llncvrlpln2 - Mathbox for Norm Megill

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Theorem llncvrlpln2 40178

Description: A lattice line under a lattice plane is covered by it. (Contributed by NM, 24-Jun-2012.)

**Hypotheses**
Ref	Expression
llncvrlpln2.l	⊢ ≤ = (le‘𝐾)
llncvrlpln2.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
llncvrlpln2.n	⊢ 𝑁 = (LLines‘𝐾)
llncvrlpln2.p	⊢ 𝑃 = (LPlanes‘𝐾)

**Assertion**
Ref	Expression
llncvrlpln2	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → 𝑋𝐶𝑌)

Proof of Theorem llncvrlpln2 Dummy variables 𝑞 𝑝 𝑟 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 488	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
2		simpl1 1205	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ HL)
3		simpl3 1207	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝑃)
4		llncvrlpln2.n	. . . . . 6 ⊢ 𝑁 = (LLines‘𝐾)
5		llncvrlpln2.p	. . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾)
6	4, 5	lplnnelln 40167	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑃) → ¬ 𝑌 ∈ 𝑁)
7	2, 3, 6	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → ¬ 𝑌 ∈ 𝑁)
8		simpl2 1206	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝑁)
9		eleq1 2850	. . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 ∈ 𝑁 ↔ 𝑌 ∈ 𝑁))
10	8, 9	syl5ibcom 247	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → (𝑋 = 𝑌 → 𝑌 ∈ 𝑁))
11	10	necon3bd 2971	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → (¬ 𝑌 ∈ 𝑁 → 𝑋 ≠ 𝑌))
12	7, 11	mpd 15	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≠ 𝑌)
13		llncvrlpln2.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
14		eqid 2762	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
15	13, 14	pltval 18362	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
16	15	adantr 484	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
17	1, 12, 16	mpbir2and 723	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → 𝑋(lt‘𝐾)𝑌)
18		simpl1 1205	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝐾 ∈ HL)
19		simpl2 1206	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋 ∈ 𝑁)
20		eqid 2762	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 4	llnbase 40130	. . . . 5 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾))
22	19, 21	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋 ∈ (Base‘𝐾))
23		simpl3 1207	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌 ∈ 𝑃)
24	20, 5	lplnbase 40155	. . . . 5 ⊢ (𝑌 ∈ 𝑃 → 𝑌 ∈ (Base‘𝐾))
25	23, 24	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌 ∈ (Base‘𝐾))
26		simpr 488	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋(lt‘𝐾)𝑌)
27		eqid 2762	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
28		llncvrlpln2.c	. . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾)
29		eqid 2762	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
30	20, 13, 14, 27, 28, 29	hlrelat3 40033	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) ∧ 𝑋(lt‘𝐾)𝑌) → ∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌))
31	18, 22, 25, 26, 30	syl31anc 1392	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → ∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌))
32	20, 13, 27, 29, 5	islpln2 40157	. . . . . . . 8 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ ¬ 𝑢 ≤ (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)))))
33	32	adantr 484	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → (𝑌 ∈ 𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ ¬ 𝑢 ≤ (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)))))
34		simp3 1151	. . . . . . . . . . 11 ⊢ ((𝑠 ≠ 𝑡 ∧ ¬ 𝑢 ≤ (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)) → 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))
35	20, 27, 29, 4	islln2 40132	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))))
36		simp3l 1215	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑋𝐶(𝑋(join‘𝐾)𝑟))
37		simp3r 1216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → (𝑋(join‘𝐾)𝑟) ≤ 𝑌)
38		simp12r 1301	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑋 = (𝑝(join‘𝐾)𝑞))
39	38	oveq1d 7411	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → (𝑋(join‘𝐾)𝑟) = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))
40		simp22 1221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))
41	37, 39, 40	3brtr3d 5131	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ≤ ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))
42		simp111 1316	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝐾 ∈ HL)
43		simp112 1317	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑝 ∈ (Atoms‘𝐾))
44		simp113 1318	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑞 ∈ (Atoms‘𝐾))
45		simp23 1222	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑟 ∈ (Atoms‘𝐾))
46	43, 44, 45	3jca 1141	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)))
47		simp13l 1302	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑠 ∈ (Atoms‘𝐾))
