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Theorem dvh4dimat 36204
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 ∧ 𝑊𝐻))
dvh4dimat.p (𝜑𝑃𝐴)
dvh4dimat.q (𝜑𝑄𝐴)
dvh4dimat.r (𝜑𝑅𝐴)
Assertion
Ref Expression
dvh4dimat (𝜑 → ∃𝑠𝐴 ¬ 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.
StepHypRef Expression
1 dvh4dimat.k . . . . 5 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
21simpld 475 . . . 4 (𝜑𝐾 ∈ HL)
3 dvh4dimat.p . . . . 5 (𝜑𝑃𝐴)
4 eqid 2621 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 dvh4dimat.h . . . . . 6 𝐻 = (LHyp‘𝐾)
6 dvh4dimat.u . . . . . 6 𝑈 = ((DVecH‘𝐾)‘𝑊)
7 eqid 2621 . . . . . 6 ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
8 dvh4dimat.a . . . . . 6 𝐴 = (LSAtoms‘𝑈)
94, 5, 6, 7, 8dihlatat 36103 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾))
101, 3, 9syl2anc 692 . . . 4 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾))
11 dvh4dimat.q . . . . 5 (𝜑𝑄𝐴)
124, 5, 6, 7, 8dihlatat 36103 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾))
131, 11, 12syl2anc 692 . . . 4 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾))
14 dvh4dimat.r . . . . 5 (𝜑𝑅𝐴)
154, 5, 6, 7, 8dihlatat 36103 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾))
161, 14, 15syl2anc 692 . . . 4 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾))
17 eqid 2621 . . . . 5 (join‘𝐾) = (join‘𝐾)
18 eqid 2621 . . . . 5 (le‘𝐾) = (le‘𝐾)
1917, 18, 43dim3 34232 . . . 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)))
202, 10, 13, 16, 19syl13anc 1325 . . 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)
221adantr 481 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
235, 6, 7, 8dih1dimat 36096 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴) → 𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
241, 3, 23syl2anc 692 . . . . . . . . . . 11 (𝜑𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
255, 7, 6, 21, 8, 1, 24, 11dihsmatrn 36202 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
2625adantr 481 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (𝑃 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
2714adantr 481 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑅𝐴)
2817, 5, 7, 6, 21, 8, 22, 26, 27dihjat4 36199 . . . . . . . 8 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → ((𝑃 𝑄) 𝑅) = (((DIsoH‘𝐾)‘𝑊)‘((((DIsoH‘𝐾)‘𝑊)‘(𝑃 𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅))))
2924adantr 481 . . . . . . . . . . 11 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
3011adantr 481 . . . . . . . . . . 11 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑄𝐴)
3117, 5, 7, 6, 21, 8, 22, 29, 30dihjat6 36200 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘(𝑃 𝑄)) = ((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄)))
3231oveq1d 6619 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → ((((DIsoH‘𝐾)‘𝑊)‘(𝑃 𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅)) = (((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅)))
3332fveq2d 6152 . . . . . . . 8 ((𝜑𝑟 ∈ (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))))
3428, 33eqtrd 2655 . . . . . . 7 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → ((𝑃 𝑄) 𝑅) = (((DIsoH‘𝐾)‘𝑊)‘(((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅))))
3534sseq2d 3612 . . . . . 6 ((𝜑𝑟 ∈ (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)))))
36 eqid 2621 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3736, 4atbase 34053 . . . . . . . 8 (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
3837adantl 482 . . . . . . 7 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑟 ∈ (Base‘𝐾))
39 hllat 34127 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ Lat)
402, 39syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Lat)
4136, 17, 4hlatjcl 34130 . . . . . . . . . 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))
422, 10, 13, 41syl3anc 1323 . . . . . . . . 9 (𝜑 → ((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄)) ∈ (Base‘𝐾))
4336, 4atbase 34053 . . . . . . . . . 10 ((((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾) → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Base‘𝐾))
4416, 43syl 17 . . . . . . . . 9 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Base‘𝐾))
4536, 17latjcl 16972 . . . . . . . . 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))
4640, 42, 44, 45syl3anc 1323 . . . . . . . 8 (𝜑 → (((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅)) ∈ (Base‘𝐾))
