Theorem lplncvrlvol2 39814

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 1192	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ HL)
3		simpl3 1194	. . . . 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 39789	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑉) → ¬ 𝑌 ∈ 𝑃)
7	2, 3, 6	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → ¬ 𝑌 ∈ 𝑃)
8		simpl2 1193	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝑃)
9		eleq1 2822	. . . . . 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 2944	. . . 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 2734	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
15	13, 14	pltval 18251	. . . 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 713	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋(lt‘𝐾)𝑌)
18		simpl1 1192	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝐾 ∈ HL)
19		simpl2 1193	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋 ∈ 𝑃)
20		eqid 2734	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 4	lplnbase 39733	. . . . 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 1194	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌 ∈ 𝑉)
24	20, 5	lvolbase 39777	. . . . 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 2734	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
28		lplncvrlvol2.c	. . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾)
29		eqid 2734	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
30	20, 13, 14, 27, 28, 29	hlrelat3 39611	. . . 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 1375	. . 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 39779	. . . . . . . 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 39735	. . . . . . . . . . . . 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 1247	. . . . . . . . . . . . . . . . . . . . 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 1248	. . . . . . . . . . . . . . . . . . . . . . . 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 1311	. . . . . . . . . . . . . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . . . . . . . . . . . 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 5127	. . . . . . . . . . . . . . . . . . . . . . 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 1204	. . . . . . . . . . . . . . . . . . . . . . . 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 1205	. . . . . . . . . . . . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . . . . . . . . . . . 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 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))) → 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 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))) → 𝑢 ∈ (Atoms‘𝐾))
48		simp22l 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))) → 𝑣 ∈ (Atoms‘𝐾))
49		simp22r 1294	. . . . . . . . . . . . . . . . . . . . . . . . 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 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))) → 𝑝 ≠ 𝑞)
52		simp132 1310	. . . . . . . . . . . . . . . . . . . . . . . 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 5127	. . . . . . . . . . . . . . . . . . . . . . . . 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 1303	. . . . . . . . . . . . . . . . . . . . . . . . . 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 39563	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 39566	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 39488	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18360	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . 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 39609	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . 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 39813	. . . . . . . . . . . . . . . . . . . . . . . 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 1387	. . . . . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . . . 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 5122	. . . . . . . . . . . . . . . . . . . 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 1354	. . . . . . . . . . . . . . . . 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 3139	. . . . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . . 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 3190	. . . . . . . . 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 3191	. . . . . . . 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 3133	. . . 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 591	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋𝐶𝑌)
88	17, 87	syldan 591	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≤ 𝑌) → 𝑋𝐶𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  lecple 17182  ltcplt 18229  joincjn 18232  Latclat 18352   ⋖ ccvr 39461  Atomscatm 39462  HLchlt 39549  LPlanesclpl 39691  LVolsclvol 39692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-proset 18215  df-poset 18234  df-plt 18249  df-lub 18265  df-glb 18266  df-join 18267  df-meet 18268  df-p0 18344  df-lat 18353  df-clat 18420  df-oposet 39375  df-ol 39377  df-oml 39378  df-covers 39465  df-ats 39466  df-atl 39497  df-cvlat 39521  df-hlat 39550  df-llines 39697  df-lplanes 39698  df-lvols 39699

This theorem is referenced by:  lplncvrlvol  39815  lvolcmp  39816  2lplnm2N  39820  2lplnmj  39821