Theorem cdlemb 37735

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 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 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.

Step	Hyp	Ref	Expression
1		simp11 1201	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝐾 ∈ HL)
2		simp12 1202	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐴)
3		simp13 1203	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐴)
4		simp2l 1197	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
5		simp2r 1198	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≠ 𝑄)
6		simp31 1207	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑋𝐶 1 )
7		simp32 1208	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (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 2738	. . . . 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)
15	8, 9, 10, 11, 12, 13, 14	1cvrat 37417	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴)
16	1, 2, 3, 4, 5, 6, 7, 15	syl133anc 1391	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴)
17	1	hllatd 37305	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝐾 ∈ Lat)
18	8, 14	atbase 37230	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
19	2, 18	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐵)
20	8, 14	atbase 37230	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
21	3, 20	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐵)
22	8, 10	latjcl 18072	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
23	17, 19, 21, 22	syl3anc 1369	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
24	8, 9, 11	latmle2 18098	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ≤ 𝑋)
25	17, 23, 4, 24	syl3anc 1369	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ≤ 𝑋)
26		eqid 2738	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
27	8, 9, 26, 12, 13, 14	1cvratlt 37415	. . . 4 ⊢ (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑋𝐶 1 ∧ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ≤ 𝑋)) → ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋)(lt‘𝐾)𝑋)
28	1, 16, 4, 6, 25, 27	syl32anc 1376	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋)(lt‘𝐾)𝑋)
29	8, 26, 14	2atlt 37380	. . 3 ⊢ (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋)(lt‘𝐾)𝑋) → ∃𝑢 ∈ 𝐴 (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))
30	1, 16, 4, 28, 29	syl31anc 1371	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑢 ∈ 𝐴 (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))
31		simpl11 1246	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝐾 ∈ HL)
32		simpl12 1247	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑃 ∈ 𝐴)
33		simprl 767	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑢 ∈ 𝐴)
34		simpl32 1253	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → ¬ 𝑃 ≤ 𝑋)
35		simprrr 778	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑢(lt‘𝐾)𝑋)
36		simpl2l 1224	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑋 ∈ 𝐵)
37	9, 26	pltle 17966	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑢 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑢(lt‘𝐾)𝑋 → 𝑢 ≤ 𝑋))
38	31, 33, 36, 37	syl3anc 1369	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (𝑢(lt‘𝐾)𝑋 → 𝑢 ≤ 𝑋))
39	35, 38	mpd 15	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑢 ≤ 𝑋)
40		breq1 5073	. . . . . . 7 ⊢ (𝑃 = 𝑢 → (𝑃 ≤ 𝑋 ↔ 𝑢 ≤ 𝑋))
41	39, 40	syl5ibrcom 246	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (𝑃 = 𝑢 → 𝑃 ≤ 𝑋))
42	41	necon3bd 2956	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (¬ 𝑃 ≤ 𝑋 → 𝑃 ≠ 𝑢))
43	34, 42	mpd 15	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → 𝑃 ≠ 𝑢)
44	9, 10, 14	hlsupr 37327	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑃 ≠ 𝑢) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))
45	31, 32, 33, 43, 44	syl31anc 1371	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → ∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))
46		eqid 2738	. . . . . . . 8 ⊢ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) = ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋)
47	8, 9, 10, 12, 13, 14, 26, 11, 46	cdlemblem 37734	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
48	47	3exp 1117	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋)) → ((𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢))) → (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))))
49	48	exp4a 431	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋)) → (𝑟 ∈ 𝐴 → ((𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)) → (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))))))
50	49	imp 406	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (𝑟 ∈ 𝐴 → ((𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)) → (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))))
51	50	reximdvai 3199	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → (∃𝑟 ∈ 𝐴 (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))))
52	45, 51	mpd 15	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑋) ∧ 𝑢(lt‘𝐾)𝑋))) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
53	30, 52	rexlimddv 3219	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  ltcplt 17941  joincjn 17944  meetcmee 17945  1.cp1 18057  Latclat 18064   ⋖ ccvr 37203  Atomscatm 37204  HLchlt 37291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292

This theorem is referenced by:  cdlemb2  37982