4746adantr 481 . . . . . . 7 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅)) ∈ (Base‘𝐾))
4836, 18, 5, 7dihord 36030 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 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))))
4922, 38, 47, 48syl3anc 1323 . . . . . 6 ((𝜑𝑟 ∈ (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))))
5035, 49bitr2d 269 . . . . 5 ((𝜑𝑟 ∈ (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)))
5150notbid 308 . . . 4 ((𝜑𝑟 ∈ (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)))
5251rexbidva 3042 . . 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)))
5320, 52mpbid 222 . 2 (𝜑 → ∃𝑟 ∈ (Atoms‘𝐾) ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅))
544, 5, 6, 7, 8dihatlat 36100 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) ∈ 𝐴)
551, 54sylan 488 . . 3 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) ∈ 𝐴)
564, 5, 6, 7, 8dihlatat 36103 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾))
571, 56sylan 488 . . . 4 ((𝜑𝑠𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾))
581adantr 481 . . . . . 6 ((𝜑𝑠𝐴) → (𝐾 ∈ HL ∧ 𝑊𝐻))
595, 6, 7, 8dih1dimat 36096 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐴) → 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊))
601, 59sylan 488 . . . . . 6 ((𝜑𝑠𝐴) → 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊))
615, 7dihcnvid2 36039 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)) = 𝑠)
6258, 60, 61syl2anc 692 . . . . 5 ((𝜑𝑠𝐴) → (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)) = 𝑠)
6362eqcomd 2627 . . . 4 ((𝜑𝑠𝐴) → 𝑠 = (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)))
64 fveq2 6148 . . . . . 6 (𝑟 = (((DIsoH‘𝐾)‘𝑊)‘𝑠) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) = (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)))
6564eqeq2d 2631 . . . . 5 (𝑟 = (((DIsoH‘𝐾)‘𝑊)‘𝑠) → (𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟) ↔ 𝑠 = (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠))))
6665rspcev 3295 . . . 4 (((((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾) ∧ 𝑠 = (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠))) → ∃𝑟 ∈ (Atoms‘𝐾)𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟))
6757, 63, 66syl2anc 692 . . 3 ((𝜑𝑠𝐴) → ∃𝑟 ∈ (Atoms‘𝐾)𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟))
68 sseq1 3605 . . . . 5 (𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟) → (𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
6968notbid 308 . . . 4 (𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟) → (¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
7069adantl 482 . . 3 ((𝜑𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟)) → (¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
7155, 67, 70rexxfrd 4841 . 2 (𝜑 → (∃𝑠𝐴 ¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ ∃𝑟 ∈ (Atoms‘𝐾) ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
7253, 71mpbird 247 1 (𝜑 → ∃𝑠𝐴 ¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  wss 3555   class class class wbr 4613  ccnv 5073  ran crn 5075  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  Latclat 16966  LSSumclsm 17970  LSAtomsclsa 33738  Atomscatm 34027  HLchlt 34114  LHypclh 34747  DVecHcdvh 35844  DIsoHcdih 35994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-riotaBAD 33716
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-tpos 7297  df-undef 7344  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-0g 16023  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-p1 16961  df-lat 16967  df-clat 17029  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-subg 17512  df-cntz 17671  df-lsm 17972  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-ring 18470  df-oppr 18544  df-dvdsr 18562  df-unit 18563  df-invr 18593  df-dvr 18604  df-drng 18670  df-lmod 18786  df-lss 18852  df-lsp 18891  df-lvec 19022  df-lsatoms 33740  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-llines 34261  df-lplanes 34262  df-lvols 34263  df-lines 34264  df-psubsp 34266  df-pmap 34267  df-padd 34559  df-lhyp 34751  df-laut 34752  df-ldil 34867  df-ltrn 34868  df-trl 34923  df-tgrp 35508  df-tendo 35520  df-edring 35522  df-dveca 35768  df-disoa 35795  df-dvech 35845  df-dib 35905  df-dic 35939  df-dih 35995  df-doch 36114  df-djh 36161
This theorem is referenced by:  dvh3dimatN  36205  dvh4dimlem  36209
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