Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmapval0 Structured version   Visualization version   GIF version

Theorem hdmapval0 36143
 Description: Value of map from vectors to functionals at zero. Note: we use dvh3dim 35753 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 36154 could be derived from the other to shorten proof. (Contributed by NM, 17-May-2015.)
Hypotheses
Ref Expression
hdmapval0.h 𝐻 = (LHyp‘𝐾)
hdmapval0.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapval0.o 0 = (0g𝑈)
hdmapval0.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmapval0.q 𝑄 = (0g𝐶)
hdmapval0.s 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapval0.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
Assertion
Ref Expression
hdmapval0 (𝜑 → (𝑆0 ) = 𝑄)

Proof of Theorem hdmapval0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hdmapval0.h . . 3 𝐻 = (LHyp‘𝐾)
2 hdmapval0.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 eqid 2610 . . 3 (Base‘𝑈) = (Base‘𝑈)
4 eqid 2610 . . 3 (LSpan‘𝑈) = (LSpan‘𝑈)
5 hdmapval0.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
6 eqid 2610 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
7 eqid 2610 . . . . 5 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
8 hdmapval0.o . . . . 5 0 = (0g𝑈)
9 eqid 2610 . . . . 5 ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
101, 6, 7, 2, 3, 8, 9, 5dvheveccl 35419 . . . 4 (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ ((Base‘𝑈) ∖ { 0 }))
1110eldifad 3552 . . 3 (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (Base‘𝑈))
121, 2, 5dvhlmod 35417 . . . 4 (𝜑𝑈 ∈ LMod)
133, 8lmod0vcl 18715 . . . 4 (𝑈 ∈ LMod → 0 ∈ (Base‘𝑈))
1412, 13syl 17 . . 3 (𝜑0 ∈ (Base‘𝑈))
151, 2, 3, 4, 5, 11, 14dvh3dim 35753 . 2 (𝜑 → ∃𝑥 ∈ (Base‘𝑈) ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
16 hdmapval0.c . . . . 5 𝐶 = ((LCDual‘𝐾)‘𝑊)
17 eqid 2610 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
18 eqid 2610 . . . . 5 ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
19 eqid 2610 . . . . 5 ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊)
20 hdmapval0.s . . . . 5 𝑆 = ((HDMap‘𝐾)‘𝑊)
2153ad2ant1 1075 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (𝐾 ∈ HL ∧ 𝑊𝐻))
22143ad2ant1 1075 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → 0 ∈ (Base‘𝑈))
23 simp2 1055 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → 𝑥 ∈ (Base‘𝑈))
24 eqid 2610 . . . . . . . . . 10 (LSubSp‘𝑈) = (LSubSp‘𝑈)
253, 24, 4, 12, 11, 14lspprcl 18799 . . . . . . . . . 10 (𝜑 → ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) ∈ (LSubSp‘𝑈))
263, 4, 12, 11, 14lspprid1 18818 . . . . . . . . . 10 (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
2724, 4, 12, 25, 26lspsnel5a 18817 . . . . . . . . 9 (𝜑 → ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ⊆ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
283, 4, 12, 11, 14lspprid2 18819 . . . . . . . . . 10 (𝜑0 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
2924, 4, 12, 25, 28lspsnel5a 18817 . . . . . . . . 9 (𝜑 → ((LSpan‘𝑈)‘{ 0 }) ⊆ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
3027, 29unssd 3751 . . . . . . . 8 (𝜑 → (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{ 0 })) ⊆ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
3130ssneld 3570 . . . . . . 7 (𝜑 → (¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) → ¬ 𝑥 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{ 0 }))))
3231adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑈)) → (¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) → ¬ 𝑥 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{ 0 }))))
33323impia 1253 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ¬ 𝑥 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{ 0 })))
