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Theorem dvaabl 40497
Description: The constructed partial vector space A for a lattice 𝐾 is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dvalvec.h 𝐻 = (LHypβ€˜πΎ)
dvalvec.v π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dvaabl ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Abel)

Proof of Theorem dvaabl
Dummy variables 𝑓 𝑠 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvalvec.h . . 3 𝐻 = (LHypβ€˜πΎ)
2 eqid 2728 . . 3 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
3 eqid 2728 . . 3 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
4 eqid 2728 . . 3 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
5 dvalvec.v . . 3 π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5dvaset 40478 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ = ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩}))
7 eqid 2728 . . . . 5 ((TGrpβ€˜πΎ)β€˜π‘Š) = ((TGrpβ€˜πΎ)β€˜π‘Š)
81, 2, 7tgrpset 40218 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((TGrpβ€˜πΎ)β€˜π‘Š) = {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩})
91, 7tgrpabl 40224 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((TGrpβ€˜πΎ)β€˜π‘Š) ∈ Abel)
108, 9eqeltrrd 2830 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩} ∈ Abel)
11 fvex 6910 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
12 eqid 2728 . . . . . . 7 {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩} = {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩}
1312grpbase 17267 . . . . . 6 (((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V β†’ ((LTrnβ€˜πΎ)β€˜π‘Š) = (Baseβ€˜{⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩}))
14 eqid 2728 . . . . . . 7 ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩}) = ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩})
1514lmodbase 17307 . . . . . 6 (((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V β†’ ((LTrnβ€˜πΎ)β€˜π‘Š) = (Baseβ€˜({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩})))
1613, 15eqtr3d 2770 . . . . 5 (((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V β†’ (Baseβ€˜{⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩}) = (Baseβ€˜({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩})))
1711, 16ax-mp 5 . . . 4 (Baseβ€˜{⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩}) = (Baseβ€˜({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩}))
1811, 11mpoex 8084 . . . . 5 (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔)) ∈ V
1912grpplusg 17269 . . . . . 6 ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔)) ∈ V β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔)) = (+gβ€˜{⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩}))
2014lmodplusg 17308 . . . . . 6 ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔)) ∈ V β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔)) = (+gβ€˜({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩})))
2119, 20eqtr3d 2770 . . . . 5 ((𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔)) ∈ V β†’ (+gβ€˜{⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩}) = (+gβ€˜({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩})))
2218, 21ax-mp 5 . . . 4 (+gβ€˜{⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩}) = (+gβ€˜({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩}))
2317, 22ablprop 19748 . . 3 ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩} ∈ Abel ↔ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩}) ∈ Abel)
2410, 23sylib 217 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (π‘ β€˜π‘“))⟩}) ∈ Abel)
256, 24eqeltrd 2829 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471   βˆͺ cun 3945  {csn 4629  {cpr 4631  {ctp 4633  βŸ¨cop 4635   ∘ ccom 5682  β€˜cfv 6548   ∈ cmpo 7422  ndxcnx 17162  Basecbs 17180  +gcplusg 17233  Scalarcsca 17236   ·𝑠 cvsca 17237  Abelcabl 19736  HLchlt 38822  LHypclh 39457  LTrncltrn 39574  TGrpctgrp 40215  TEndoctendo 40225  EDRingcedring 40226  DVecAcdveca 40475
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-riotaBAD 38425
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-undef 8279  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-n0 12504  df-z 12590  df-uz 12854  df-fz 13518  df-struct 17116  df-slot 17151  df-ndx 17163  df-base 17181  df-plusg 17246  df-sca 17249  df-vsca 17250  df-0g 17423  df-proset 18287  df-poset 18305  df-plt 18322  df-lub 18338  df-glb 18339  df-join 18340  df-meet 18341  df-p0 18417  df-p1 18418  df-lat 18424  df-clat 18491  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18893  df-cmn 19737  df-abl 19738  df-oposet 38648  df-ol 38650  df-oml 38651  df-covers 38738  df-ats 38739  df-atl 38770  df-cvlat 38794  df-hlat 38823  df-llines 38971  df-lplanes 38972  df-lvols 38973  df-lines 38974  df-psubsp 38976  df-pmap 38977  df-padd 39269  df-lhyp 39461  df-laut 39462  df-ldil 39577  df-ltrn 39578  df-trl 39632  df-tgrp 40216  df-dveca 40476
This theorem is referenced by:  dvalveclem  40498
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