Theorem lplnexllnN 37505

Description: Given an atom on a lattice plane, there is a lattice line whose join with the atom equals the plane. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lplnexat.l	⊢ ≤ = (le‘𝐾)
lplnexat.j	⊢ ∨ = (join‘𝐾)
lplnexat.a	⊢ 𝐴 = (Atoms‘𝐾)
lplnexat.n	⊢ 𝑁 = (LLines‘𝐾)
lplnexat.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
lplnexllnN	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))

Distinct variable groups:   𝑦, ∨   𝑦, ≤   𝑦,𝑁   𝑦,𝑄   𝑦,𝑋

Allowed substitution hints:   𝐴(𝑦)   𝑃(𝑦)   𝐾(𝑦)

Proof of Theorem lplnexllnN

Dummy variables 𝑠 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1190	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → 𝑋 ∈ 𝑃)
2		simpl1 1189	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → 𝐾 ∈ HL)
3		eqid 2738	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		lplnexat.p	. . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾)
5	3, 4	lplnbase 37475	. . . . 5 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾))
6	1, 5	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → 𝑋 ∈ (Base‘𝐾))
7		lplnexat.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
8		lplnexat.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
9		lplnexat.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
10		lplnexat.n	. . . . 5 ⊢ 𝑁 = (LLines‘𝐾)
11	3, 7, 8, 9, 10, 4	islpln3 37474	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝑃 ↔ ∃𝑧 ∈ 𝑁 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))))
12	2, 6, 11	syl2anc 583	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → (𝑋 ∈ 𝑃 ↔ ∃𝑧 ∈ 𝑁 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))))
13	1, 12	mpbid 231	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → ∃𝑧 ∈ 𝑁 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))
14		simpll1 1210	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝐾 ∈ HL)
15		simpr2l 1230	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑧 ∈ 𝑁)
16		simpll3 1212	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑄 ∈ 𝐴)
17		simpr1 1192	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑄 ≤ 𝑧)
18	7, 8, 9, 10	llnexatN 37462	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ 𝑁 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑧) → ∃𝑠 ∈ 𝐴 (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))
19	14, 15, 16, 17, 18	syl31anc 1371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → ∃𝑠 ∈ 𝐴 (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))
20		simp1l1 1264	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝐾 ∈ HL)
21		simp22r 1291	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑟 ∈ 𝐴)
22		simp3l 1199	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑠 ∈ 𝐴)
23		simp1l3 1266	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑄 ∈ 𝐴)
24		simp23l 1292	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → ¬ 𝑟 ≤ 𝑧)
25		simp3rr 1245	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑧 = (𝑄 ∨ 𝑠))
26	25	breq2d 5082	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → (𝑟 ≤ 𝑧 ↔ 𝑟 ≤ (𝑄 ∨ 𝑠)))
27	24, 26	mtbid 323	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → ¬ 𝑟 ≤ (𝑄 ∨ 𝑠))
28	7, 8, 9	atnlej2 37321	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ ¬ 𝑟 ≤ (𝑄 ∨ 𝑠)) → 𝑟 ≠ 𝑠)
29	20, 21, 23, 22, 27, 28	syl131anc 1381	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑟 ≠ 𝑠)
30	8, 9, 10	llni2 37453	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ 𝑟 ≠ 𝑠) → (𝑟 ∨ 𝑠) ∈ 𝑁)
31	20, 21, 22, 29, 30	syl31anc 1371	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → (𝑟 ∨ 𝑠) ∈ 𝑁)
32		simp3rl 1244	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑄 ≠ 𝑠)
33	7, 8, 9	hlatcon2 37393	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑄 ≠ 𝑠 ∧ ¬ 𝑟 ≤ (𝑄 ∨ 𝑠))) → ¬ 𝑄 ≤ (𝑟 ∨ 𝑠))