341, 9, 2, 3, 4, 16, 17, 18, 19, 20, 21, 22, 23, 33hdmapval2 36142 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (𝑆0 ) = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑥, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑥⟩), 0 ⟩))
35 hdmapval0.q . . . . 5 𝑄 = (0g𝐶)
36 eqid 2610 . . . . . 6 (LSpan‘𝐶) = (LSpan‘𝐶)
37 eqid 2610 . . . . . 6 ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
381, 2, 3, 8, 16, 17, 35, 18, 5, 10hvmapcl2 36073 . . . . . . . 8 (𝜑 → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ ((Base‘𝐶) ∖ {𝑄}))
3938eldifad 3552 . . . . . . 7 (𝜑 → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ (Base‘𝐶))
40393ad2ant1 1075 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ (Base‘𝐶))
411, 2, 3, 8, 4, 16, 36, 37, 18, 5, 10mapdhvmap 36076 . . . . . . 7 (𝜑 → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩})) = ((LSpan‘𝐶)‘{(((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)}))
42413ad2ant1 1075 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩})) = ((LSpan‘𝐶)‘{(((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)}))
431, 2, 5dvhlvec 35416 . . . . . . . . . 10 (𝜑𝑈 ∈ LVec)
44433ad2ant1 1075 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → 𝑈 ∈ LVec)
45113ad2ant1 1075 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (Base‘𝑈))
46 simp3 1056 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
473, 4, 44, 23, 45, 22, 46lspindpi 18953 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((LSpan‘𝑈)‘{𝑥}) ≠ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∧ ((LSpan‘𝑈)‘{𝑥}) ≠ ((LSpan‘𝑈)‘{ 0 })))
4847simpld 474 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ((LSpan‘𝑈)‘{𝑥}) ≠ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}))
4948necomd 2837 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ≠ ((LSpan‘𝑈)‘{𝑥}))
50103ad2ant1 1075 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ ((Base‘𝑈) ∖ { 0 }))
511, 2, 3, 8, 4, 16, 17, 36, 37, 19, 21, 40, 42, 49, 50, 23hdmap1cl 36112 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑥⟩) ∈ (Base‘𝐶))
521, 2, 3, 8, 16, 17, 35, 19, 21, 51, 23hdmap1val0 36107 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((HDMap1‘𝐾)‘𝑊)‘⟨𝑥, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑥⟩), 0 ⟩) = 𝑄)
5334, 52eqtrd 2644 . . 3 ((𝜑𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (𝑆0 ) = 𝑄)
5453rexlimdv3a 3015 . 2 (𝜑 → (∃𝑥 ∈ (Base‘𝑈) ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) → (𝑆0 ) = 𝑄))
5515, 54mpd 15 1 (𝜑 → (𝑆0 ) = 𝑄)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537   ∪ cun 3538  {csn 4125  {cpr 4127  ⟨cop 4131  ⟨cotp 4133   I cid 4948   ↾ cres 5040  ‘cfv 5804  Basecbs 15695  0gc0g 15923  LModclmod 18686  LSubSpclss 18753  LSpanclspn 18792  LVecclvec 18923  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  DVecHcdvh 35385  LCDualclcd 35893  mapdcmpd 35931  HVMapchvm 36063  HDMap1chdma1 36099  HDMapchdma 36100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-undef 7286  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-0g 15925  df-mre 16069  df-mrc 16070  df-acs 16072  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-cntz 17573  df-oppg 17599  df-lsm 17874  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lvec 18924  df-lsatoms 33281  df-lshyp 33282  df-lcv 33324  df-lfl 33363  df-lkr 33391  df-ldual 33429  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464  df-tgrp 35049  df-tendo 35061  df-edring 35063  df-dveca 35309  df-disoa 35336  df-dvech 35386  df-dib 35446  df-dic 35480  df-dih 35536  df-doch 35655  df-djh 35702  df-lcdual 35894  df-mapd 35932  df-hvmap 36064  df-hdmap1 36101  df-hdmap 36102 This theorem is referenced by:  hdmapval3N  36148  hdmap10  36150  hdmaprnlem17N  36173  hdmap14lem2a  36177  hdmap14lem6  36183  hdmap14lem13  36190  hgmapval0  36202
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