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Theorem dvh4dimat 37326
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 488 . . . 4 (𝜑𝐾 ∈ HL)
3 dvh4dimat.p . . . . 5 (𝜑𝑃𝐴)
4 eqid 2765 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 dvh4dimat.h . . . . . 6 𝐻 = (LHyp‘𝐾)
6 dvh4dimat.u . . . . . 6 𝑈 = ((DVecH‘𝐾)‘𝑊)
7 eqid 2765 . . . . . 6 ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
8 dvh4dimat.a . . . . . 6 𝐴 = (LSAtoms‘𝑈)
94, 5, 6, 7, 8dihlatat 37225 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾))
101, 3, 9syl2anc 579 . . . 4 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾))
11 dvh4dimat.q . . . . 5 (𝜑𝑄𝐴)
124, 5, 6, 7, 8dihlatat 37225 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾))
131, 11, 12syl2anc 579 . . . 4 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾))
14 dvh4dimat.r . . . . 5 (𝜑𝑅𝐴)
154, 5, 6, 7, 8dihlatat 37225 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾))
161, 14, 15syl2anc 579 . . . 4 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾))
17 eqid 2765 . . . . 5 (join‘𝐾) = (join‘𝐾)
18 eqid 2765 . . . . 5 (le‘𝐾) = (le‘𝐾)
1917, 18, 43dim3 35357 . . . 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 1491 . . 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 472 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
235, 6, 7, 8dih1dimat 37218 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴) → 𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
241, 3, 23syl2anc 579 . . . . . . . . . . 11 (𝜑𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
255, 7, 6, 21, 8, 1, 24, 11dihsmatrn 37324 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
2625adantr 472 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (𝑃 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
2714adantr 472 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑅𝐴)
2817, 5, 7, 6, 21, 8, 22, 26, 27dihjat4 37321 . . . . . . . 8 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → ((𝑃 𝑄) 𝑅) = (((DIsoH‘𝐾)‘𝑊)‘((((DIsoH‘𝐾)‘𝑊)‘(𝑃 𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅))))
2924adantr 472 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
3011adantr 472 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑄𝐴)
3117, 5, 7, 6, 21, 8, 22, 29, 30dihjat6 37322 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘(𝑃 𝑄)) = ((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄)))
3231fvoveq1d 6864 . . . . . . . 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))))
3328, 32eqtrd 2799 . . . . . . 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))))
3433sseq2d 3793 . . . . . 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)))))
35 eqid 2765 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3635, 4atbase 35177 . . . . . . . 8 (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
3736adantl 473 . . . . . . 7 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑟 ∈ (Base‘𝐾))
382hllatd 35252 . . . . . . . . 9 (𝜑𝐾 ∈ Lat)
3935, 17, 4hlatjcl 35255 . . . . . . . . . 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))
402, 10, 13, 39syl3anc 1490 . . . . . . . . 9 (𝜑 → ((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄)) ∈ (Base‘𝐾))
4135, 4atbase 35177 . . . . . . . . . 10 ((((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾) → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Base‘𝐾))
4216, 41syl 17 . . . . . . . . 9 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Base‘𝐾))
4335, 17latjcl 17319 . . . . . . . . 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))
4438, 40, 42, 43syl3anc 1490 . . . . . . . 8 (𝜑 → (((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅)) ∈ (Base‘𝐾))
4544adantr 472 . . . . . . 7 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅)) ∈ (Base‘𝐾))
4635, 18, 5, 7dihord 37152 . . . . . . 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))))
4722, 37, 45, 46syl3anc 1490 . . . . . 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))))
4834, 47bitr2d 271 . . . . 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)))
4948notbid 309 . . . 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)))
5049rexbidva 3196 . . 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)))
5120, 50mpbid 223 . 2 (𝜑 → ∃𝑟 ∈ (Atoms‘𝐾) ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅))
524, 5, 6, 7, 8dihatlat 37222 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) ∈ 𝐴)
531, 52sylan 575 . . 3 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) ∈ 𝐴)
544, 5, 6, 7, 8dihlatat 37225 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾))
551, 54sylan 575 . . . 4 ((𝜑𝑠𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾))
561adantr 472 . . . . . 6 ((𝜑𝑠𝐴) → (𝐾 ∈ HL ∧ 𝑊𝐻))
575, 6, 7, 8dih1dimat 37218 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐴) → 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊))
581, 57sylan 575 . . . . . 6 ((𝜑𝑠𝐴) → 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊))
595, 7dihcnvid2 37161 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)) = 𝑠)
6056, 58, 59syl2anc 579 . . . . 5 ((𝜑𝑠𝐴) → (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)) = 𝑠)
6160eqcomd 2771 . . . 4 ((𝜑𝑠𝐴) → 𝑠 = (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)))
62 fveq2 6375 . . . . 5 (𝑟 = (((DIsoH‘𝐾)‘𝑊)‘𝑠) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) = (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)))
6362rspceeqv 3479 . . . 4 (((((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾) ∧ 𝑠 = (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠))) → ∃𝑟 ∈ (Atoms‘𝐾)𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟))
6455, 61, 63syl2anc 579 . . 3 ((𝜑𝑠𝐴) → ∃𝑟 ∈ (Atoms‘𝐾)𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟))
65 sseq1 3786 . . . . 5 (𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟) → (𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
6665notbid 309 . . . 4 (𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟) → (¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
6766adantl 473 . . 3 ((𝜑𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟)) → (¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
6853, 64, 67rexxfrd 5044 . 2 (𝜑 → (∃𝑠𝐴 ¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ ∃𝑟 ∈ (Atoms‘𝐾) ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
6951, 68mpbird 248 1 (𝜑 → ∃𝑠𝐴 ¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wrex 3056  wss 3732   class class class wbr 4809  ccnv 5276  ran crn 5278  cfv 6068  (class class class)co 6842  Basecbs 16132  lecple 16223  joincjn 17212  Latclat 17313  LSSumclsm 18315  LSAtomsclsa 34862  Atomscatm 35151  HLchlt 35238  LHypclh 35872  DVecHcdvh 36966  DIsoHcdih 37116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-riotaBAD 34841
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-tpos 7555  df-undef 7602  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-mulr 16230  df-sca 16232  df-vsca 16233  df-0g 16370  df-proset 17196  df-poset 17214  df-plt 17226  df-lub 17242  df-glb 17243  df-join 17244  df-meet 17245  df-p0 17307  df-p1 17308  df-lat 17314  df-clat 17376  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-submnd 17604  df-grp 17694  df-minusg 17695  df-sbg 17696  df-subg 17857  df-cntz 18015  df-lsm 18317  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-oppr 18890  df-dvdsr 18908  df-unit 18909  df-invr 18939  df-dvr 18950  df-drng 19018  df-lmod 19134  df-lss 19202  df-lsp 19244  df-lvec 19375  df-lsatoms 34864  df-oposet 35064  df-ol 35066  df-oml 35067  df-covers 35154  df-ats 35155  df-atl 35186  df-cvlat 35210  df-hlat 35239  df-llines 35386  df-lplanes 35387  df-lvols 35388  df-lines 35389  df-psubsp 35391  df-pmap 35392  df-padd 35684  df-lhyp 35876  df-laut 35877  df-ldil 35992  df-ltrn 35993  df-trl 36047  df-tgrp 36631  df-tendo 36643  df-edring 36645  df-dveca 36891  df-disoa 36917  df-dvech 36967  df-dib 37027  df-dic 37061  df-dih 37117  df-doch 37236  df-djh 37283
This theorem is referenced by:  dvh3dimatN  37327  dvh4dimlem  37331
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