Theorem lplnexllnN 40502

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 1211	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → 𝑋 ∈ 𝑃)
2		simpl1 1210	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → 𝐾 ∈ HL)
3		eqid 2760	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		lplnexat.p	. . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾)
5	3, 4	lplnbase 40472	. . . . 5 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾))
6	1, 5	syl 18	. . . 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 40471	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝑃 ↔ ∃𝑧 ∈ 𝑁 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))))
12	2, 6, 11	syl2anc 596	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → (𝑋 ∈ 𝑃 ↔ ∃𝑧 ∈ 𝑁 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟))))
13	1, 12	mpbid 235	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → ∃𝑧 ∈ 𝑁 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))
14		simpll1 1231	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝐾 ∈ HL)
15		simpr2l 1251	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑧 ∈ 𝑁)
16		simpll3 1233	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑄 ∈ 𝐴)
17		simpr1 1213	. . . . . . 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 40459	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ 𝑁 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑧) → ∃𝑠 ∈ 𝐴 (𝑄 ≠ 𝑠 ∧ 𝑧 = (𝑄 ∨ 𝑠)))
19	14, 15, 16, 17, 18	syl31anc 1400	. . . . . 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 1285	. . . . . . . . . . 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 1312	. . . . . . . . . . 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 1220	. . . . . . . . . . 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 1287	. . . . . . . . . . . 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 1313	. . . . . . . . . . . . 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 1266	. . . . . . . . . . . . . 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 5115	. . . . . . . . . . . . 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 327	. . . . . . . . . . . 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 40318	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ ¬ 𝑟 ≤ (𝑄 ∨ 𝑠)) → 𝑟 ≠ 𝑠)
29	20, 21, 23, 22, 27, 28	syl131anc 1410	. . . . . . . . . . 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 40450	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ 𝑟 ≠ 𝑠) → (𝑟 ∨ 𝑠) ∈ 𝑁)
31	20, 21, 22, 29, 30	syl31anc 1400	. . . . . . . . . 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 1265	. . . . . . . . . . 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 40390	. . . . . . . . . . 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 1415	. . . . . . . . . 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 1314	. . . . . . . . . . 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 7431	. . . . . . . . . . 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 40302	. . . . . . . . . . . 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 40227	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
39	23, 38	syl 18	. . . . . . . . . . . 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 40227	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (Base‘𝐾))
41	22, 40	syl 18	. . . . . . . . . . . 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 40227	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾))
43	21, 42	syl 18	. . . . . . . . . . . 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 18600	. . . . . . . . . . . 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 1399	. . . . . . . . . . 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 2799	. . . . . . . . . 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 5107	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → (𝑄 ≤ 𝑦 ↔ 𝑄 ≤ (𝑟 ∨ 𝑠)))
48	47	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → (¬ 𝑄 ≤ 𝑦 ↔ ¬ 𝑄 ≤ (𝑟 ∨ 𝑠)))
49		oveq1 7423	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → (𝑦 ∨ 𝑄) = ((𝑟 ∨ 𝑠) ∨ 𝑄))
50	49	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → (𝑋 = (𝑦 ∨ 𝑄) ↔ 𝑋 = ((𝑟 ∨ 𝑠) ∨ 𝑄)))
51	48, 50	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑟 ∨ 𝑠) → ((¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)) ↔ (¬ 𝑄 ≤ (𝑟 ∨ 𝑠) ∧ 𝑋 = ((𝑟 ∨ 𝑠) ∨ 𝑄))))
52	51	rspcev 3576	. . . . . . . . . 10 ⊢ (((𝑟 ∨ 𝑠) ∈ 𝑁 ∧ (¬ 𝑄 ≤ (𝑟 ∨ 𝑠) ∧ 𝑋 = ((𝑟 ∨ 𝑠) ∨ 𝑄))) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
