Theorem dihmeetlem13N 42179

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihmeetlem13.b	⊢ 𝐵 = (Base‘𝐾)
dihmeetlem13.l	⊢ ≤ = (le‘𝐾)
dihmeetlem13.j	⊢ ∨ = (join‘𝐾)
dihmeetlem13.a	⊢ 𝐴 = (Atoms‘𝐾)
dihmeetlem13.h	⊢ 𝐻 = (LHyp‘𝐾)
dihmeetlem13.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dihmeetlem13.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihmeetlem13.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihmeetlem13.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
dihmeetlem13.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihmeetlem13.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihmeetlem13.z	⊢ 0 = (0g‘𝑈)
dihmeetlem13.f	⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
dihmeetlem13.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)

Assertion

Ref	Expression
dihmeetlem13N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) = { 0 })

Distinct variable groups:   ≤ ,ℎ   𝐴,ℎ   𝐵,ℎ   ℎ,𝐻   ℎ,𝐾   𝑃,ℎ   𝑄,ℎ   𝑅,ℎ   𝑇,ℎ   ℎ,𝑊

Allowed substitution hints:   𝑈(ℎ)   𝐸(ℎ)   𝐹(ℎ)   𝐺(ℎ)   𝐼(ℎ)   ∨ (ℎ)   𝑂(ℎ)   0 (ℎ)

Proof of Theorem dihmeetlem13N

Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihmeetlem13.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		dihmeetlem13.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
3	1, 2	dihvalrel 42139	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑄))
4	3	3ad2ant1 1151	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → Rel (𝐼‘𝑄))
5		relin1 5797	. . . 4 ⊢ (Rel (𝐼‘𝑄) → Rel ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
6	4, 5	syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → Rel ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
7		elin 3918	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅)))
8		simp1 1154	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simp2l 1218	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
10		dihmeetlem13.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
11		dihmeetlem13.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
12		dihmeetlem13.p	. . . . . . . . 9 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
13		dihmeetlem13.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
14		dihmeetlem13.e	. . . . . . . . 9 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
15		dihmeetlem13.f	. . . . . . . . 9 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
16		vex 3457	. . . . . . . . 9 ⊢ 𝑓 ∈ V
17		vex 3457	. . . . . . . . 9 ⊢ 𝑠 ∈ V
18	10, 11, 1, 12, 13, 14, 2, 15, 16, 17	dihopelvalcqat 42106	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ↔ (𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸)))
19	8, 9, 18	syl2anc 596	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ↔ (𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸)))
20		simp2r 1219	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
21		dihmeetlem13.g	. . . . . . . . 9 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)
22	10, 11, 1, 12, 13, 14, 2, 21, 16, 17	dihopelvalcqat 42106	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅) ↔ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
23	8, 20, 22	syl2anc 596	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅) ↔ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
24	19, 23	anbi12d 644	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅)) ↔ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))))
25	7, 24	bitrid 286	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ↔ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))))
26		simprll 791	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → 𝑓 = (𝑠‘𝐹))
27		simpl3 1212	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → 𝑄 ≠ 𝑅)
28		fveq1 6881	. . . . . . . . . . . . 13 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃))
29		simpl1 1210	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30	10, 11, 1, 12	lhpocnel2 40879	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
31	29, 30	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
32		simpl2l 1245	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
33	10, 11, 1, 13, 15	ltrniotaval 41441	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑄)
34	29, 31, 32, 33	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝐹‘𝑃) = 𝑄)
35		simpl2r 1246	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
36	10, 11, 1, 13, 21	ltrniotaval 41441	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐺‘𝑃) = 𝑅)
37	29, 31, 35, 36	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝐺‘𝑃) = 𝑅)
38	34, 37	eqeq12d 2778	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → ((𝐹‘𝑃) = (𝐺‘𝑃) ↔ 𝑄 = 𝑅))
39	28, 38	imbitrid 247	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝐹 = 𝐺 → 𝑄 = 𝑅))
40	39	necon3d 2978	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝑄 ≠ 𝑅 → 𝐹 ≠ 𝐺))
41	27, 40	mpd 16	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → 𝐹 ≠ 𝐺)
42		simp2ll 1259	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → 𝑓 = (𝑠‘𝐹))
43		simp2rl 1261	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → 𝑓 = (𝑠‘𝐺))
44	42, 43	eqtr3d 2799	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → (𝑠‘𝐹) = (𝑠‘𝐺))
45		simp11 1222	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46		simp2rr 1262	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → 𝑠 ∈ 𝐸)
47		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → 𝑠 ≠ 𝑂)
48	45, 30	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
49		simp12l 1305	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
50	10, 11, 1, 13, 15	ltrniotacl 41439	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
51	45, 48, 49, 50	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → 𝐹 ∈ 𝑇)
52		simp12r 1306	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
53	10, 11, 1, 13, 21	ltrniotacl 41439	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
54	45, 48, 52, 53	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → 𝐺 ∈ 𝑇)
55		dihmeetlem13.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝐾)
56		dihmeetlem13.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
57	55, 1, 13, 14, 56	tendospcanN 41883	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑠‘𝐹) = (𝑠‘𝐺) ↔ 𝐹 = 𝐺))
58	45, 46, 47, 51, 54, 57	syl122anc 1406	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → ((𝑠‘𝐹) = (𝑠‘𝐺) ↔ 𝐹 = 𝐺))
59	44, 58	mpbid 235	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ∧ 𝑠 ≠ 𝑂) → 𝐹 = 𝐺)
60	59	3expia 1139	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝑠 ≠ 𝑂 → 𝐹 = 𝐺))
61	60	necon1d 2979	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝐹 ≠ 𝐺 → 𝑠 = 𝑂))
62	41, 61	mpd 16	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → 𝑠 = 𝑂)
63	62	fveq1d 6884	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝑠‘𝐹) = (𝑂‘𝐹))
64	29, 31, 32, 50	syl3anc 1398	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → 𝐹 ∈ 𝑇)
65	56, 55	tendo02 41647	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵))
66	64, 65	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝑂‘𝐹) = ( I ↾ 𝐵))
67	26, 63, 66	3eqtrd 2801	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → 𝑓 = ( I ↾ 𝐵))
68	67, 62	jca 521	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) ∧ ((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
69	68	ex 418	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (((𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸) ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
70	25, 69	sylbid 243	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
71		opex 5443	. . . . . . 7 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
72	71	elsn 4602	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
73	16, 17	opth 5456	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
74	72, 73	bitr2i 279	. . . . 5 ⊢ ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩})
75		dihmeetlem13.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
76		dihmeetlem13.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑈)
77	55, 1, 13, 75, 76, 56	dvh0g 41971	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
78	77	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
79	78	sneqd 4599	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
80	79	eleq2d 2848	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (⟨𝑓, 𝑠⟩ ∈ { 0 } ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩}))
81	74, 80	bitr4id 293	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ { 0 }))
82	70, 81	sylibd 242	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) → ⟨𝑓, 𝑠⟩ ∈ { 0 }))
83	6, 82	relssdv 5772	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ⊆ { 0 })
84	1, 75, 8	dvhlmod 41970	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑈 ∈ LMod)
85		simp2ll 1259	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ∈ 𝐴)
86	55, 11	atbase 40149	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
87	85, 86	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ∈ 𝐵)
88		eqid 2762	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
89	55, 1, 2, 75, 88	dihlss 42110	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
90	8, 87, 89	syl2anc 596	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
91		simp2rl 1261	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑅 ∈ 𝐴)
92	55, 11	atbase 40149	. . . . . 6 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ 𝐵)
93	91, 92	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑅 ∈ 𝐵)
94	55, 1, 2, 75, 88	dihlss 42110	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐵) → (𝐼‘𝑅) ∈ (LSubSp‘𝑈))
95	8, 93, 94	syl2anc 596	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐼‘𝑅) ∈ (LSubSp‘𝑈))
96	88	lssincl 21150	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝐼‘𝑄) ∈ (LSubSp‘𝑈) ∧ (𝐼‘𝑅) ∈ (LSubSp‘𝑈)) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈))
97	84, 90, 95, 96	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈))
98	76, 88	lss0ss 21134	. . 3 ⊢ ((𝑈 ∈ LMod ∧ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈)) → { 0 } ⊆ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
99	84, 97, 98	syl2anc 596	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → { 0 } ⊆ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
100	83, 99	eqssd 3951	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) = { 0 })

