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Theorem dvhgrp 38733
Description: The full vector space 𝑈 constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane 𝑊) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
dvhgrp.b 𝐵 = (Base‘𝐾)
dvhgrp.h 𝐻 = (LHyp‘𝐾)
dvhgrp.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhgrp.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhgrp.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhgrp.d 𝐷 = (Scalar‘𝑈)
dvhgrp.p = (+g𝐷)
dvhgrp.a + = (+g𝑈)
dvhgrp.o 0 = (0g𝐷)
dvhgrp.i 𝐼 = (invg𝐷)
Assertion
Ref Expression
dvhgrp ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)

Proof of Theorem dvhgrp
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhgrp.h . . . 4 𝐻 = (LHyp‘𝐾)
2 dvhgrp.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
3 dvhgrp.e . . . 4 𝐸 = ((TEndo‘𝐾)‘𝑊)
4 dvhgrp.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
5 eqid 2738 . . . 4 (Base‘𝑈) = (Base‘𝑈)
61, 2, 3, 4, 5dvhvbase 38713 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
76eqcomd 2744 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑇 × 𝐸) = (Base‘𝑈))
8 dvhgrp.a . . 3 + = (+g𝑈)
98a1i 11 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → + = (+g𝑈))
10 dvhgrp.d . . . 4 𝐷 = (Scalar‘𝑈)
11 dvhgrp.p . . . 4 = (+g𝐷)
121, 2, 3, 4, 10, 11, 8dvhvaddcl 38721 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13123impb 1116 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
141, 2, 3, 4, 10, 11, 8dvhvaddass 38723 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸) ∧ ∈ (𝑇 × 𝐸))) → ((𝑓 + 𝑔) + ) = (𝑓 + (𝑔 + )))
15 dvhgrp.b . . . 4 𝐵 = (Base‘𝐾)
1615, 1, 2idltrn 37776 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17 eqid 2738 . . . . . . . 8 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
181, 17, 4, 10dvhsca 38708 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
191, 17erngdv 38619 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
2018, 19eqeltrd 2833 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
21 drnggrp 19622 . . . . . 6 (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
2220, 21syl 17 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
23 eqid 2738 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
24 dvhgrp.o . . . . . 6 0 = (0g𝐷)
2523, 24grpidcl 18242 . . . . 5 (𝐷 ∈ Grp → 0 ∈ (Base‘𝐷))
2622, 25syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 ∈ (Base‘𝐷))
271, 3, 4, 10, 23dvhbase 38709 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
2826, 27eleqtrd 2835 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0𝐸)
29 opelxpi 5556 . . 3 ((( I ↾ 𝐵) ∈ 𝑇0𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
3016, 28, 29syl2anc 587 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31 simpl 486 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3216adantr 484 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
3328adantr 484 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0𝐸)
34 xp1st 7739 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (1st𝑓) ∈ 𝑇)
3534adantl 485 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
36 xp2nd 7740 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
3736adantl 485 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ 𝐸)
381, 2, 3, 4, 10, 8, 11dvhopvadd 38719 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (( I ↾ 𝐵) ∈ 𝑇0𝐸) ∧ ((1st𝑓) ∈ 𝑇 ∧ (2nd𝑓) ∈ 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩)
3931, 32, 33, 35, 37, 38syl122anc 1380 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩)
4015, 1, 2ltrn1o 37750 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓):𝐵1-1-onto𝐵)
4135, 40syldan 594 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓):𝐵1-1-onto𝐵)
42 f1of 6612 . . . . . 6 ((1st𝑓):𝐵1-1-onto𝐵 → (1st𝑓):𝐵𝐵)
43 fcoi2 6547 . . . . . 6 ((1st𝑓):𝐵𝐵 → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4441, 42, 433syl 18 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4522adantr 484 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
4627adantr 484 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
4737, 46eleqtrrd 2836 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ (Base‘𝐷))