34	20, 23, 22, 21, 32, 27, 33	syl132anc 1386	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → ¬ 𝑄 ≤ (𝑟 ∨ 𝑠))
35		simp23r 1293	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑋 = (𝑧 ∨ 𝑟))
36	25	oveq1d 7270	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → (𝑧 ∨ 𝑟) = ((𝑄 ∨ 𝑠) ∨ 𝑟))
37	20	hllatd 37305	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝐾 ∈ Lat)
38	3, 9	atbase 37230	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
39	23, 38	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑄 ∈ (Base‘𝐾))
40	3, 9	atbase 37230	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (Base‘𝐾))
41	22, 40	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑠 ∈ (Base‘𝐾))
42	3, 9	atbase 37230	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾))
43	21, 42	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑟 ∈ (Base‘𝐾))
44	3, 8	latj31 18120	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾))) → ((𝑄 ∨ 𝑠) ∨ 𝑟) = ((𝑟 ∨ 𝑠) ∨ 𝑄))
45	37, 39, 41, 43, 44	syl13anc 1370	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → ((𝑄 ∨ 𝑠) ∨ 𝑟) = ((𝑟 ∨ 𝑠) ∨ 𝑄))
46	35, 36, 45	3eqtrd 2782	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → 𝑋 = ((𝑟 ∨ 𝑠) ∨ 𝑄))
47		breq2 5074	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → (𝑄 ≤ 𝑦 ↔ 𝑄 ≤ (𝑟 ∨ 𝑠)))
48	47	notbid 317	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → (¬ 𝑄 ≤ 𝑦 ↔ ¬ 𝑄 ≤ (𝑟 ∨ 𝑠)))
49		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → (𝑦 ∨ 𝑄) = ((𝑟 ∨ 𝑠) ∨ 𝑄))
50	49	eqeq2d 2749	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → (𝑋 = (𝑦 ∨ 𝑄) ↔ 𝑋 = ((𝑟 ∨ 𝑠) ∨ 𝑄)))
51	48, 50	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → ((¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)) ↔ (¬ 𝑄 ≤ (𝑟 ∨ 𝑠) ∧ 𝑋 = ((𝑟 ∨ 𝑠) ∨ 𝑄))))
52	51	rspcev 3552	. . . . . . . . . 10 ⊢ (((𝑟 ∨ 𝑠) ∈ 𝑁 ∧ (¬ 𝑄 ≤ (𝑟 ∨ 𝑠) ∧ 𝑋 = ((𝑟 ∨ 𝑠) ∨ 𝑄))) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
53	31, 34, 46, 52	syl12anc 833	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))) ∧ (𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
54	53	3expia 1119	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → ((𝑠 ∈ 𝐴 ∧ (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠))) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))
55	54	expd 415	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → (𝑠 ∈ 𝐴 → ((𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))))
56	55	rexlimdv 3211	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → (∃𝑠 ∈ 𝐴 (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))
57	19, 56	mpd 15	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
58	57	3exp2 1352	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → (𝑄 ≤ 𝑧 → ((𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) → ((¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))))
59		simpr2l 1230	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑧 ∈ 𝑁)
60		simpr1 1192	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → ¬ 𝑄 ≤ 𝑧)
61		simpll1 1210	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝐾 ∈ HL)
62	61	hllatd 37305	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝐾 ∈ Lat)
63	3, 10	llnbase 37450	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑁 → 𝑧 ∈ (Base‘𝐾))
64	59, 63	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑧 ∈ (Base‘𝐾))
65		simpr2r 1231	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑟 ∈ 𝐴)
66	65, 42	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑟 ∈ (Base‘𝐾))
67	3, 7, 8	latlej1 18081	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → 𝑧 ≤ (𝑧 ∨ 𝑟))
68	62, 64, 66, 67	syl3anc 1369	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑧 ≤ (𝑧 ∨ 𝑟))
69		simpr3r 1233	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑋 = (𝑧 ∨ 𝑟))
70	68, 69	breqtrrd 5098	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑧 ≤ 𝑋)
