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Theorem cdlemb 35576
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 1253 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ HL)
2 simp12 1254 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐴)
3 simp13 1255 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐴)
4 simp2l 1249 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋𝐵)
5 simp2r 1250 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝑄)
6 simp31 1259 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋𝐶 1 )
7 simp32 1260 . . . 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 2813 . . . . 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 35258 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴)
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1505 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴)
171hllatd 35146 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ Lat)
188, 14atbase 35071 . . . . . . 7 (𝑃𝐴𝑃𝐵)
192, 18syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐵)
208, 14atbase 35071 . . . . . . 7 (𝑄𝐴𝑄𝐵)
213, 20syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐵)
228, 10latjcl 17259 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
2317, 19, 21, 22syl3anc 1483 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
248, 9, 11latmle2 17285 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑄)(meet‘𝐾)𝑋) 𝑋)
2517, 23, 4, 24syl3anc 1483 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋) 𝑋)
26 eqid 2813 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
278, 9, 26, 12, 13, 141cvratlt 35256 . . . 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 1490 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄)(meet‘𝐾)𝑋)(lt‘𝐾)𝑋)
298, 26, 142atlt 35221 . . 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 1485 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑢𝐴 (𝑢 ≠ ((𝑃 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))
31 simpl11 1322 . . . 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 1324 . . . 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 778 . . . 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 1336 . . . . 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 791 . . . . . . . 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 1290 . . . . . . . . 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 17169 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑢𝐴𝑋𝐵) → (𝑢(lt‘𝐾)𝑋𝑢 𝑋))
3831, 33, 36, 37syl3anc 1483 . . . . . . . 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 4854 . . . . . . 7 (𝑃 = 𝑢 → (𝑃 𝑋𝑢 𝑋))
4139, 40syl5ibrcom 238 . . . . . 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 2999 . . . . 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 35168 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑢𝐴) ∧ 𝑃𝑢) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))
4531, 32, 33, 43, 44syl31anc 1485 . . 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 2813 . . . . . . . 8 ((𝑃 𝑄)(meet‘𝐾)𝑋) = ((𝑃 𝑄)(meet‘𝐾)𝑋)
478, 9, 10, 12, 13, 14, 26, 11, 46cdlemblem 35575 . . . . . . 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 1141 . . . . . 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 420 . . . . 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 395 . . . 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 3209 . . 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 3230 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 384  w3a 1100   = wceq 1637  wcel 2157  wne 2985  wrex 3104   class class class wbr 4851  cfv 6104  (class class class)co 6877  Basecbs 16071  lecple 16163  ltcplt 17149  joincjn 17152  meetcmee 17153  1.cp1 17246  Latclat 17253  ccvr 35044  Atomscatm 35045  HLchlt 35132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-proset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34958  df-ol 34960  df-oml 34961  df-covers 35048  df-ats 35049  df-atl 35080  df-cvlat 35104  df-hlat 35133
This theorem is referenced by:  cdlemb2  35823
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