Theorem dvh4dimat 41930

Description: There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)

Hypotheses

Ref	Expression
dvh4dimat.h	⊢ 𝐻 = (LHyp‘𝐾)
dvh4dimat.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvh4dimat.s	⊢ ⊕ = (LSSum‘𝑈)
dvh4dimat.a	⊢ 𝐴 = (LSAtoms‘𝑈)
dvh4dimat.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dvh4dimat.p	⊢ (𝜑 → 𝑃 ∈ 𝐴)
dvh4dimat.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)
dvh4dimat.r	⊢ (𝜑 → 𝑅 ∈ 𝐴)

Assertion

Ref	Expression
dvh4dimat	⊢ (𝜑 → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅))

Distinct variable groups:   𝐴,𝑠   𝐾,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   ⊕ ,𝑠   𝑊,𝑠   𝜑,𝑠

Allowed substitution hints:   𝑈(𝑠)   𝐻(𝑠)

Proof of Theorem dvh4dimat

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvh4dimat.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2	1	simpld 495	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
3		dvh4dimat.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
4		eqid 2739	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		dvh4dimat.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
6		dvh4dimat.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2739	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
8		dvh4dimat.a	. . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑈)
9	4, 5, 6, 7, 8	dihlatat 41829	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴) → (◡((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾))
10	1, 3, 9	syl2anc 590	. . . 4 ⊢ (𝜑 → (◡((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾))
11		dvh4dimat.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
12	4, 5, 6, 7, 8	dihlatat 41829	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → (◡((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾))
13	1, 11, 12	syl2anc 590	. . . 4 ⊢ (𝜑 → (◡((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾))
14		dvh4dimat.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
15	4, 5, 6, 7, 8	dihlatat 41829	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐴) → (◡((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾))
16	1, 14, 15	syl2anc 590	. . . 4 ⊢ (𝜑 → (◡((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾))
17		eqid 2739	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
18		eqid 2739	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
19	17, 18, 4	3dim3 39961	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((◡((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾) ∧ (◡((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾) ∧ (◡((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾))) → ∃𝑟 ∈ (Atoms‘𝐾) ¬ 𝑟(le‘𝐾)(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)))
20	2, 10, 13, 16, 19	syl13anc 1380	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ (Atoms‘𝐾) ¬ 𝑟(le‘𝐾)(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)))
21		dvh4dimat.s	. . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝑈)
22	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	5, 6, 7, 8	dih1dimat 41822	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
24	1, 3, 23	syl2anc 590	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	5, 7, 6, 21, 8, 1, 24, 11	dihsmatrn 41928	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ⊕ 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26	25	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑃 ⊕ 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	14	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → 𝑅 ∈ 𝐴)
28	17, 5, 7, 6, 21, 8, 22, 26, 27	dihjat4 41925	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((𝑃 ⊕ 𝑄) ⊕ 𝑅) = (((DIsoH‘𝐾)‘𝑊)‘((◡((DIsoH‘𝐾)‘𝑊)‘(𝑃 ⊕ 𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅))))
29	24	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → 𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	11	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → 𝑄 ∈ 𝐴)
31	17, 5, 7, 6, 21, 8, 22, 29, 30	dihjat6 41926	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → (◡((DIsoH‘𝐾)‘𝑊)‘(𝑃 ⊕ 𝑄)) = ((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄)))
32	31	fvoveq1d 7378	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘((◡((DIsoH‘𝐾)‘𝑊)‘(𝑃 ⊕ 𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅))) = (((DIsoH‘𝐾)‘𝑊)‘(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅))))
33	28, 32	eqtrd 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((𝑃 ⊕ 𝑄) ⊕ 𝑅) = (((DIsoH‘𝐾)‘𝑊)‘(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅))))
34	33	sseq2d 3947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅) ↔ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ (((DIsoH‘𝐾)‘𝑊)‘(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)))))
35		eqid 2739	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
36	35, 4	atbase 39781	. . . . . . . 8 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
37	36	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → 𝑟 ∈ (Base‘𝐾))
38	2	hllatd 39856	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Lat)
39	35, 17, 4	hlatjcl 39859	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (◡((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾) ∧ (◡((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾)) → ((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄)) ∈ (Base‘𝐾))
40	2, 10, 13, 39	syl3anc 1379	. . . . . . . . 9 ⊢ (𝜑 → ((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄)) ∈ (Base‘𝐾))
41	35, 4	atbase 39781	. . . . . . . . . 10 ⊢ ((◡((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾) → (◡((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Base‘𝐾))
42	16, 41	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (◡((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Base‘𝐾))
