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Theorem dvhgrp 39570
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 2736 . . . 4 (Base‘𝑈) = (Base‘𝑈)
61, 2, 3, 4, 5dvhvbase 39550 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
76eqcomd 2742 . 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 39558 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13123impb 1115 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
141, 2, 3, 4, 10, 11, 8dvhvaddass 39560 . 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 38613 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17 eqid 2736 . . . . . . . 8 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
181, 17, 4, 10dvhsca 39545 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
191, 17erngdv 39456 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
2018, 19eqeltrd 2838 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
21 drnggrp 20195 . . . . . 6 (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
2220, 21syl 17 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
23 eqid 2736 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
24 dvhgrp.o . . . . . 6 0 = (0g𝐷)
2523, 24grpidcl 18778 . . . . 5 (𝐷 ∈ Grp → 0 ∈ (Base‘𝐷))
2622, 25syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 ∈ (Base‘𝐷))
271, 3, 4, 10, 23dvhbase 39546 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
2826, 27eleqtrd 2840 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0𝐸)
29 opelxpi 5670 . . 3 ((( I ↾ 𝐵) ∈ 𝑇0𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
3016, 28, 29syl2anc 584 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31 simpl 483 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3216adantr 481 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
3328adantr 481 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0𝐸)
34 xp1st 7953 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (1st𝑓) ∈ 𝑇)
3534adantl 482 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
36 xp2nd 7954 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
3736adantl 482 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ 𝐸)
381, 2, 3, 4, 10, 8, 11dvhopvadd 39556 . . . . 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 1379 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩)
4015, 1, 2ltrn1o 38587 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓):𝐵1-1-onto𝐵)
4135, 40syldan 591 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓):𝐵1-1-onto𝐵)
42 f1of 6784 . . . . . 6 ((1st𝑓):𝐵1-1-onto𝐵 → (1st𝑓):𝐵𝐵)
43 fcoi2 6717 . . . . . 6 ((1st𝑓):𝐵𝐵 → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4441, 42, 433syl 18 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4522adantr 481 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
4627adantr 481 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
4737, 46eleqtrrd 2841 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ (Base‘𝐷))
4823, 11, 24grplid 18780 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ( 0 (2nd𝑓)) = (2nd𝑓))
4945, 47, 48syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 (2nd𝑓)) = (2nd𝑓))
5044, 49opeq12d 4838 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩ = ⟨(1st𝑓), (2nd𝑓)⟩)
5139, 50eqtrd 2776 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(1st𝑓), (2nd𝑓)⟩)
52 1st2nd2 7960 . . . . 5 (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5352adantl 482 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5453oveq2d 7373 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩))
5551, 54, 533eqtr4d 2786 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = 𝑓)
561, 2ltrncnv 38609 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓) ∈ 𝑇)
5735, 56syldan 591 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
58 dvhgrp.i . . . . . 6 𝐼 = (invg𝐷)
5923, 58grpinvcl 18798 . . . . 5 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6045, 47, 59syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6160, 46eleqtrd 2840 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ 𝐸)
62 opelxpi 5670 . . 3 (((1st𝑓) ∈ 𝑇 ∧ (𝐼‘(2nd𝑓)) ∈ 𝐸) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6357, 61, 62syl2anc 584 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6453oveq2d 7373 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩))
651, 2, 3, 4, 10, 8, 11dvhopvadd 39556 . . . . 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 1379 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩)
67 f1ococnv1 6813 . . . . . 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 18800 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7045, 47, 69syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7168, 70opeq12d 4838 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩ = ⟨( I ↾ 𝐵), 0 ⟩)
7266, 71eqtrd 2776 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨( I ↾ 𝐵), 0 ⟩)
7364, 72eqtrd 2776 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = ⟨( I ↾ 𝐵), 0 ⟩)
747, 9, 13, 14, 30, 55, 63, 73isgrpd 18772 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cop 4592   I cid 5530   × cxp 5631  ccnv 5632  cres 5635  ccom 5637  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  +gcplusg 17133  Scalarcsca 17136  0gc0g 17321  Grpcgrp 18748  invgcminusg 18749  DivRingcdr 20185  HLchlt 37812  LHypclh 38447  LTrncltrn 38564  TEndoctendo 39215  EDRingcedring 39216  DVecHcdvh 39541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-riotaBAD 37415
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-undef 8204  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-0g 17323  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622  df-tendo 39218  df-edring 39220  df-dvech 39542
This theorem is referenced by:  dvhlveclem  39571
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