16, 27, 32lattrd 17259 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅 (𝑋 𝑌))
341, 2, 11latjle12 17263 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑅𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → ((𝑋 (𝑋 𝑌) ∧ 𝑅 (𝑋 𝑌)) ↔ (𝑋 𝑅) (𝑋 𝑌)))
355, 6, 10, 16, 34syl13anc 1479 . . 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)))
3618, 33, 35mpbi2and 994 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) (𝑋 𝑌))
371, 2, 11latlej1 17261 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑅𝐵) → 𝑋 (𝑋 𝑅))
385, 6, 10, 37syl3anc 1477 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑋 (𝑋 𝑅))
39 simp331 1411 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝐹𝑌) ∈ 𝑁)
40 simp332 1412 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) ∈ 𝐴)
41 simp333 1413 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → ¬ 𝑄 𝑋)
42 cdlema1.m . . . . . . . . . 10 = (meet‘𝐾)
431, 2, 42latmle1 17277 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑋)
445, 6, 14, 43syl3anc 1477 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) 𝑋)
45 breq1 4807 . . . . . . . 8 (𝑄 = (𝑋 𝑌) → (𝑄 𝑋 ↔ (𝑋 𝑌) 𝑋))
4644, 45syl5ibrcom 237 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑄 = (𝑋 𝑌) → 𝑄 𝑋))
4746necon3bd 2946 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (¬ 𝑄 𝑋𝑄 ≠ (𝑋 𝑌)))
4841, 47mpd 15 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 ≠ (𝑋 𝑌))
491, 2, 42latmle2 17278 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
505, 6, 14, 49syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) 𝑌)
51 cdlema1.n . . . . . 6 𝑁 = (Lines‘𝐾)
52 cdlema1.f . . . . . 6 𝐹 = (pmap‘𝐾)
531, 2, 11, 8, 51, 52lneq2at 35567 . . . . 5 (((𝐾 ∈ HL ∧ 𝑌𝐵 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑄𝐴 ∧ (𝑋 𝑌) ∈ 𝐴𝑄 ≠ (𝑋 𝑌)) ∧ (𝑄 𝑌 ∧ (𝑋 𝑌) 𝑌)) → 𝑌 = (𝑄 (𝑋 𝑌)))
543, 14, 39, 22, 40, 48, 29, 50, 53syl332anc 1508 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑌 = (𝑄 (𝑋 𝑌)))
551, 11latjcl 17252 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑅𝐵) → (𝑃 𝑅) ∈ 𝐵)
565, 21, 10, 55syl3anc 1477 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑅) ∈ 𝐵)
577, 22, 193jca 1123 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑅𝐴𝑄𝐴𝑃𝐴))
58 simp31l 1381 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅𝑃)
593, 57, 583jca 1123 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃))
602, 11, 8hlatexch1 35184 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃) → (𝑅 (𝑃 𝑄) → 𝑄 (𝑃 𝑅)))
6159, 27, 60sylc 65 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 (𝑃 𝑅))
6221, 6, 103jca 1123 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃𝐵𝑋𝐵𝑅𝐵))
635, 62jca 555 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝐾 ∈ Lat ∧ (𝑃𝐵𝑋𝐵𝑅𝐵)))
641, 2, 11latjlej1 17266 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑋𝐵𝑅𝐵)) → (𝑃 𝑋 → (𝑃 𝑅) (𝑋 𝑅)))
6563, 28, 64sylc 65 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑅) (𝑋 𝑅))
661, 2, 5, 24, 56, 13, 61, 65lattrd 17259 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 (𝑋 𝑅))
671, 2, 11, 42latmlej11 17291 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑅𝐵)) → (𝑋 𝑌) (𝑋 𝑅))
685, 6, 14, 10, 67syl13anc 1479 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) (𝑋 𝑅))
691, 42latmcl 17253 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
705, 6, 14, 69syl3anc 1477 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) ∈ 𝐵)
711, 2, 11latjle12 17263 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄𝐵 ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑅) ∈ 𝐵)) → ((𝑄 (𝑋 𝑅) ∧ (𝑋 𝑌) (𝑋 𝑅)) ↔ (𝑄 (𝑋 𝑌)) (𝑋 𝑅)))
725, 24, 70, 13, 71syl13anc 1479 . . . . 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)))
7366, 68, 72mpbi2and 994 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑄 (𝑋 𝑌)) (𝑋 𝑅))
7454, 73eqbrtrd 4826 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑌 (𝑋 𝑅))
751, 2, 11latjle12 17263 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 𝑅) ∈ 𝐵)) → ((𝑋 (𝑋 𝑅) ∧ 𝑌 (𝑋 𝑅)) ↔ (𝑋 𝑌) (𝑋 𝑅)))
765, 6, 14, 13, 75syl13anc 1479 . . 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)))
7738, 74, 76mpbi2and 994 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) (𝑋 𝑅))
781, 2, 5, 13, 16, 36, 77latasymd 17258 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 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932   class class class wbr 4804  ‘cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  Latclat 17246  Atomscatm 35053  HLchlt 35140  Linesclines 35283  pmapcpmap 35286 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-lines 35290  df-pmap 35293 This theorem is referenced by: (None)
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