4823, 11, 24grplid 18244 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ( 0 (2nd𝑓)) = (2nd𝑓))
4945, 47, 48syl2anc 587 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 (2nd𝑓)) = (2nd𝑓))
5044, 49opeq12d 4766 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩ = ⟨(1st𝑓), (2nd𝑓)⟩)
5139, 50eqtrd 2773 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(1st𝑓), (2nd𝑓)⟩)
52 1st2nd2 7746 . . . . 5 (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5352adantl 485 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5453oveq2d 7180 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩))
5551, 54, 533eqtr4d 2783 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = 𝑓)
561, 2ltrncnv 37772 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓) ∈ 𝑇)
5735, 56syldan 594 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
58 dvhgrp.i . . . . . 6 𝐼 = (invg𝐷)
5923, 58grpinvcl 18262 . . . . 5 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6045, 47, 59syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6160, 46eleqtrd 2835 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ 𝐸)
62 opelxpi 5556 . . 3 (((1st𝑓) ∈ 𝑇 ∧ (𝐼‘(2nd𝑓)) ∈ 𝐸) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6357, 61, 62syl2anc 587 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6453oveq2d 7180 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩))
651, 2, 3, 4, 10, 8, 11dvhopvadd 38719 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((1st𝑓) ∈ 𝑇 ∧ (𝐼‘(2nd𝑓)) ∈ 𝐸) ∧ ((1st𝑓) ∈ 𝑇 ∧ (2nd𝑓) ∈ 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩)
6631, 57, 61, 35, 37, 65syl122anc 1380 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩)
67 f1ococnv1 6640 . . . . . 6 ((1st𝑓):𝐵1-1-onto𝐵 → ((1st𝑓) ∘ (1st𝑓)) = ( I ↾ 𝐵))
6841, 67syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((1st𝑓) ∘ (1st𝑓)) = ( I ↾ 𝐵))
6923, 11, 24, 58grplinv 18263 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7045, 47, 69syl2anc 587 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7168, 70opeq12d 4766 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩ = ⟨( I ↾ 𝐵), 0 ⟩)
7266, 71eqtrd 2773 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨( I ↾ 𝐵), 0 ⟩)
7364, 72eqtrd 2773 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = ⟨( I ↾ 𝐵), 0 ⟩)
747, 9, 13, 14, 30, 55, 63, 73isgrpd 18236 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  cop 4519   I cid 5424   × cxp 5517  ccnv 5518  cres 5521  ccom 5523  wf 6329  1-1-ontowf1o 6332  cfv 6333  (class class class)co 7164  1st c1st 7705  2nd c2nd 7706  Basecbs 16579  +gcplusg 16661  Scalarcsca 16664  0gc0g 16809  Grpcgrp 18212  invgcminusg 18213  DivRingcdr 19614  HLchlt 36976  LHypclh 37610  LTrncltrn 37727  TEndoctendo 38378  EDRingcedring 38379  DVecHcdvh 38704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685  ax-riotaBAD 36579
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-iin 4881  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-tpos 7914  df-undef 7961  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-er 8313  df-map 8432  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-2 11772  df-3 11773  df-4 11774  df-5 11775  df-6 11776  df-n0 11970  df-z 12056  df-uz 12318  df-fz 12975  df-struct 16581  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-ress 16587  df-plusg 16674  df-mulr 16675  df-sca 16677  df-vsca 16678  df-0g 16811  df-proset 17647  df-poset 17665  df-plt 17677  df-lub 17693  df-glb 17694  df-join 17695  df-meet 17696  df-p0 17758  df-p1 17759  df-lat 17765  df-clat 17827  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-grp 18215  df-minusg 18216  df-mgp 19352  df-ur 19364  df-ring 19411  df-oppr 19488  df-dvdsr 19506  df-unit 19507  df-invr 19537  df-dvr 19548  df-drng 19616  df-oposet 36802  df-ol 36804  df-oml 36805  df-covers 36892  df-ats 36893  df-atl 36924  df-cvlat 36948  df-hlat 36977  df-llines 37124  df-lplanes 37125  df-lvols 37126  df-lines 37127  df-psubsp 37129  df-pmap 37130  df-padd 37422  df-lhyp 37614  df-laut 37615  df-ldil 37730  df-ltrn 37731  df-trl 37785  df-tendo 38381  df-edring 38383  df-dvech 38705
This theorem is referenced by:  dvhlveclem  38734
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