Theorem ps-2b 40067

Description: Variation of projective geometry axiom ps-2 40063. (Contributed by NM, 3-Jul-2012.)

Hypotheses

Ref	Expression
ps-2b.l	⊢ ≤ = (le‘𝐾)
ps-2b.j	⊢ ∨ = (join‘𝐾)
ps-2b.m	⊢ ∧ = (meet‘𝐾)
ps-2b.z	⊢ 0 = (0.‘𝐾)
ps-2b.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
ps-2b	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ≠ 0 )

Proof of Theorem ps-2b

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp11 1216	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → 𝐾 ∈ HL)
2		simp12 1217	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → 𝑃 ∈ 𝐴)
3		simp13 1218	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → 𝑄 ∈ 𝐴)
4		simp21 1219	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → 𝑅 ∈ 𝐴)
5	2, 3, 4	3jca 1140	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
6		simp22 1220	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → 𝑆 ∈ 𝐴)
7		simp23 1221	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → 𝑇 ∈ 𝐴)
8	6, 7	jca 519	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴))
9		simp31 1222	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑅))
10		simp32 1223	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → 𝑆 ≠ 𝑇)
11	9, 10	jca 519	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇))
12		simp33 1224	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))
13		ps-2b.l	. . . 4 ⊢ ≤ = (le‘𝐾)
14		ps-2b.j	. . . 4 ⊢ ∨ = (join‘𝐾)
15		ps-2b.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
16	13, 14, 15	ps-2 40063	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)))
17	1, 5, 8, 11, 12, 16	syl32anc 1396	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)))
18		simp111 1315	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ HL)
19		hlatl 39945	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
20	18, 19	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ AtLat)
21	18	hllatd 39949	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ Lat)
22		simp112 1316	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝑃 ∈ 𝐴)
23		simp121 1318	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝑅 ∈ 𝐴)
24		eqid 2761	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 14, 15	hlatjcl 39952	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
26	18, 22, 23, 25	syl3anc 1389	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
27		simp122 1319	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝑆 ∈ 𝐴)
28		simp123 1320	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝑇 ∈ 𝐴)
29	24, 14, 15	hlatjcl 39952	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
30	18, 27, 28, 29	syl3anc 1389	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
31		ps-2b.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
32	24, 31	latmcl 18463	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ∈ (Base‘𝐾))
33	21, 26, 30, 32	syl3anc 1389	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ∈ (Base‘𝐾))
34		simp2 1149	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝑢 ∈ 𝐴)
35		simp3 1150	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)))
36	24, 15	atbase 39874	. . . . . . 7 ⊢ (𝑢 ∈ 𝐴 → 𝑢 ∈ (Base‘𝐾))
37	34, 36	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝑢 ∈ (Base‘𝐾))
38	24, 13, 31	latlem12 18489	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑢 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))) → ((𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)) ↔ 𝑢 ≤ ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇))))
39	21, 37, 26, 30, 38	syl13anc 1390	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → ((𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)) ↔ 𝑢 ≤ ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇))))
40	35, 39	mpbid 234	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → 𝑢 ≤ ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)))
41		ps-2b.z	. . . . 5 ⊢ 0 = (0.‘𝐾)
42	24, 13, 41, 15	atlen0 39895	. . . 4 ⊢ (((𝐾 ∈ AtLat ∧ ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ∈ (Base‘𝐾) ∧ 𝑢 ∈ 𝐴) ∧ 𝑢 ≤ ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇))) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ≠ 0 )
43	20, 33, 34, 40, 42	syl31anc 1391	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) ∧ 𝑢 ∈ 𝐴 ∧ (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ≠ 0 )
44	43	rexlimdv3a 3166	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → (∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ≠ 0 ))
45	17, 44	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ((𝑃 ∨ 𝑅) ∧ (𝑆 ∨ 𝑇)) ≠ 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085   class class class wbr 5097  ‘cfv 6516  (class class class)co 7391  Basecbs 17236  lecple 17284  joincjn 18334  meetcmee 18335  0.cp0 18444  Latclat 18454  Atomscatm 39848  AtLatcal 39849  HLchlt 39935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-proset 18317  df-poset 18336  df-plt 18351  df-lub 18367  df-glb 18368  df-join 18369  df-meet 18370  df-p0 18446  df-lat 18455  df-clat 18522  df-oposet 39761  df-ol 39763  df-oml 39764  df-covers 39851  df-ats 39852  df-atl 39883  df-cvlat 39907  df-hlat 39936

This theorem is referenced by:  ps-2c  40113