48		simp13r 1303	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑡 ∈ (Atoms‘𝐾))
49		simp21 1220	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑢 ∈ (Atoms‘𝐾))
50	47, 48, 49	3jca 1141	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)))
51	36, 38, 39	3brtr3d 5131	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → (𝑝(join‘𝐾)𝑞)𝐶((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))
52	20, 27, 29	hlatjcl 39988	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
53	42, 43, 44, 52	syl3anc 1390	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
54	20, 13, 27, 28, 29	cvr1 40031	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ↔ (𝑝(join‘𝐾)𝑞)𝐶((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)))
55	42, 53, 45, 54	syl3anc 1390	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → (¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ↔ (𝑝(join‘𝐾)𝑞)𝐶((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)))
56	51, 55	mpbird 259	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → ¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞))
57		simp12l 1300	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑝 ≠ 𝑞)
58	13, 27, 29	3at 40111	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑟 ≤ (𝑝(join‘𝐾)𝑞) ∧ 𝑝 ≠ 𝑞)) → (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ≤ ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ↔ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)))
59	42, 46, 50, 56, 57, 58	syl32anc 1397	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ≤ ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ↔ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)))
60	41, 59	mpbid 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))
61	60, 39, 40	3eqtr4d 2807	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → (𝑋(join‘𝐾)𝑟) = 𝑌)
62	36, 61	breqtrd 5126	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑋𝐶𝑌)
63	62	3exp 1132	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → ((𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌)))
64	63	3expd 1367	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌)))))
65	64	3exp 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌)))))))
66	65	3expib 1135	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ HL → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌))))))))
67	66	rexlimdvv 3218	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌)))))))
68	67	adantld 494	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞))) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌)))))))
69	35, 68	sylbid 242	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌)))))))
70	69	imp31 421	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌)))))
71	34, 70	syl7 74	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → (𝑢 ∈ (Atoms‘𝐾) → ((𝑠 ≠ 𝑡 ∧ ¬ 𝑢 ≤ (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌)))))
72	71	rexlimdv 3161	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → (∃𝑢 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ ¬ 𝑢 ≤ (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌))))
73	72	rexlimdvva 3219	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → (∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ ¬ 𝑢 ≤ (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌))))
74	73	adantld 494	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)(𝑠 ≠ 𝑡 ∧ ¬ 𝑢 ≤ (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌))))
75	33, 74	sylbid 242	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → (𝑌 ∈ 𝑃 → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌))))
76	75	3impia 1130	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌)))
77	76	rexlimdv 3161	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) → (∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌) → 𝑋𝐶𝑌))
78	77	imp 410	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ ∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) ≤ 𝑌)) → 𝑋𝐶𝑌)
79	31, 78	syldan 600	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋𝐶𝑌)
80	17, 79	syldan 600	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋 ≤ 𝑌) → 𝑋𝐶𝑌)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∃wrex 3086 class class class wbr 5100 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 lecple 17293 ltcplt 18340 joincjn 18343 ⋖ ccvr 39883 Atomscatm 39884 HLchlt 39971 LLinesclln 40112 LPlanesclpl 40113

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-proset 18326 df-poset 18345 df-plt 18360 df-lub 18376 df-glb 18377 df-join 18378 df-meet 18379 df-p0 18455 df-lat 18464 df-clat 18531 df-oposet 39797 df-ol 39799 df-oml 39800 df-covers 39887 df-ats 39888 df-atl 39919 df-cvlat 39943 df-hlat 39972 df-llines 40119 df-lplanes 40120

This theorem is referenced by: llncvrlpln 40179 2llnmj 40181 lplncmp 40183 lplnexatN 40184 2llnm2N 40189 2lplnmj 40243

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