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Theorem cdlema1N 34095
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 1084 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝐾 ∈ HL)
4 hllat 33668 . . 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 1085 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑋𝐵)
7 simp23 1089 . . . 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 33594 . . . 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 16874 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑅𝐵) → (𝑋 𝑅) ∈ 𝐵)
135, 6, 10, 12syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) ∈ 𝐵)
14 simp13 1086 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑌𝐵)
151, 11latjcl 16874 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
165, 6, 14, 15syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) ∈ 𝐵)
171, 2, 11latlej1 16883 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))
185, 6, 14, 17syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑋 (𝑋 𝑌))
19 simp21 1087 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑃𝐴)
201, 8atbase 33594 . . . . . 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 1088 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄𝐴)
231, 8atbase 33594 . . . . . 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 16874 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
265, 21, 24, 25syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑄) ∈ 𝐵)
27 simp31r 1178 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅 (𝑃 𝑄))
28 simp32l 1179 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑃 𝑋)
29 simp32r 1180 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 𝑌)
301, 2, 11latjlej12 16890 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑋𝐵) ∧ (𝑄𝐵𝑌𝐵)) → ((𝑃 𝑋𝑄 𝑌) → (𝑃 𝑄) (𝑋 𝑌)))
315, 21, 6, 24, 14, 30syl122anc 1327 . . . . 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 711 . . . 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 16881 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅 (𝑋 𝑌))
341, 2, 11latjle12 16885 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑅𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → ((𝑋 (𝑋 𝑌) ∧ 𝑅 (𝑋 𝑌)) ↔ (𝑋 𝑅) (𝑋 𝑌)))
355, 6, 10, 16, 34syl13anc 1320 . . 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 958 . 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 16883 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑅𝐵) → 𝑋 (𝑋 𝑅))
385, 6, 10, 37syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑋 (𝑋 𝑅))
39 simp331 1207 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝐹𝑌) ∈ 𝑁)
40 simp332 1208 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) ∈ 𝐴)
41 simp333 1209 . . . . . 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 16899 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑋)
445, 6, 14, 43syl3anc 1318 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) 𝑋)
45 breq1 4586 . . . . . . . 8 (𝑄 = (𝑋 𝑌) → (𝑄 𝑋 ↔ (𝑋 𝑌) 𝑋))
4644, 45syl5ibrcom 236 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑄 = (𝑋 𝑌) → 𝑄 𝑋))
4746necon3bd 2796 . . . . . 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 16900 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
505, 6, 14, 49syl3anc 1318 . . . . 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 34082 . . . . 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 1349 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑌 = (𝑄 (𝑋 𝑌)))
551, 11latjcl 16874 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑅𝐵) → (𝑃 𝑅) ∈ 𝐵)
565, 21, 10, 55syl3anc 1318 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃 𝑅) ∈ 𝐵)
577, 22, 193jca 1235 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑅𝐴𝑄𝐴𝑃𝐴))
58 simp31l 1177 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑅𝑃)
593, 57, 583jca 1235 . . . . . . 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 33699 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ 𝑅𝑃) → (𝑅 (𝑃 𝑄) → 𝑄 (𝑃 𝑅)))
6159, 27, 60sylc 63 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑄 (𝑃 𝑅))
6221, 6, 103jca 1235 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑃𝐵𝑋𝐵𝑅𝐵))
635, 62jca 553 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝐾 ∈ Lat ∧ (𝑃𝐵𝑋𝐵𝑅𝐵)))
641, 2, 11latjlej1 16888 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑋𝐵𝑅𝐵)) → (𝑃 𝑋 → (𝑃 𝑅) (𝑋 𝑅)))
6563, 28, 64sylc 63 . . . . . 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 16881 . . . . 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 16913 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑅𝐵)) → (𝑋 𝑌) (𝑋 𝑅))
685, 6, 14, 10, 67syl13anc 1320 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) (𝑋 𝑅))
691, 42latmcl 16875 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
705, 6, 14, 69syl3anc 1318 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑌) ∈ 𝐵)
711, 2, 11latjle12 16885 . . . . . 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 1320 . . . . 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 958 . . . 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 4605 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → 𝑌 (𝑋 𝑅))
751, 2, 11latjle12 16885 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 𝑅) ∈ 𝐵)) → ((𝑋 (𝑋 𝑅) ∧ 𝑌 (𝑋 𝑅)) ↔ (𝑋 𝑌) (𝑋 𝑅)))
765, 6, 14, 13, 75syl13anc 1320 . . 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 958 . 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 16880 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780   class class class wbr 4583  cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  HLchlt 33655  Linesclines 33798  pmapcpmap 33801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-lines 33805  df-pmap 33808
This theorem is referenced by: (None)
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