Theorem dihmeetlem13N 39580

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 39540	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑄))
4	3	3ad2ant1 1132	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → Rel (𝐼‘𝑄))
5		relin1 5748	. . . 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 3913	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅)))
8		simp1 1135	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simp2l 1198	. . . . . . . 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 3445	. . . . . . . . 9 ⊢ 𝑓 ∈ V
17		vex 3445	. . . . . . . . 9 ⊢ 𝑠 ∈ V
18	10, 11, 1, 12, 13, 14, 2, 15, 16, 17	dihopelvalcqat 39507	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ↔ (𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸)))
19	8, 9, 18	syl2anc 584	. . . . . . 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 1199	. . . . . . . 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 39507	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅) ↔ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
23	8, 20, 22	syl2anc 584	. . . . . . 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 631	. . . . . 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 282	. . . . 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 776	. . . . . . . 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 1192	. . . . . . . . . . 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 6818	. . . . . . . . . . . . 13 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃))
29		simpl1 1190	. . . . . . . . . . . . . . 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 38280	. . . . . . . . . . . . . . . 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 1225	. . . . . . . . . . . . . . 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 38842	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑄)
34	29, 31, 32, 33	syl3anc 1370	. . . . . . . . . . . . . 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 1226	. . . . . . . . . . . . . . 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 38842	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐺‘𝑃) = 𝑅)
37	29, 31, 35, 36	syl3anc 1370	. . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 2961	. . . . . . . . . . 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 1239	. . . . . . . . . . . . . 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 1241	. . . . . . . . . . . . . 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 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)) ∧ 𝑠 ≠ 𝑂) → (𝑠‘𝐹) = (𝑠‘𝐺))
45		simp11 1202	. . . . . . . . . . . . . 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 1242	. . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . 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 1285	. . . . . . . . . . . . . . 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 38840	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
51	45, 48, 49, 50	syl3anc 1370	. . . . . . . . . . . . . 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 1286	. . . . . . . . . . . . . . 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 38840	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
54	45, 48, 52, 53	syl3anc 1370	. . . . . . . . . . . . . 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 39284	. . . . . . . . . . . . . 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 1378	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 1120	. . . . . . . . . . 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 2962	. . . . . . . . . 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 6821	. . . . . . . 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 1370	. . . . . . . . 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 39048	. . . . . . . . 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 2780	. . . . . . 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 512	. . . . . 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 413	. . . . 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 239	. . . 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 5403	. . . . . . 7 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
72	71	elsn 4587	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
73	16, 17	opth 5415	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
74	72, 73	bitr2i 275	. . . . 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 39372	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
78	77	3ad2ant1 1132	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
79	78	sneqd 4584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
80	79	eleq2d 2822	. . . . 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 289	. . . 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 238	. . 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 5724	. 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 39371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑈 ∈ LMod)
85		simp2ll 1239	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ∈ 𝐴)
86	55, 11	atbase 37549	. . . . . 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 2736	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
89	55, 1, 2, 75, 88	dihlss 39511	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
90	8, 87, 89	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
91		simp2rl 1241	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑅 ∈ 𝐴)
92	55, 11	atbase 37549	. . . . . 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, 75, 88	dihlss 39511	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐵) → (𝐼‘𝑅) ∈ (LSubSp‘𝑈))
95	8, 93, 94	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐼‘𝑅) ∈ (LSubSp‘𝑈))
96	88	lssincl 20325	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝐼‘𝑄) ∈ (LSubSp‘𝑈) ∧ (𝐼‘𝑅) ∈ (LSubSp‘𝑈)) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈))
97	84, 90, 95, 96	syl3anc 1370	. . 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 20308	. . 3 ⊢ ((𝑈 ∈ LMod ∧ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈)) → { 0 } ⊆ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
99	84, 97, 98	syl2anc 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → { 0 } ⊆ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
100	83, 99	eqssd 3948	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2940   ∩ cin 3896   ⊆ wss 3897  {csn 4572  ⟨cop 4578   class class class wbr 5089   ↦ cmpt 5172   I cid 5511   ↾ cres 5616  Rel wrel 5619  ‘cfv 6473  ℩crio 7285  Basecbs 17001  lecple 17058  occoc 17059  0_gc0g 17239  joincjn 18118  LModclmod 20221  LSubSpclss 20291  Atomscatm 37523  HLchlt 37610  LHypclh 38245  LTrncltrn 38362  TEndoctendo 39013  DVecHcdvh 39339  DIsoHcdih 39489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041  ax-riotaBAD 37213

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-iin 4941  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-tpos 8104  df-undef 8151  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-er 8561  df-map 8680  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-3 12130  df-4 12131  df-5 12132  df-6 12133  df-n0 12327  df-z 12413  df-uz 12676  df-fz 13333  df-struct 16937  df-sets 16954  df-slot 16972  df-ndx 16984  df-base 17002  df-ress 17031  df-plusg 17064  df-mulr 17065  df-sca 17067  df-vsca 17068  df-0g 17241  df-proset 18102  df-poset 18120  df-plt 18137  df-lub 18153  df-glb 18154  df-join 18155  df-meet 18156  df-p0 18232  df-p1 18233  df-lat 18239  df-clat 18306  df-mgm 18415  df-sgrp 18464  df-mnd 18475  df-submnd 18520  df-grp 18668  df-minusg 18669  df-sbg 18670  df-subg 18840  df-cntz 19011  df-lsm 19329  df-cmn 19475  df-abl 19476  df-mgp 19808  df-ur 19825  df-ring 19872  df-oppr 19949  df-dvdsr 19970  df-unit 19971  df-invr 20001  df-dvr 20012  df-drng 20087  df-lmod 20223  df-lss 20292  df-lsp 20332  df-lvec 20463  df-oposet 37436  df-ol 37438  df-oml 37439  df-covers 37526  df-ats 37527  df-atl 37558  df-cvlat 37582  df-hlat 37611  df-llines 37759  df-lplanes 37760  df-lvols 37761  df-lines 37762  df-psubsp 37764  df-pmap 37765  df-padd 38057  df-lhyp 38249  df-laut 38250  df-ldil 38365  df-ltrn 38366  df-trl 38420  df-tendo 39016  df-edring 39018  df-disoa 39290  df-dvech 39340  df-dib 39400  df-dic 39434  df-dih 39490

This theorem is referenced by:  dihmeetlem15N  39582