Theorem dihmeetlem4preN 41249

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 41222	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑄))
4		relin1 5804	. . . 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 41222	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘(0.‘𝐾)))
8		eqid 2734	. . . . . 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 41223	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘(0.‘𝐾)) = { 0 })
12	11	releqd 5770	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Rel (𝐼‘(0.‘𝐾)) ↔ Rel { 0 }))
13	7, 12	mpbid 232	. . 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 3949	. . . 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 3468	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
24		vex 3468	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
25	17, 18, 1, 19, 20, 21, 2, 22, 23, 24	dihopelvalcqat 41189	. . . . . . . . 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 39306	. . . . . . . . . 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 39941	. . . . . . . . . . 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 18459	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
37	29, 30, 34, 36	syl3anc 1372	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
38	32, 17, 35	latmle2 18484	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
39	29, 30, 34, 38	syl3anc 1372	. . . . . . . . 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 41181	. . . . . . . . 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 836	. . . . . . . 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 632	. . . . . . 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 778	. . . . . . . . . 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 781	. . . . . . . . . . 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 6889	. . . . . . . . . 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 39962	. . . . . . . . . . . . 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 40522	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
53	48, 50, 51, 52	syl3anc 1372	. . . . . . . . . . 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 40730	. . . . . . . . . . 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 2773	. . . . . . . . 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 511	. . . . . . . 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 1372	. . . . . . . . . . 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 772	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑠 = 𝑂)
64	63	fveq1d 6889	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑠‘𝐺) = (𝑂‘𝐺))
65		simprl 770	. . . . . . . . . 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 2780	. . . . . . . . 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 40733	. . . . . . . . . . 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 2833	. . . . . . . . 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 40093	. . . . . . . . . . . 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 2833	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑓 ∈ 𝑇)
73	65	fveq2d 6891	. . . . . . . . . . . 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 40119	. . . . . . . . . . . . 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 2769	. . . . . . . . . . 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 39302	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 39246	. . . . . . . . . . . 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 5147	. . . . . . . . . 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 514	. . . . . . . . 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 514	. . . . . . . 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 800	. . . . . . 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 279	. . . . . 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 5451	. . . . . . . 8 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
89	88	elsn 4623	. . . . . . 7 ⊢ (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
90	23, 24	opth 5463	. . . . . . 7 ⊢ (⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
91	89, 90	bitr2i 276	. . . . . 6 ⊢ ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩})
92	87, 91	bitrdi 287	. . . . 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 41054	. . . . . . . 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 4620	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
96	95	eleq2d 2819	. . . . 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 282	. . . 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 283	. . 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 5787	. 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 837	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ∩ cin 3932  {csn 4608  ⟨cop 4614   class class class wbr 5125   ↦ cmpt 5207   I cid 5559   ↾ cres 5669  Rel wrel 5672  ‘cfv 6542  ℩crio 7370  (class class class)co 7414  Basecbs 17230  lecple 17284  occoc 17285  0_gc0g 17460  meetcmee 18333  0.cp0 18442  Latclat 18450  Atomscatm 39205  AtLatcal 39206  HLchlt 39292  LHypclh 39927  LTrncltrn 40044  trLctrl 40101  TEndoctendo 40695  DVecHcdvh 41021  DIsoHcdih 41171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-riotaBAD 38895

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-tp 4613  df-op 4615  df-uni 4890  df-int 4929  df-iun 4975  df-iin 4976  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-1st 7997  df-2nd 7998  df-tpos 8234  df-undef 8281  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-1o 8489  df-er 8728  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11477  df-neg 11478  df-nn 12250  df-2 12312  df-3 12313  df-4 12314  df-5 12315  df-6 12316  df-n0 12511  df-z 12598  df-uz 12862  df-fz 13531  df-struct 17167  df-sets 17184  df-slot 17202  df-ndx 17214  df-base 17231  df-ress 17257  df-plusg 17290  df-mulr 17291  df-sca 17293  df-vsca 17294  df-0g 17462  df-proset 18315  df-poset 18334  df-plt 18349  df-lub 18365  df-glb 18366  df-join 18367  df-meet 18368  df-p0 18444  df-p1 18445  df-lat 18451  df-clat 18518  df-mgm 18627  df-sgrp 18706  df-mnd 18722  df-submnd 18771  df-grp 18928  df-minusg 18929  df-sbg 18930  df-subg 19115  df-cntz 19309  df-lsm 19627  df-cmn 19773  df-abl 19774  df-mgp 20111  df-rng 20123  df-ur 20152  df-ring 20205  df-oppr 20307  df-dvdsr 20330  df-unit 20331  df-invr 20361  df-dvr 20374  df-drng 20704  df-lmod 20833  df-lss 20903  df-lsp 20943  df-lvec 21075  df-oposet 39118  df-ol 39120  df-oml 39121  df-covers 39208  df-ats 39209  df-atl 39240  df-cvlat 39264  df-hlat 39293  df-llines 39441  df-lplanes 39442  df-lvols 39443  df-lines 39444  df-psubsp 39446  df-pmap 39447  df-padd 39739  df-lhyp 39931  df-laut 39932  df-ldil 40047  df-ltrn 40048  df-trl 40102  df-tendo 40698  df-edring 40700  df-disoa 40972  df-dvech 41022  df-dib 41082  df-dic 41116  df-dih 41172

This theorem is referenced by:  dihmeetlem4N  41250