Theorem dihmeetlem13N 37933

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 37893	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑄))
4	3	3ad2ant1 1114	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → Rel (𝐼‘𝑄))
5		relin1 5532	. . . 4 ⊢ (Rel (𝐼‘𝑄) → Rel ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
6	4, 5	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → Rel ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
7		elin 4052	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅)))
8		simp1 1117	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simp2l 1180	. . . . . . . 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 3413	. . . . . . . . 9 ⊢ 𝑓 ∈ V
17		vex 3413	. . . . . . . . 9 ⊢ 𝑠 ∈ V
18	10, 11, 1, 12, 13, 14, 2, 15, 16, 17	dihopelvalcqat 37860	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ↔ (𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸)))
19	8, 9, 18	syl2anc 576	. . . . . . 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 1181	. . . . . . . 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 37860	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅) ↔ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
23	8, 20, 22	syl2anc 576	. . . . . . 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 622	. . . . . 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	syl5bb 275	. . . . 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 767	. . . . . . . 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 1174	. . . . . . . . . . 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 6496	. . . . . . . . . . . . 13 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃))
29		simpl1 1172	. . . . . . . . . . . . . . 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 36633	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
31	29, 30	syl 17	. . . . . . . . . . . . . . 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 1207	. . . . . . . . . . . . . . 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 37195	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑄)
34	29, 31, 32, 33	syl3anc 1352	. . . . . . . . . . . . . 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 1208	. . . . . . . . . . . . . . 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 37195	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐺‘𝑃) = 𝑅)
37	29, 31, 35, 36	syl3anc 1352	. . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . 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	syl5ib 236	. . . . . . . . . . . 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 2983	. . . . . . . . . . 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 15	. . . . . . . . . 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 1221	. . . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . 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 1184	. . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 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 1267	. . . . . . . . . . . . . . 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 37193	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
51	45, 48, 49, 50	syl3anc 1352	. . . . . . . . . . . . . 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 1268	. . . . . . . . . . . . . . 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 37193	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
54	45, 48, 52, 53	syl3anc 1352	. . . . . . . . . . . . . 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 37637	. . . . . . . . . . . . . 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 1360	. . . . . . . . . . . . 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 224	. . . . . . . . . . . 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 1102	. . . . . . . . . . 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 2984	. . . . . . . . . 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 15	. . . . . . . . 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 6499	. . . . . . . 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 1352	. . . . . . . . 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 37401	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵))
66	64, 65	syl 17	. . . . . . . 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 2813	. . . . . . 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 504	. . . . . 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 405	. . . . 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 232	. . . 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		dihmeetlem13.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
72		dihmeetlem13.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑈)
73	55, 1, 13, 71, 72, 56	dvh0g 37725	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
74	73	3ad2ant1 1114	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
75	74	sneqd 4448	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
76	75	eleq2d 2846	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (⟨𝑓, 𝑠⟩ ∈ { 0 } ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩}))
77		opex 5210	. . . . . . 7 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
78	77	elsn 4451	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
79	16, 17	opth 5222	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
80	78, 79	bitr2i 268	. . . . 5 ⊢ ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩})
81	76, 80	syl6rbbr 282	. . . 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 231	. . 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 5508	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ⊆ { 0 })
84	1, 71, 8	dvhlmod 37724	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑈 ∈ LMod)
85		simp2ll 1221	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ∈ 𝐴)
86	55, 11	atbase 35903	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
87	85, 86	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ∈ 𝐵)
88		eqid 2773	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
89	55, 1, 2, 71, 88	dihlss 37864	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
90	8, 87, 89	syl2anc 576	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
91		simp2rl 1223	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑅 ∈ 𝐴)
92	55, 11	atbase 35903	. . . . . 6 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ 𝐵)
93	91, 92	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑅 ∈ 𝐵)
94	55, 1, 2, 71, 88	dihlss 37864	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐵) → (𝐼‘𝑅) ∈ (LSubSp‘𝑈))
95	8, 93, 94	syl2anc 576	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐼‘𝑅) ∈ (LSubSp‘𝑈))
96	88	lssincl 19472	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝐼‘𝑄) ∈ (LSubSp‘𝑈) ∧ (𝐼‘𝑅) ∈ (LSubSp‘𝑈)) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈))
97	84, 90, 95, 96	syl3anc 1352	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈))
98	72, 88	lss0ss 19455	. . 3 ⊢ ((𝑈 ∈ LMod ∧ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈)) → { 0 } ⊆ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
99	84, 97, 98	syl2anc 576	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → { 0 } ⊆ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
100	83, 99	eqssd 3870	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) = { 0 })

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   ≠ wne 2962   ∩ cin 3823   ⊆ wss 3824  {csn 4436  ⟨cop 4442   class class class wbr 4926   ↦ cmpt 5005   I cid 5308   ↾ cres 5406  Rel wrel 5409  ‘cfv 6186  ℩crio 6935  Basecbs 16338  lecple 16427  occoc 16428  0_gc0g 16568  joincjn 17425  LModclmod 19369  LSubSpclss 19438  Atomscatm 35877  HLchlt 35964  LHypclh 36598  LTrncltrn 36715  TEndoctendo 37366  DVecHcdvh 37692  DIsoHcdih 37842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-riotaBAD 35567

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-iin 4792  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-1st 7500  df-2nd 7501  df-tpos 7694  df-undef 7741  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-oadd 7908  df-er 8088  df-map 8207  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-nn 11439  df-2 11502  df-3 11503  df-4 11504  df-5 11505  df-6 11506  df-n0 11707  df-z 11793  df-uz 12058  df-fz 12708  df-struct 16340  df-ndx 16341  df-slot 16342  df-base 16344  df-sets 16345  df-ress 16346  df-plusg 16433  df-mulr 16434  df-sca 16436  df-vsca 16437  df-0g 16570  df-proset 17409  df-poset 17427  df-plt 17439  df-lub 17455  df-glb 17456  df-join 17457  df-meet 17458  df-p0 17520  df-p1 17521  df-lat 17527  df-clat 17589  df-mgm 17723  df-sgrp 17765  df-mnd 17776  df-submnd 17817  df-grp 17907  df-minusg 17908  df-sbg 17909  df-subg 18073  df-cntz 18231  df-lsm 18535  df-cmn 18681  df-abl 18682  df-mgp 18976  df-ur 18988  df-ring 19035  df-oppr 19109  df-dvdsr 19127  df-unit 19128  df-invr 19158  df-dvr 19169  df-drng 19240  df-lmod 19371  df-lss 19439  df-lsp 19479  df-lvec 19610  df-oposet 35790  df-ol 35792  df-oml 35793  df-covers 35880  df-ats 35881  df-atl 35912  df-cvlat 35936  df-hlat 35965  df-llines 36112  df-lplanes 36113  df-lvols 36114  df-lines 36115  df-psubsp 36117  df-pmap 36118  df-padd 36410  df-lhyp 36602  df-laut 36603  df-ldil 36718  df-ltrn 36719  df-trl 36773  df-tendo 37369  df-edring 37371  df-disoa 37643  df-dvech 37693  df-dib 37753  df-dic 37787  df-dih 37843

This theorem is referenced by:  dihmeetlem15N  37935