53	31, 34, 46, 52	syl12anc 850	. . . . . . . . 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 1139	. . . . . . . 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 421	. . . . . . 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 3161	. . . . . 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 16	. . . . 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 1373	. . . 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 1251	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑧 ∈ 𝑁)
60		simpr1 1213	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → ¬ 𝑄 ≤ 𝑧)
61		simpll1 1231	. . . . . . . . . . . 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 40302	. . . . . . . . . . 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 40447	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑁 → 𝑧 ∈ (Base‘𝐾))
64	59, 63	syl 18	. . . . . . . . . . 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 1252	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 18561	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → 𝑧 ≤ (𝑧 ∨ 𝑟))
68	62, 64, 66, 67	syl3anc 1398	. . . . . . . . . 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 1254	. . . . . . . . . 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 5133	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑧 ≤ 𝑋)
71		simplr 781	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) ∧ (¬ 𝑄 ≤ 𝑧 ∧ (𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴) ∧ (¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑟)))) → 𝑄 ≤ 𝑋)
72		simpll3 1233	. . . . . . . . . . 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 18	. . . . . . . . . 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 1232	. . . . . . . . . . 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 18	. . . . . . . . . 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 18563	. . . . . . . . . 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 1399	. . . . . . . . 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 725	. . . . . . . 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 18552	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑧 ∨ 𝑄) ∈ (Base‘𝐾))
80	62, 64, 73, 79	syl3anc 1398	. . . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
82	3, 7, 8, 81, 9	cvr1 40348	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ 𝐴) → (¬ 𝑄 ≤ 𝑧 ↔ 𝑧( ⋖ ‘𝐾)(𝑧 ∨ 𝑄)))
83	61, 64, 72, 82	syl3anc 1398	. . . . . . . . . . 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 235	. . . . . . . . . 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 40470	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑧 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ 𝑁) ∧ 𝑧( ⋖ ‘𝐾)(𝑧 ∨ 𝑄)) → (𝑧 ∨ 𝑄) ∈ 𝑃)
86	61, 80, 59, 84, 85	syl31anc 1400	. . . . . . . . 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 40500	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑧 ∨ 𝑄) ∈ 𝑃 ∧ 𝑋 ∈ 𝑃) → ((𝑧 ∨ 𝑄) ≤ 𝑋 ↔ (𝑧 ∨ 𝑄) = 𝑋))
88	61, 86, 74, 87	syl3anc 1398	. . . . . . . 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 235	. . . . . . 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 2766	. . . . . 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 5107	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑄 ≤ 𝑦 ↔ 𝑄 ≤ 𝑧))
92	91	notbid 321	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (¬ 𝑄 ≤ 𝑦 ↔ ¬ 𝑄 ≤ 𝑧))
93		oveq1 7423	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ∨ 𝑄) = (𝑧 ∨ 𝑄))
94	93	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑋 = (𝑦 ∨ 𝑄) ↔ 𝑋 = (𝑧 ∨ 𝑄)))
95	92, 94	anbi12d 644	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)) ↔ (¬ 𝑄 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑄))))
96	95	rspcev 3576	. . . . . 6 ⊢ ((𝑧 ∈ 𝑁 ∧ (¬ 𝑄 ≤ 𝑧 ∧ 𝑋 = (𝑧 ∨ 𝑄))) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))
97	59, 60, 90, 96	syl12anc 850	. . . . 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 1373	. . . 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 181	. . 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 3218	. 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 16	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄 ≤ 𝑋) → ∃𝑦 ∈ 𝑁 (¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = (𝑦 ∨ 𝑄)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086   class class class wbr 5103  ‘cfv 6535  (class class class)co 7416  Basecbs 17326  lecple 17374  joincjn 18424  Latclat 18544   ⋖ ccvr 40200  Atomscatm 40201  HLchlt 40288  LLinesclln 40429  LPlanesclpl 40430

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-proset 18407  df-poset 18426  df-plt 18441  df-lub 18457  df-glb 18458  df-join 18459  df-meet 18460  df-p0 18536  df-lat 18545  df-clat 18612  df-oposet 40114  df-ol 40116  df-oml 40117  df-covers 40204  df-ats 40205  df-atl 40236  df-cvlat 40260  df-hlat 40289  df-llines 40436  df-lplanes 40437

This theorem is used by: (None)