Theorem lplncvrlvol2 39572

Description: A lattice line under a lattice plane is covered by it. (Contributed by NM, 12-Jul-2012.)

Hypotheses

Ref	Expression
lplncvrlvol2.l	⊢ ≤ = (le‘𝐾)
lplncvrlvol2.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
lplncvrlvol2.p	⊢ 𝑃 = (LPlanes‘𝐾)
lplncvrlvol2.v	⊢ 𝑉 = (LVols‘𝐾)

Assertion

Ref	Expression
lplncvrlvol2	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋𝐶𝑌)

Proof of Theorem lplncvrlvol2

Dummy variables 𝑞 𝑝 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
2		simpl1 1191	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ HL)
3		simpl3 1193	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝑉)
4		lplncvrlvol2.p	. . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾)
5		lplncvrlvol2.v	. . . . . 6 ⊢ 𝑉 = (LVols‘𝐾)
6	4, 5	lvolnelpln 39547	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑉) → ¬ 𝑌 ∈ 𝑃)
7	2, 3, 6	syl2anc 583	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → ¬ 𝑌 ∈ 𝑃)
8		simpl2 1192	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝑃)
9		eleq1 2832	. . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃))
10	8, 9	syl5ibcom 245	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → (𝑋 = 𝑌 → 𝑌 ∈ 𝑃))
11	10	necon3bd 2960	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → (¬ 𝑌 ∈ 𝑃 → 𝑋 ≠ 𝑌))
12	7, 11	mpd 15	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≠ 𝑌)
13		lplncvrlvol2.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
14		eqid 2740	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
15	13, 14	pltval 18402	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
16	15	adantr 480	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
17	1, 12, 16	mpbir2and 712	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋(lt‘𝐾)𝑌)
18		simpl1 1191	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝐾 ∈ HL)
19		simpl2 1192	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋 ∈ 𝑃)
20		eqid 2740	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 4	lplnbase 39491	. . . . 5 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾))
22	19, 21	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋 ∈ (Base‘𝐾))
23		simpl3 1193	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌 ∈ 𝑉)
24	20, 5	lvolbase 39535	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ (Base‘𝐾))
25	23, 24	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌 ∈ (Base‘𝐾))
26		simpr 484	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋(lt‘𝐾)𝑌)
27		eqid 2740	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
28		lplncvrlvol2.c	. . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾)
29		eqid 2740	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
30	20, 13, 14, 27, 28, 29	hlrelat3 39369	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) ∧ 𝑋(lt‘𝐾)𝑌) → ∃𝑠 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))
31	18, 22, 25, 26, 30	syl31anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → ∃𝑠 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))
32	20, 13, 27, 29, 5	islvol2 39537	. . . . . . . 8 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑉 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ≤ ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)))))
33	32	adantr 480	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → (𝑌 ∈ 𝑉 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ≤ ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)))))
34		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ≤ ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))
35	20, 13, 27, 29, 4	islpln2 39493	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)))))
36		simp3rl 1246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑋𝐶(𝑋(join‘𝐾)𝑠))
37		simp3rr 1247	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (𝑋(join‘𝐾)𝑠) ≤ 𝑌)
38		simp133 1310	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))
39	38	oveq1d 7463	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (𝑋(join‘𝐾)𝑠) = (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠))
40		simp23 1208	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))
41	37, 39, 40	3brtr3d 5197	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) ≤ (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))
42		simp11 1203	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)))
43		simp12 1204	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑟 ∈ (Atoms‘𝐾))
44		simp3l 1201	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑠 ∈ (Atoms‘𝐾))
45		simp21l 1290	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑡 ∈ (Atoms‘𝐾))
46	43, 44, 45	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)))
47		simp21r 1291	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑢 ∈ (Atoms‘𝐾))
48		simp22l 1292	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑣 ∈ (Atoms‘𝐾))
49		simp22r 1293	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑤 ∈ (Atoms‘𝐾))
50	47, 48, 49	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)))
51		simp131 1308	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑝 ≠ 𝑞)
52		simp132 1309	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞))
53	36, 38, 39	3brtr3d 5197	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)𝐶(((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠))
54		simp111 1302	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝐾 ∈ HL)
55	54	hllatd 39320	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝐾 ∈ Lat)
56	20, 27, 29	hlatjcl 39323	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
57	42, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
58	20, 29	atbase 39245	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
59	43, 58	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑟 ∈ (Base‘𝐾))
60	20, 27	latjcl 18509	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐾 ∈ Lat ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ∈ (Base‘𝐾))
61	55, 57, 59, 60	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ∈ (Base‘𝐾))
62	20, 13, 27, 28, 29	cvr1 39367	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐾 ∈ HL ∧ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) → (¬ 𝑠 ≤ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ↔ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)𝐶(((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠)))
63	54, 61, 44, 62	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (¬ 𝑠 ≤ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ↔ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)𝐶(((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠)))
64	53, 63	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → ¬ 𝑠 ≤ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))
65	13, 27, 29	4at2 39571	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾))) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ ¬ 𝑠 ≤ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → ((((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) ≤ (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) ↔ (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)))
66	42, 46, 50, 51, 52, 64, 65	syl33anc 1385	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → ((((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) ≤ (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) ↔ (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)))
67	41, 66	mpbid 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))
68	67, 39, 40	3eqtr4d 2790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → (𝑋(join‘𝐾)𝑠) = 𝑌)
69	36, 68	breqtrd 5192	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌))) → 𝑋𝐶𝑌)
70	69	3exp 1119	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → (((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → ((𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌)) → 𝑋𝐶𝑌)))
71	70	exp4a 431	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → (((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌))))
72	71	3expd 1353	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌))))))
73	72	rexlimdv3a 3165	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (∃𝑟 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌)))))))
74	73	3expib 1122	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ HL → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (∃𝑟 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌))))))))
75	74	rexlimdvv 3218	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌)))))))
76	75	adantld 490	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌)))))))
77	35, 76	sylbid 240	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌)))))))
78	77	imp31 417	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾))) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌)))))
79	34, 78	syl7 74	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾))) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (((𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ≤ ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌)))))
80	79	rexlimdvv 3218	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾))) → (∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ≤ ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌))))
81	80	rexlimdvva 3219	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ≤ ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌))))
82	81	adantld 490	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ≤ ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌))))
83	33, 82	sylbid 240	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → (𝑌 ∈ 𝑉 → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌))))
84	83	3impia 1117	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌)))
85	84	rexlimdv 3159	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) → (∃𝑠 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌) → 𝑋𝐶𝑌))
86	85	imp 406	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ ∃𝑠 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) ≤ 𝑌)) → 𝑋𝐶𝑌)
87	31, 86	syldan 590	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋𝐶𝑌)
88	17, 87	syldan 590	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋𝐶𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  ltcplt 18378  joincjn 18381  Latclat 18501   ⋖ ccvr 39218  Atomscatm 39219  HLchlt 39306  LPlanesclpl 39449  LVolsclvol 39450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-lat 18502  df-clat 18569  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307  df-llines 39455  df-lplanes 39456  df-lvols 39457

This theorem is referenced by:  lplncvrlvol  39573  lvolcmp  39574  2lplnm2N  39578  2lplnmj  39579