Theorem dihmeetlem4preN 39842

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihmeetlem4.b	⊢ 𝐵 = (Base‘𝐾)
dihmeetlem4.l	⊢ ≤ = (le‘𝐾)
dihmeetlem4.m	⊢ ∧ = (meet‘𝐾)
dihmeetlem4.a	⊢ 𝐴 = (Atoms‘𝐾)
dihmeetlem4.h	⊢ 𝐻 = (LHyp‘𝐾)
dihmeetlem4.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihmeetlem4.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihmeetlem4.z	⊢ 0 = (0g‘𝑈)
dihmeetlem4.g	⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
dihmeetlem4.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dihmeetlem4.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihmeetlem4.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dihmeetlem4.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihmeetlem4.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))

Assertion

Ref	Expression
dihmeetlem4preN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) = { 0 })

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

Allowed substitution hints:   𝐴(ℎ)   𝐵(𝑔)   𝑃(ℎ)   𝑄(ℎ)   𝑅(𝑔,ℎ)   𝑈(𝑔,ℎ)   𝐸(𝑔,ℎ)   𝐺(𝑔,ℎ)   𝐼(𝑔,ℎ)   ≤ (ℎ)   ∧ (𝑔,ℎ)   𝑂(𝑔,ℎ)   𝑋(𝑔,ℎ)   0 (𝑔,ℎ)

Proof of Theorem dihmeetlem4preN

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

Step	Hyp	Ref	Expression
1		dihmeetlem4.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		dihmeetlem4.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
3	1, 2	dihvalrel 39815	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑄))
4		relin1 5773	. . . 4 ⊢ (Rel (𝐼‘𝑄) → Rel ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))))
5	3, 4	syl 17	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))))
6	5	3ad2ant1 1133	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → Rel ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))))
7	1, 2	dihvalrel 39815	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘(0.‘𝐾)))
8		eqid 2731	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
9		dihmeetlem4.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
10		dihmeetlem4.z	. . . . . 6 ⊢ 0 = (0g‘𝑈)
11	8, 1, 2, 9, 10	dih0 39816	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘(0.‘𝐾)) = { 0 })
12	11	releqd 5739	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Rel (𝐼‘(0.‘𝐾)) ↔ Rel { 0 }))
13	7, 12	mpbid 231	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel { 0 })
14	13	3ad2ant1 1133	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → Rel { 0 })
15		id 22	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
16		elin 3929	. . . 4 ⊢ (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊))))
17		dihmeetlem4.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
18		dihmeetlem4.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
19		dihmeetlem4.p	. . . . . . . . . 10 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
20		dihmeetlem4.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
21		dihmeetlem4.e	. . . . . . . . . 10 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
22		dihmeetlem4.g	. . . . . . . . . 10 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
23		vex 3450	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
24		vex 3450	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
25	17, 18, 1, 19, 20, 21, 2, 22, 23, 24	dihopelvalcqat 39782	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ↔ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
26	25	3adant2 1131	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ↔ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
27		simp1 1136	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28		simp1l 1197	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐾 ∈ HL)
29	28	hllatd 37899	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐾 ∈ Lat)
30		simp2l 1199	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
31		simp1r 1198	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
32		dihmeetlem4.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐾)
33	32, 1	lhpbase 38534	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
34	31, 33	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
35		dihmeetlem4.m	. . . . . . . . . . 11 ⊢ ∧ = (meet‘𝐾)
36	32, 35	latmcl 18343	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
37	29, 30, 34, 36	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
38	32, 17, 35	latmle2 18368	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
39	29, 30, 34, 38	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑋 ∧ 𝑊) ≤ 𝑊)
40		dihmeetlem4.r	. . . . . . . . . 10 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
41		dihmeetlem4.o	. . . . . . . . . 10 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
42	32, 17, 1, 20, 40, 41, 2	dihopelvalbN 39774	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊)) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂)))
43	27, 37, 39, 42	syl12anc 835	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊)) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂)))
44	26, 43	anbi12d 631	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊))) ↔ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))))
45		simprll 777	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → 𝑓 = (𝑠‘𝐺))
46		simprrr 780	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → 𝑠 = 𝑂)
47	46	fveq1d 6849	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑠‘𝐺) = (𝑂‘𝐺))
48		simpl1 1191	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
49	17, 18, 1, 19	lhpocnel2 38555	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
51		simpl3 1193	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
52	17, 18, 1, 20, 22	ltrniotacl 39115	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
53	48, 50, 51, 52	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → 𝐺 ∈ 𝑇)
54	41, 32	tendo02 39323	. . . . . . . . . . 11 ⊢ (𝐺 ∈ 𝑇 → (𝑂‘𝐺) = ( I ↾ 𝐵))
55	53, 54	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑂‘𝐺) = ( I ↾ 𝐵))
56	45, 47, 55	3eqtrd 2775	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → 𝑓 = ( I ↾ 𝐵))
57	56, 46	jca 512	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
58		simpl1 1191	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
59	58, 49	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
60		simpl3 1193	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
61	58, 59, 60, 52	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝐺 ∈ 𝑇)
62	61, 54	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑂‘𝐺) = ( I ↾ 𝐵))
63		simprr 771	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑠 = 𝑂)
64	63	fveq1d 6849	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑠‘𝐺) = (𝑂‘𝐺))
65		simprl 769	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑓 = ( I ↾ 𝐵))
66	62, 64, 65	3eqtr4rd 2782	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑓 = (𝑠‘𝐺))
67	32, 1, 20, 21, 41	tendo0cl 39326	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸)
68	58, 67	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑂 ∈ 𝐸)
69	63, 68	eqeltrd 2832	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑠 ∈ 𝐸)
70	32, 1, 20	idltrn 38686	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
71	58, 70	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → ( I ↾ 𝐵) ∈ 𝑇)
72	65, 71	eqeltrd 2832	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑓 ∈ 𝑇)
73	65	fveq2d 6851	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑅‘𝑓) = (𝑅‘( I ↾ 𝐵)))
74	32, 8, 1, 40	trlid0 38712	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑅‘( I ↾ 𝐵)) = (0.‘𝐾))
75	58, 74	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑅‘( I ↾ 𝐵)) = (0.‘𝐾))
76	73, 75	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑅‘𝑓) = (0.‘𝐾))
77		simpl1l 1224	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝐾 ∈ HL)
78		hlatl 37895	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
79	77, 78	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝐾 ∈ AtLat)
80	37	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
81	32, 17, 8	atl0le 37839	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ AtLat ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → (0.‘𝐾) ≤ (𝑋 ∧ 𝑊))
82	79, 80, 81	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (0.‘𝐾) ≤ (𝑋 ∧ 𝑊))
83	76, 82	eqbrtrd 5132	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))
84	72, 83, 63	jca31 515	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂))
85	66, 69, 84	jca31 515	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂)))
86	57, 85	impbida 799	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑠 = 𝑂)) ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
87	44, 86	bitrd 278	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊))) ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
88		opex 5426	. . . . . . . 8 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
89	88	elsn 4606	. . . . . . 7 ⊢ (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
90	23, 24	opth 5438	. . . . . . 7 ⊢ (⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
91	89, 90	bitr2i 275	. . . . . 6 ⊢ ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩})
92	87, 91	bitrdi 286	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊))) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩}))
93	32, 1, 20, 9, 10, 41	dvh0g 39647	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
94	93	3ad2ant1 1133	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
95	94	sneqd 4603	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
96	95	eleq2d 2818	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ { 0 } ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩}))
97	92, 96	bitr4d 281	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊))) ↔ ⟨𝑓, 𝑠⟩ ∈ { 0 }))
98	16, 97	bitrid 282	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) ↔ ⟨𝑓, 𝑠⟩ ∈ { 0 }))
99	98	eqrelrdv2 5756	. 2 ⊢ (((Rel ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) ∧ Rel { 0 }) ∧ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) = { 0 })
100	6, 14, 15, 99	syl21anc 836	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) = { 0 })

