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Theorem dvaabl 39537
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 2733 . . 3 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
3 eqid 2733 . . 3 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
4 eqid 2733 . . 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 39518 . 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 2733 . . . . 5 ((TGrpβ€˜πΎ)β€˜π‘Š) = ((TGrpβ€˜πΎ)β€˜π‘Š)
81, 2, 7tgrpset 39258 . . . 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 39264 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((TGrpβ€˜πΎ)β€˜π‘Š) ∈ Abel)
108, 9eqeltrrd 2835 . . 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 6859 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
12 eqid 2733 . . . . . . 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 17175 . . . . . 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 2733 . . . . . . 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 17215 . . . . . 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 2775 . . . . 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 8016 . . . . 5 (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∘ 𝑔)) ∈ V
1912grpplusg 17177 . . . . . 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 17216 . . . . . 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 2775 . . . . 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 19583 . . 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 2834 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447   βˆͺ cun 3912  {csn 4590  {cpr 4592  {ctp 4594  βŸ¨cop 4596   ∘ ccom 5641  β€˜cfv 6500   ∈ cmpo 7363  ndxcnx 17073  Basecbs 17091  +gcplusg 17141  Scalarcsca 17144   ·𝑠 cvsca 17145  Abelcabl 19571  HLchlt 37862  LHypclh 38497  LTrncltrn 38614  TGrpctgrp 39255  TEndoctendo 39265  EDRingcedring 39266  DVecAcdveca 39515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-riotaBAD 37465
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-undef 8208  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-struct 17027  df-slot 17062  df-ndx 17074  df-base 17092  df-plusg 17154  df-sca 17157  df-vsca 17158  df-0g 17331  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-p1 18323  df-lat 18329  df-clat 18396  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-cmn 19572  df-abl 19573  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-llines 38011  df-lplanes 38012  df-lvols 38013  df-lines 38014  df-psubsp 38016  df-pmap 38017  df-padd 38309  df-lhyp 38501  df-laut 38502  df-ldil 38617  df-ltrn 38618  df-trl 38672  df-tgrp 39256  df-dveca 39516
This theorem is referenced by:  dvalveclem  39538
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