71		simplr 765	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑄 ≤ 𝑋)
72		simpll3 1212	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑄 ∈ 𝐴)
73	72, 38	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑄 ∈ (Base‘𝐾))
74		simpll2 1211	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑋 ∈ 𝑃)
75	74, 5	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑋 ∈ (Base‘𝐾))
76	3, 7, 8	latjle12 18083	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾))) → ((𝑧 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) ↔ (𝑧 ∨ 𝑄) ≤ 𝑋))
77	62, 64, 73, 75, 76	syl13anc 1370	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → ((𝑧 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) ↔ (𝑧 ∨ 𝑄) ≤ 𝑋))
78	70, 71, 77	mpbi2and 708	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → (𝑧 ∨ 𝑄) ≤ 𝑋)
79	3, 8	latjcl 18072	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑧 ∨ 𝑄) ∈ (Base‘𝐾))
80	62, 64, 73, 79	syl3anc 1369	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → (𝑧 ∨ 𝑄) ∈ (Base‘𝐾))
81		eqid 2738	. . . . . . . . . . . . 13 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
82	3, 7, 8, 81, 9	cvr1 37351	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ 𝐴) → (¬ 𝑄 ≤ 𝑧 ↔ 𝑧( ⋖ ‘𝐾)(𝑧 ∨ 𝑄)))
83	61, 64, 72, 82	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → (¬ 𝑄 ≤ 𝑧 ↔ 𝑧( ⋖ ‘𝐾)(𝑧 ∨ 𝑄)))
84	60, 83	mpbid 231	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑧( ⋖ ‘𝐾)(𝑧 ∨ 𝑄))
85	3, 81, 10, 4	lplni 37473	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑧 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ 𝑁) ∧ 𝑧( ⋖ ‘𝐾)(𝑧 ∨ 𝑄)) → (𝑧 ∨ 𝑄) ∈ 𝑃)
86	61, 80, 59, 84, 85	syl31anc 1371	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → (𝑧 ∨ 𝑄) ∈ 𝑃)
87	7, 4	lplncmp 37503	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑧 ∨ 𝑄) ∈ 𝑃 ∧ 𝑋 ∈ 𝑃) → ((𝑧 ∨ 𝑄) ≤ 𝑋 ↔ (𝑧 ∨ 𝑄) = 𝑋))
88	61, 86, 74, 87	syl3anc 1369	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → ((𝑧 ∨ 𝑄) ≤ 𝑋 ↔ (𝑧 ∨ 𝑄) = 𝑋))
89	78, 88	mpbid 231	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → (𝑧 ∨ 𝑄) = 𝑋)
90	89	eqcomd 2744	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑋 = (𝑧 ∨ 𝑄))
91		breq2 5074	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑄 ≤ 𝑦 ↔ 𝑄 ≤ 𝑧))
92	91	notbid 317	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (¬ 𝑄 ≤ 𝑦 ↔ ¬ 𝑄 ≤ 𝑧))
93		oveq1 7262	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ∨ 𝑄) = (𝑧 ∨ 𝑄))
94	93	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑋 = (𝑦 ∨ 𝑄) ↔ 𝑋 = (𝑧 ∨ 𝑄)))
95	92, 94	anbi12d 630	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)) ↔ (¬ 𝑄 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑄))))
96	95	rspcev 3552	. . . . . 6 ⊢ ((𝑧 ∈ 𝑁 ∧ (¬ 𝑄 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑄))) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
97	59, 60, 90, 96	syl12anc 833	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
98	97	3exp2 1352	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → (¬ 𝑄 ≤ 𝑧 → ((𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) → ((¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))))
99	58, 98	pm2.61d 179	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → ((𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) → ((¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))))
100	99	rexlimdvv 3221	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → (∃𝑧 ∈ 𝑁 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄))))
101	13, 100	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ 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  joincjn 17944  Latclat 18064   ⋖ ccvr 37203  Atomscatm 37204  HLchlt 37291  LLinesclln 37432  LPlanesclpl 37433

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-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  df-llines 37439  df-lplanes 37440

This theorem is referenced by: (None)