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Theorem cdlemb 36482
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 1196 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ HL)
2 simp12 1197 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐴)
3 simp13 1198 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐴)
4 simp2l 1192 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋𝐵)
5 simp2r 1193 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝑄)
6 simp31 1202 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋𝐶 1 )
7 simp32 1203 . . . 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 2797 . . . . 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 36164 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴)
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1386 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴)
171hllatd 36052 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ Lat)
188, 14atbase 35977 . . . . . . 7 (𝑃𝐴𝑃𝐵)
192, 18syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐵)
208, 14atbase 35977 . . . . . . 7 (𝑄𝐴𝑄𝐵)
213, 20syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐵)
228, 10latjcl 17494 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
2317, 19, 21, 22syl3anc 1364 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
248, 9, 11latmle2 17520 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑄)(meet‘𝐾)𝑋) 𝑋)
2517, 23, 4, 24syl3anc 1364 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) 𝑋)
26 eqid 2797 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
278, 9, 26, 12, 13, 141cvratlt 36162 . . . 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 1371 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋)(lt‘𝐾)𝑋)
298, 26, 142atlt 36127 . . 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 1366 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑢𝐴 (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))
31 simpl11 1241 . . . 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 1242 . . . 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 767 . . . 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 1248 . . . . 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 778 . . . . . . . 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 1219 . . . . . . . . 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 17404 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑢𝐴𝑋𝐵) → (𝑢(lt‘𝐾)𝑋𝑢 𝑋))
3831, 33, 36, 37syl3anc 1364 . . . . . . . 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 4971 . . . . . . 7 (𝑃 = 𝑢 → (𝑃 𝑋𝑢 𝑋))
4139, 40syl5ibrcom 248 . . . . . 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 3000 . . . . 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 36074 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑢𝐴) ∧ 𝑃𝑢) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))
4531, 32, 33, 43, 44syl31anc 1366 . . 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 2797 . . . . . . . 8 ((𝑃 𝑄)(meet‘𝐾)𝑋) = ((𝑃 𝑄)(meet‘𝐾)𝑋)
478, 9, 10, 12, 13, 14, 26, 11, 46cdlemblem 36481 . . . . . . 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 1112 . . . . . 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 432 . . . . 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 407 . . . 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 3237 . . 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 3256 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 396  w3a 1080   = wceq 1525  wcel 2083  wne 2986  wrex 3108   class class class wbr 4968  cfv 6232  (class class class)co 7023  Basecbs 16316  lecple 16405  ltcplt 17384  joincjn 17387  meetcmee 17388  1.cp1 17481  Latclat 17488  ccvr 35950  Atomscatm 35951  HLchlt 36038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-proset 17371  df-poset 17389  df-plt 17401  df-lub 17417  df-glb 17418  df-join 17419  df-meet 17420  df-p0 17482  df-p1 17483  df-lat 17489  df-clat 17551  df-oposet 35864  df-ol 35866  df-oml 35867  df-covers 35954  df-ats 35955  df-atl 35986  df-cvlat 36010  df-hlat 36039
This theorem is referenced by:  cdlemb2  36729
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