Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvh4dimat Structured version   Visualization version   GIF version

Theorem dvh4dimat 38566
 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 497 . . . 4 (𝜑𝐾 ∈ HL)
3 dvh4dimat.p . . . . 5 (𝜑𝑃𝐴)
4 eqid 2819 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 dvh4dimat.h . . . . . 6 𝐻 = (LHyp‘𝐾)
6 dvh4dimat.u . . . . . 6 𝑈 = ((DVecH‘𝐾)‘𝑊)
7 eqid 2819 . . . . . 6 ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
8 dvh4dimat.a . . . . . 6 𝐴 = (LSAtoms‘𝑈)
94, 5, 6, 7, 8dihlatat 38465 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾))
101, 3, 9syl2anc 586 . . . 4 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑃) ∈ (Atoms‘𝐾))
11 dvh4dimat.q . . . . 5 (𝜑𝑄𝐴)
124, 5, 6, 7, 8dihlatat 38465 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾))
131, 11, 12syl2anc 586 . . . 4 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑄) ∈ (Atoms‘𝐾))
14 dvh4dimat.r . . . . 5 (𝜑𝑅𝐴)
154, 5, 6, 7, 8dihlatat 38465 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾))
161, 14, 15syl2anc 586 . . . 4 (𝜑 → (((DIsoH‘𝐾)‘𝑊)‘𝑅) ∈ (Atoms‘𝐾))
17 eqid 2819 . . . . 5 (join‘𝐾) = (join‘𝐾)
18 eqid 2819 . . . . 5 (le‘𝐾) = (le‘𝐾)
1917, 18, 43dim3 36597 . . . 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 1367 . . 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 483 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
235, 6, 7, 8dih1dimat 38458 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴) → 𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
241, 3, 23syl2anc 586 . . . . . . . . . . 11 (𝜑𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
255, 7, 6, 21, 8, 1, 24, 11dihsmatrn 38564 . . . . . . . . . 10 (𝜑 → (𝑃 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
2625adantr 483 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (𝑃 𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
2714adantr 483 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑅𝐴)
2817, 5, 7, 6, 21, 8, 22, 26, 27dihjat4 38561 . . . . . . . 8 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → ((𝑃 𝑄) 𝑅) = (((DIsoH‘𝐾)‘𝑊)‘((((DIsoH‘𝐾)‘𝑊)‘(𝑃 𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅))))
2924adantr 483 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑃 ∈ ran ((DIsoH‘𝐾)‘𝑊))
3011adantr 483 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑄𝐴)
3117, 5, 7, 6, 21, 8, 22, 29, 30dihjat6 38562 . . . . . . . . 9 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘(𝑃 𝑄)) = ((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄)))
3231fvoveq1d 7170 . . . . . . . 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 2854 . . . . . . 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 3997 . . . . . 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 2819 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3635, 4atbase 36417 . . . . . . . 8 (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
3736adantl 484 . . . . . . 7 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → 𝑟 ∈ (Base‘𝐾))
382hllatd 36492 . . . . . . . . 9 (𝜑𝐾 ∈ Lat)
3935, 17, 4hlatjcl 36495 . . . . . . . . . 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 1366 . . . . . . . . 9 (𝜑 → ((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄)) ∈ (Base‘𝐾))
4135, 4atbase 36417 . . . . . . . . . 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 17653 . . . . . . . . 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 1366 . . . . . . . 8 (𝜑 → (((((DIsoH‘𝐾)‘𝑊)‘𝑃)(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑄))(join‘𝐾)(((DIsoH‘𝐾)‘𝑊)‘𝑅)) ∈ (Base‘𝐾))
4544adantr 483 . . . . . . 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 38392 . . . . . . 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 1366 . . . . . 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 282 . . . . 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 320 . . . 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 3294 . . 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 234 . 2 (𝜑 → ∃𝑟 ∈ (Atoms‘𝐾) ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅))
524, 5, 6, 7, 8dihatlat 38462 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) ∈ 𝐴)
531, 52sylan 582 . . 3 ((𝜑𝑟 ∈ (Atoms‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) ∈ 𝐴)
544, 5, 6, 7, 8dihlatat 38465 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾))
551, 54sylan 582 . . . 4 ((𝜑𝑠𝐴) → (((DIsoH‘𝐾)‘𝑊)‘𝑠) ∈ (Atoms‘𝐾))
561adantr 483 . . . . . 6 ((𝜑𝑠𝐴) → (𝐾 ∈ HL ∧ 𝑊𝐻))
575, 6, 7, 8dih1dimat 38458 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐴) → 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊))
581, 57sylan 582 . . . . . 6 ((𝜑𝑠𝐴) → 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊))
595, 7dihcnvid2 38401 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)) = 𝑠)
6056, 58, 59syl2anc 586 . . . . 5 ((𝜑𝑠𝐴) → (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)) = 𝑠)
6160eqcomd 2825 . . . 4 ((𝜑𝑠𝐴) → 𝑠 = (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)))
62 fveq2 6663 . . . . 5 (𝑟 = (((DIsoH‘𝐾)‘𝑊)‘𝑠) → (((DIsoH‘𝐾)‘𝑊)‘𝑟) = (((DIsoH‘𝐾)‘𝑊)‘(((DIsoH‘𝐾)‘𝑊)‘𝑠)))
6362rspceeqv 3636 . . . 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 586 . . 3 ((𝜑𝑠𝐴) → ∃𝑟 ∈ (Atoms‘𝐾)𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟))
65 sseq1 3990 . . . . 5 (𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟) → (𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
6665notbid 320 . . . 4 (𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟) → (¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
6766adantl 484 . . 3 ((𝜑𝑠 = (((DIsoH‘𝐾)‘𝑊)‘𝑟)) → (¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
6853, 64, 67rexxfrd 5300 . 2 (𝜑 → (∃𝑠𝐴 ¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅) ↔ ∃𝑟 ∈ (Atoms‘𝐾) ¬ (((DIsoH‘𝐾)‘𝑊)‘𝑟) ⊆ ((𝑃 𝑄) 𝑅)))
6951, 68mpbird 259 1 (𝜑 → ∃𝑠𝐴 ¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∃wrex 3137   ⊆ wss 3934   class class class wbr 5057  ◡ccnv 5547  ran crn 5549  ‘cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  Latclat 17647  LSSumclsm 18751  LSAtomsclsa 36102  Atomscatm 36391  HLchlt 36478  LHypclh 37112  DVecHcdvh 38206  DIsoHcdih 38356 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-riotaBAD 36081 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-tpos 7884  df-undef 7931  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-0g 16707  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-cntz 18439  df-lsm 18753  df-cmn 18900  df-abl 18901  df-mgp 19232  df-ur 19244  df-ring 19291  df-oppr 19365  df-dvdsr 19383  df-unit 19384  df-invr 19414  df-dvr 19425  df-drng 19496  df-lmod 19628  df-lss 19696  df-lsp 19736  df-lvec 19867  df-lsatoms 36104  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-llines 36626  df-lplanes 36627  df-lvols 36628  df-lines 36629  df-psubsp 36631  df-pmap 36632  df-padd 36924  df-lhyp 37116  df-laut 37117  df-ldil 37232  df-ltrn 37233  df-trl 37287  df-tgrp 37871  df-tendo 37883  df-edring 37885  df-dveca 38131  df-disoa 38157  df-dvech 38207  df-dib 38267  df-dic 38301  df-dih 38357  df-doch 38476  df-djh 38523 This theorem is referenced by:  dvh3dimatN  38567  dvh4dimlem  38571
 Copyright terms: Public domain W3C validator