43	35, 17	latjcl 18396	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄)) ∈ (Base‘𝐾) ∧ (◡((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Base‘𝐾)) → (((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)) ∈ (Base‘𝐾))
44	38, 40, 42, 43	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → (((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)) ∈ (Base‘𝐾))
45	44	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → (((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)) ∈ (Base‘𝐾))
46	35, 18, 5, 7	dihord 41756	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑟 ∈ (Base‘𝐾) ∧ (((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)) ∈ (Base‘𝐾)) → ((((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ (((DIsoH‘𝐾)‘𝑊)‘(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅))) ↔ 𝑟(le‘𝐾)(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅))))
47	22, 37, 45, 46	syl3anc 1379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ (((DIsoH‘𝐾)‘𝑊)‘(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅))) ↔ 𝑟(le‘𝐾)(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅))))
48	34, 47	bitr2d 281	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)) ↔ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅)))
49	48	notbid 319	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → (¬ 𝑟(le‘𝐾)(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)) ↔ ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅)))
50	49	rexbidva 3161	. . 3 ⊢ (𝜑 → (∃𝑟 ∈ (Atoms‘𝐾) ¬ 𝑟(le‘𝐾)(((◡((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(◡((DIsoH‘𝐾)‘𝑊)‘𝑅)) ↔ ∃𝑟 ∈ (Atoms‘𝐾) ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅)))
51	20, 50	mpbid 233	. 2 ⊢ (𝜑 → ∃𝑟 ∈ (Atoms‘𝐾) ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅))
52	4, 5, 6, 7, 8	dihatlat 41826	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) ∈ 𝐴)
53	1, 52	sylan 586	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) ∈ 𝐴)
54	4, 5, 6, 7, 8	dihlatat 41829	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐴) → (◡((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾))
55	1, 54	sylan 586	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (◡((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾))
56	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
57	5, 6, 7, 8	dih1dimat 41822	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊))
58	1, 57	sylan 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊))
59	5, 7	dihcnvid2 41765	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (((DIsoH‘𝐾)‘𝑊)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑠)) = 𝑠)
60	56, 58, 59	syl2anc 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (((DIsoH‘𝐾)‘𝑊)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑠)) = 𝑠)
61	60	eqcomd 2745	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 = (((DIsoH‘𝐾)‘𝑊)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑠)))
62		fveq2 6827	. . . . 5 ⊢ (𝑟 = (◡((DIsoH‘𝐾)‘𝑊)‘𝑠) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) = (((DIsoH‘𝐾)‘𝑊)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑠)))
63	62	rspceeqv 3583	. . . 4 ⊢ (((◡((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾) ∧ 𝑠 = (((DIsoH‘𝐾)‘𝑊)‘(◡((DIsoH‘𝐾)‘𝑊)‘𝑠))) → ∃𝑟 ∈ (Atoms‘𝐾)𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟))
64	55, 61, 63	syl2anc 590	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ∃𝑟 ∈ (Atoms‘𝐾)𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟))
65		sseq1 3940	. . . . 5 ⊢ (𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟) → (𝑠 ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅) ↔ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅)))
66	65	notbid 319	. . . 4 ⊢ (𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟) → (¬ 𝑠 ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅) ↔ ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅)))
67	66	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟)) → (¬ 𝑠 ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅) ↔ ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅)))
68	53, 64, 67	rexxfrd 5338	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝐴 ¬ 𝑠 ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅) ↔ ∃𝑟 ∈ (Atoms‘𝐾) ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅)))
69	51, 68	mpbird 258	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3063   ⊆ wss 3883   class class class wbr 5072  ^◡ccnv 5617  ran crn 5619  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  lecple 17218  joincjn 18268  Latclat 18388  LSSumclsm 19600  LSAtomsclsa 39466  Atomscatm 39755  HLchlt 39842  LHypclh 40476  DVecHcdvh 41570  DIsoHcdih 41720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-riotaBAD 39445

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-undef 8213  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-0g 17395  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18389  df-clat 18456  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-subg 19090  df-cntz 19283  df-lsm 19602  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-oppr 20308  df-dvdsr 20328  df-unit 20329  df-invr 20359  df-dvr 20372  df-drng 20703  df-lmod 20852  df-lss 20922  df-lsp 20962  df-lvec 21093  df-lsatoms 39468  df-oposet 39668  df-ol 39670  df-oml 39671  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843  df-llines 39990  df-lplanes 39991  df-lvols 39992  df-lines 39993  df-psubsp 39995  df-pmap 39996  df-padd 40288  df-lhyp 40480  df-laut 40481  df-ldil 40596  df-ltrn 40597  df-trl 40651  df-tgrp 41235  df-tendo 41247  df-edring 41249  df-dveca 41495  df-disoa 41521  df-dvech 41571  df-dib 41631  df-dic 41665  df-dih 41721  df-doch 41840  df-djh 41887

This theorem is referenced by:  dvh3dimatN  41931  dvh4dimlem  41935