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   ∩ cin 3901   ⊆ wss 3902  {csn 4587  ⟨cop 4593   class class class wbr 5107   ↦ cmpt 5190   I cid 5553   ↾ cres 5661  Rel wrel 5664  ‘cfv 6537  ℩crio 7372  Basecbs 17305  lecple 17353  occoc 17354  0_gc0g 17528  joincjn 18403  LModclmod 21045  LSubSpclss 21116  Atomscatm 40123  HLchlt 40210  LHypclh 40844  LTrncltrn 40961  TEndoctendo 41612  DVecHcdvh 41938  DIsoHcdih 42088

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-riotaBAD 39813

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-tpos 8227  df-undef 8274  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13564  df-struct 17243  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-mulr 17360  df-sca 17362  df-vsca 17363  df-0g 17530  df-proset 18386  df-poset 18405  df-plt 18420  df-lub 18436  df-glb 18437  df-join 18438  df-meet 18439  df-p0 18515  df-p1 18516  df-lat 18524  df-clat 18591  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-submnd 18893  df-grp 19061  df-minusg 19062  df-sbg 19063  df-subg 19247  df-cntz 19445  df-lsm 19764  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-ring 20375  df-oppr 20479  df-dvdsr 20499  df-unit 20500  df-invr 20530  df-dvr 20543  df-drng 20893  df-lmod 21047  df-lss 21117  df-lsp 21157  df-lvec 21288  df-oposet 40036  df-ol 40038  df-oml 40039  df-covers 40126  df-ats 40127  df-atl 40158  df-cvlat 40182  df-hlat 40211  df-llines 40358  df-lplanes 40359  df-lvols 40360  df-lines 40361  df-psubsp 40363  df-pmap 40364  df-padd 40656  df-lhyp 40848  df-laut 40849  df-ldil 40964  df-ltrn 40965  df-trl 41019  df-tendo 41615  df-edring 41617  df-disoa 41889  df-dvech 41939  df-dib 41999  df-dic 42033  df-dih 42089

This theorem is used by:  dihmeetlem15N  42181