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Theorem dvhgrp 38247
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 2824 . . . 4 (Base‘𝑈) = (Base‘𝑈)
61, 2, 3, 4, 5dvhvbase 38227 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
76eqcomd 2830 . 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 38235 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13123impb 1111 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
141, 2, 3, 4, 10, 11, 8dvhvaddass 38237 . 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 37290 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17 eqid 2824 . . . . . . . 8 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
181, 17, 4, 10dvhsca 38222 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
191, 17erngdv 38133 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
2018, 19eqeltrd 2916 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
21 drnggrp 19513 . . . . . 6 (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
2220, 21syl 17 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
23 eqid 2824 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
24 dvhgrp.o . . . . . 6 0 = (0g𝐷)
2523, 24grpidcl 18134 . . . . 5 (𝐷 ∈ Grp → 0 ∈ (Base‘𝐷))
2622, 25syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 ∈ (Base‘𝐷))
271, 3, 4, 10, 23dvhbase 38223 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
2826, 27eleqtrd 2918 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0𝐸)
29 opelxpi 5595 . . 3 ((( I ↾ 𝐵) ∈ 𝑇0𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
3016, 28, 29syl2anc 586 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31 simpl 485 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3216adantr 483 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
3328adantr 483 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0𝐸)
34 xp1st 7724 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (1st𝑓) ∈ 𝑇)
3534adantl 484 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
36 xp2nd 7725 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
3736adantl 484 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ 𝐸)
381, 2, 3, 4, 10, 8, 11dvhopvadd 38233 . . . . 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 1375 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩)
4015, 1, 2ltrn1o 37264 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓):𝐵1-1-onto𝐵)
4135, 40syldan 593 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓):𝐵1-1-onto𝐵)
42 f1of 6618 . . . . . 6 ((1st𝑓):𝐵1-1-onto𝐵 → (1st𝑓):𝐵𝐵)
43 fcoi2 6556 . . . . . 6 ((1st𝑓):𝐵𝐵 → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4441, 42, 433syl 18 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4522adantr 483 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
4627adantr 483 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
4737, 46eleqtrrd 2919 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ (Base‘𝐷))
4823, 11, 24grplid 18136 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ( 0 (2nd𝑓)) = (2nd𝑓))
4945, 47, 48syl2anc 586 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 (2nd𝑓)) = (2nd𝑓))
5044, 49opeq12d 4814 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩ = ⟨(1st𝑓), (2nd𝑓)⟩)
5139, 50eqtrd 2859 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(1st𝑓), (2nd𝑓)⟩)
52 1st2nd2 7731 . . . . 5 (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5352adantl 484 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5453oveq2d 7175 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩))
5551, 54, 533eqtr4d 2869 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = 𝑓)
561, 2ltrncnv 37286 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓) ∈ 𝑇)
5735, 56syldan 593 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
58 dvhgrp.i . . . . . 6 𝐼 = (invg𝐷)
5923, 58grpinvcl 18154 . . . . 5 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6045, 47, 59syl2anc 586 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6160, 46eleqtrd 2918 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ 𝐸)
62 opelxpi 5595 . . 3 (((1st𝑓) ∈ 𝑇 ∧ (𝐼‘(2nd𝑓)) ∈ 𝐸) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6357, 61, 62syl2anc 586 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6453oveq2d 7175 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩))
651, 2, 3, 4, 10, 8, 11dvhopvadd 38233 . . . . 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 1375 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩)
67 f1ococnv1 6646 . . . . . 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 18155 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7045, 47, 69syl2anc 586 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7168, 70opeq12d 4814 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩ = ⟨( I ↾ 𝐵), 0 ⟩)
7266, 71eqtrd 2859 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨( I ↾ 𝐵), 0 ⟩)
7364, 72eqtrd 2859 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = ⟨( I ↾ 𝐵), 0 ⟩)
747, 9, 13, 14, 30, 55, 63, 73isgrpd 18128 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  cop 4576   I cid 5462   × cxp 5556  ccnv 5557  cres 5560  ccom 5562  wf 6354  1-1-ontowf1o 6357  cfv 6358  (class class class)co 7159  1st c1st 7690  2nd c2nd 7691  Basecbs 16486  +gcplusg 16568  Scalarcsca 16571  0gc0g 16716  Grpcgrp 18106  invgcminusg 18107  DivRingcdr 19505  HLchlt 36490  LHypclh 37124  LTrncltrn 37241  TEndoctendo 37892  EDRingcedring 37893  DVecHcdvh 38218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-riotaBAD 36093
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-tpos 7895  df-undef 7942  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-sca 16584  df-vsca 16585  df-0g 16718  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-p1 17653  df-lat 17659  df-clat 17721  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-grp 18109  df-minusg 18110  df-mgp 19243  df-ur 19255  df-ring 19302  df-oppr 19376  df-dvdsr 19394  df-unit 19395  df-invr 19425  df-dvr 19436  df-drng 19507  df-oposet 36316  df-ol 36318  df-oml 36319  df-covers 36406  df-ats 36407  df-atl 36438  df-cvlat 36462  df-hlat 36491  df-llines 36638  df-lplanes 36639  df-lvols 36640  df-lines 36641  df-psubsp 36643  df-pmap 36644  df-padd 36936  df-lhyp 37128  df-laut 37129  df-ldil 37244  df-ltrn 37245  df-trl 37299  df-tendo 37895  df-edring 37897  df-dvech 38219
This theorem is referenced by:  dvhlveclem  38248
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