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Theorem cdlemb 39394
Description: Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)
Hypotheses
Ref Expression
cdlemb.b 𝐵 = (Base‘𝐾)
cdlemb.l = (le‘𝐾)
cdlemb.j = (join‘𝐾)
cdlemb.u 1 = (1.‘𝐾)
cdlemb.c 𝐶 = ( ⋖ ‘𝐾)
cdlemb.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cdlemb (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))
Distinct variable groups:   𝐴,𝑟   𝐵,𝑟   𝐶,𝑟   ,𝑟   𝐾,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   1 ,𝑟   𝑋,𝑟

Proof of Theorem cdlemb
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 simp11 1200 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ HL)
2 simp12 1201 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐴)
3 simp13 1202 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐴)
4 simp2l 1196 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋𝐵)
5 simp2r 1197 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝑄)
6 simp31 1206 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋𝐶 1 )
7 simp32 1207 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ¬ 𝑃 𝑋)
8 cdlemb.b . . . . 5 𝐵 = (Base‘𝐾)
9 cdlemb.l . . . . 5 = (le‘𝐾)
10 cdlemb.j . . . . 5 = (join‘𝐾)
11 eqid 2725 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
12 cdlemb.u . . . . 5 1 = (1.‘𝐾)
13 cdlemb.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
14 cdlemb.a . . . . 5 𝐴 = (Atoms‘𝐾)
158, 9, 10, 11, 12, 13, 141cvrat 39076 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴)
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1390 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴)
171hllatd 38963 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ Lat)
188, 14atbase 38888 . . . . . . 7 (𝑃𝐴𝑃𝐵)
192, 18syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐵)
208, 14atbase 38888 . . . . . . 7 (𝑄𝐴𝑄𝐵)
213, 20syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐵)
228, 10latjcl 18434 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
2317, 19, 21, 22syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
248, 9, 11latmle2 18460 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑄)(meet‘𝐾)𝑋) 𝑋)
2517, 23, 4, 24syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) 𝑋)
26 eqid 2725 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
278, 9, 26, 12, 13, 141cvratlt 39074 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴𝑋𝐵) ∧ (𝑋𝐶 1 ∧ ((𝑃 𝑄)(meet‘𝐾)𝑋) 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋)(lt‘𝐾)𝑋)
281, 16, 4, 6, 25, 27syl32anc 1375 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋)(lt‘𝐾)𝑋)
298, 26, 142atlt 39039 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴𝑋𝐵) ∧ ((𝑃 𝑄)(meet‘𝐾)𝑋)(lt‘𝐾)𝑋) → ∃𝑢𝐴 (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))
301, 16, 4, 28, 29syl31anc 1370 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑢𝐴 (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))
31 simpl11 1245 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝐾 ∈ HL)
32 simpl12 1246 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑃𝐴)
33 simprl 769 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑢𝐴)
34 simpl32 1252 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → ¬ 𝑃 𝑋)
35 simprrr 780 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑢(lt‘𝐾)𝑋)
36 simpl2l 1223 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑋𝐵)
379, 26pltle 18328 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑢𝐴𝑋𝐵) → (𝑢(lt‘𝐾)𝑋𝑢 𝑋))
3831, 33, 36, 37syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (𝑢(lt‘𝐾)𝑋𝑢 𝑋))
3935, 38mpd 15 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑢 𝑋)
40 breq1 5152 . . . . . . 7 (𝑃 = 𝑢 → (𝑃 𝑋𝑢 𝑋))
4139, 40syl5ibrcom 246 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (𝑃 = 𝑢𝑃 𝑋))
4241necon3bd 2943 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (¬ 𝑃 𝑋𝑃𝑢))
4334, 42mpd 15 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑃𝑢)
449, 10, 14hlsupr 38986 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑢𝐴) ∧ 𝑃𝑢) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))
4531, 32, 33, 43, 44syl31anc 1370 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))
46 eqid 2725 . . . . . . . 8 ((𝑃 𝑄)(meet‘𝐾)𝑋) = ((𝑃 𝑄)(meet‘𝐾)𝑋)
478, 9, 10, 12, 13, 14, 26, 11, 46cdlemblem 39393 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (¬ 𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))
48473exp 1116 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋)) → ((𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢))) → (¬ 𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))))
4948exp4a 430 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋)) → (𝑟𝐴 → ((𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)) → (¬ 𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄))))))
5049imp 405 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (𝑟𝐴 → ((𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)) → (¬ 𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))))
5150reximdvai 3154 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (∃𝑟𝐴 (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)) → ∃𝑟𝐴𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄))))
5245, 51mpd 15 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → ∃𝑟𝐴𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))
5330, 52rexlimddv 3150 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wrex 3059   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  ltcplt 18303  joincjn 18306  meetcmee 18307  1.cp1 18419  Latclat 18426  ccvr 38861  Atomscatm 38862  HLchlt 38949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-proset 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-p1 18421  df-lat 18427  df-clat 18494  df-oposet 38775  df-ol 38777  df-oml 38778  df-covers 38865  df-ats 38866  df-atl 38897  df-cvlat 38921  df-hlat 38950
This theorem is referenced by:  cdlemb2  39641
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