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3912  {csn 4591  ⟨cop 4597   class class class wbr 5110   ↦ cmpt 5193   I cid 5535   ↾ cres 5640  Rel wrel 5643  ‘cfv 6501  ℩crio 7317  (class class class)co 7362  Basecbs 17094  lecple 17154  occoc 17155  0_gc0g 17335  meetcmee 18215  0.cp0 18326  Latclat 18334  Atomscatm 37798  AtLatcal 37799  HLchlt 37885  LHypclh 38520  LTrncltrn 38637  trLctrl 38694  TEndoctendo 39288  DVecHcdvh 39614  DIsoHcdih 39764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-riotaBAD 37488

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-tpos 8162  df-undef 8209  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-n0 12423  df-z 12509  df-uz 12773  df-fz 13435  df-struct 17030  df-sets 17047  df-slot 17065  df-ndx 17077  df-base 17095  df-ress 17124  df-plusg 17160  df-mulr 17161  df-sca 17163  df-vsca 17164  df-0g 17337  df-proset 18198  df-poset 18216  df-plt 18233  df-lub 18249  df-glb 18250  df-join 18251  df-meet 18252  df-p0 18328  df-p1 18329  df-lat 18335  df-clat 18402  df-mgm 18511  df-sgrp 18560  df-mnd 18571  df-submnd 18616  df-grp 18765  df-minusg 18766  df-sbg 18767  df-subg 18939  df-cntz 19111  df-lsm 19432  df-cmn 19578  df-abl 19579  df-mgp 19911  df-ur 19928  df-ring 19980  df-oppr 20063  df-dvdsr 20084  df-unit 20085  df-invr 20115  df-dvr 20126  df-drng 20227  df-lmod 20380  df-lss 20450  df-lsp 20490  df-lvec 20621  df-oposet 37711  df-ol 37713  df-oml 37714  df-covers 37801  df-ats 37802  df-atl 37833  df-cvlat 37857  df-hlat 37886  df-llines 38034  df-lplanes 38035  df-lvols 38036  df-lines 38037  df-psubsp 38039  df-pmap 38040  df-padd 38332  df-lhyp 38524  df-laut 38525  df-ldil 38640  df-ltrn 38641  df-trl 38695  df-tendo 39291  df-edring 39293  df-disoa 39565  df-dvech 39615  df-dib 39675  df-dic 39709  df-dih 39765

This theorem is referenced by:  dihmeetlem4N  39843