Theorem dihmeetlem13N 41020

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 40980	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑄))
4	3	3ad2ant1 1130	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → Rel (𝐼‘𝑄))
5		relin1 5820	. . . 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 3963	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅)))
8		simp1 1133	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simp2l 1196	. . . . . . . 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 3466	. . . . . . . . 9 ⊢ 𝑓 ∈ V
17		vex 3466	. . . . . . . . 9 ⊢ 𝑠 ∈ V
18	10, 11, 1, 12, 13, 14, 2, 15, 16, 17	dihopelvalcqat 40947	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ↔ (𝑓 = (𝑠‘𝐹) ∧ 𝑠 ∈ 𝐸)))
19	8, 9, 18	syl2anc 582	. . . . . . 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 1197	. . . . . . . 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 40947	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑅) ↔ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
23	8, 20, 22	syl2anc 582	. . . . . . 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 630	. . . . . 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 777	. . . . . . . 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 1190	. . . . . . . . . . 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 6902	. . . . . . . . . . . . 13 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃))
29		simpl1 1188	. . . . . . . . . . . . . . 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 39720	. . . . . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . . 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 40282	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑄)
34	29, 31, 32, 33	syl3anc 1368	. . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . . 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 40282	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐺‘𝑃) = 𝑅)
37	29, 31, 35, 36	syl3anc 1368	. . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . 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 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 2951	. . . . . . . . . . 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 1237	. . . . . . . . . . . . . 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 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‘𝐺))
44	42, 43	eqtr3d 2768	. . . . . . . . . . . . 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 1200	. . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . 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 1283	. . . . . . . . . . . . . . 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 40280	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
51	45, 48, 49, 50	syl3anc 1368	. . . . . . . . . . . . . 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 1284	. . . . . . . . . . . . . . 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 40280	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
54	45, 48, 52, 53	syl3anc 1368	. . . . . . . . . . . . . 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 40724	. . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . 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 1118	. . . . . . . . . . 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 2952	. . . . . . . . . 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 6905	. . . . . . . 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 1368	. . . . . . . . 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 40488	. . . . . . . . 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 2770	. . . . . . 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 510	. . . . . 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 411	. . . . 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 5472	. . . . . . 7 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
72	71	elsn 4648	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
73	16, 17	opth 5484	. . . . . 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 40812	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
78	77	3ad2ant1 1130	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
79	78	sneqd 4645	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
80	79	eleq2d 2812	. . . . 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 5796	. 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 40811	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑈 ∈ LMod)
85		simp2ll 1237	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ∈ 𝐴)
86	55, 11	atbase 38989	. . . . . 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 2726	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
89	55, 1, 2, 75, 88	dihlss 40951	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
90	8, 87, 89	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
91		simp2rl 1239	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → 𝑅 ∈ 𝐴)
92	55, 11	atbase 38989	. . . . . 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 40951	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐵) → (𝐼‘𝑅) ∈ (LSubSp‘𝑈))
95	8, 93, 94	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → (𝐼‘𝑅) ∈ (LSubSp‘𝑈))
96	88	lssincl 20944	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝐼‘𝑄) ∈ (LSubSp‘𝑈) ∧ (𝐼‘𝑅) ∈ (LSubSp‘𝑈)) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈))
97	84, 90, 95, 96	syl3anc 1368	. . 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 20928	. . 3 ⊢ ((𝑈 ∈ LMod ∧ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) ∈ (LSubSp‘𝑈)) → { 0 } ⊆ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
99	84, 97, 98	syl2anc 582	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → { 0 } ⊆ ((𝐼‘𝑄) ∩ (𝐼‘𝑅)))
100	83, 99	eqssd 3997	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 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ≠ wne 2930   ∩ cin 3946   ⊆ wss 3947  {csn 4633  ⟨cop 4639   class class class wbr 5155   ↦ cmpt 5238   I cid 5581   ↾ cres 5686  Rel wrel 5689  ‘cfv 6556  ℩crio 7381  Basecbs 17215  lecple 17275  occoc 17276  0_gc0g 17456  joincjn 18338  LModclmod 20838  LSubSpclss 20910  Atomscatm 38963  HLchlt 39050  LHypclh 39685  LTrncltrn 39802  TEndoctendo 40453  DVecHcdvh 40779  DIsoHcdih 40929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11216  ax-resscn 11217  ax-1cn 11218  ax-icn 11219  ax-addcl 11220  ax-addrcl 11221  ax-mulcl 11222  ax-mulrcl 11223  ax-mulcom 11224  ax-addass 11225  ax-mulass 11226  ax-distr 11227  ax-i2m1 11228  ax-1ne0 11229  ax-1rid 11230  ax-rnegex 11231  ax-rrecex 11232  ax-cnre 11233  ax-pre-lttri 11234  ax-pre-lttrn 11235  ax-pre-ltadd 11236  ax-pre-mulgt0 11237  ax-riotaBAD 38653

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4916  df-int 4957  df-iun 5005  df-iin 5006  df-br 5156  df-opab 5218  df-mpt 5239  df-tr 5273  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6314  df-ord 6381  df-on 6382  df-lim 6383  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8005  df-2nd 8006  df-tpos 8243  df-undef 8290  df-frecs 8298  df-wrecs 8329  df-recs 8403  df-rdg 8442  df-1o 8498  df-er 8736  df-map 8859  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11498  df-neg 11499  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-n0 12527  df-z 12613  df-uz 12877  df-fz 13541  df-struct 17151  df-sets 17168  df-slot 17186  df-ndx 17198  df-base 17216  df-ress 17245  df-plusg 17281  df-mulr 17282  df-sca 17284  df-vsca 17285  df-0g 17458  df-proset 18322  df-poset 18340  df-plt 18357  df-lub 18373  df-glb 18374  df-join 18375  df-meet 18376  df-p0 18452  df-p1 18453  df-lat 18459  df-clat 18526  df-mgm 18635  df-sgrp 18714  df-mnd 18730  df-submnd 18776  df-grp 18933  df-minusg 18934  df-sbg 18935  df-subg 19119  df-cntz 19313  df-lsm 19636  df-cmn 19782  df-abl 19783  df-mgp 20120  df-rng 20138  df-ur 20167  df-ring 20220  df-oppr 20318  df-dvdsr 20341  df-unit 20342  df-invr 20372  df-dvr 20385  df-drng 20711  df-lmod 20840  df-lss 20911  df-lsp 20951  df-lvec 21083  df-oposet 38876  df-ol 38878  df-oml 38879  df-covers 38966  df-ats 38967  df-atl 38998  df-cvlat 39022  df-hlat 39051  df-llines 39199  df-lplanes 39200  df-lvols 39201  df-lines 39202  df-psubsp 39204  df-pmap 39205  df-padd 39497  df-lhyp 39689  df-laut 39690  df-ldil 39805  df-ltrn 39806  df-trl 39860  df-tendo 40456  df-edring 40458  df-disoa 40730  df-dvech 40780  df-dib 40840  df-dic 40874  df-dih 40930

This theorem is referenced by:  dihmeetlem15N  41022