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Theorem dvhgrp 36910
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 2770 . . . 4 (Base‘𝑈) = (Base‘𝑈)
61, 2, 3, 4, 5dvhvbase 36890 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
76eqcomd 2776 . 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 36898 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13123impb 1106 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
141, 2, 3, 4, 10, 11, 8dvhvaddass 36900 . 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 35951 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17 eqid 2770 . . . . . . . 8 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
181, 17, 4, 10dvhsca 36885 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
191, 17erngdv 36795 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
2018, 19eqeltrd 2849 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
21 drnggrp 18964 . . . . . 6 (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
2220, 21syl 17 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
23 eqid 2770 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
24 dvhgrp.o . . . . . 6 0 = (0g𝐷)
2523, 24grpidcl 17657 . . . . 5 (𝐷 ∈ Grp → 0 ∈ (Base‘𝐷))
2622, 25syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 ∈ (Base‘𝐷))
271, 3, 4, 10, 23dvhbase 36886 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
2826, 27eleqtrd 2851 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0𝐸)
29 opelxpi 5288 . . 3 ((( I ↾ 𝐵) ∈ 𝑇0𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
3016, 28, 29syl2anc 565 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31 simpl 468 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3216adantr 466 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
3328adantr 466 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0𝐸)
34 xp1st 7346 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (1st𝑓) ∈ 𝑇)
3534adantl 467 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
36 xp2nd 7347 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
3736adantl 467 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ 𝐸)
381, 2, 3, 4, 10, 8, 11dvhopvadd 36896 . . . . 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 1484 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩)
4015, 1, 2ltrn1o 35925 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓):𝐵1-1-onto𝐵)
4135, 40syldan 571 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓):𝐵1-1-onto𝐵)
42 f1of 6278 . . . . . 6 ((1st𝑓):𝐵1-1-onto𝐵 → (1st𝑓):𝐵𝐵)
43 fcoi2 6219 . . . . . 6 ((1st𝑓):𝐵𝐵 → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4441, 42, 433syl 18 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4522adantr 466 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
4627adantr 466 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
4737, 46eleqtrrd 2852 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ (Base‘𝐷))
4823, 11, 24grplid 17659 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ( 0 (2nd𝑓)) = (2nd𝑓))
4945, 47, 48syl2anc 565 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 (2nd𝑓)) = (2nd𝑓))
5044, 49opeq12d 4545 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩ = ⟨(1st𝑓), (2nd𝑓)⟩)
5139, 50eqtrd 2804 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(1st𝑓), (2nd𝑓)⟩)
52 1st2nd2 7353 . . . . 5 (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5352adantl 467 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5453oveq2d 6808 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩))
5551, 54, 533eqtr4d 2814 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = 𝑓)
561, 2ltrncnv 35947 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓) ∈ 𝑇)
5735, 56syldan 571 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
58 dvhgrp.i . . . . . 6 𝐼 = (invg𝐷)
5923, 58grpinvcl 17674 . . . . 5 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6045, 47, 59syl2anc 565 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6160, 46eleqtrd 2851 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ 𝐸)
62 opelxpi 5288 . . 3 (((1st𝑓) ∈ 𝑇 ∧ (𝐼‘(2nd𝑓)) ∈ 𝐸) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6357, 61, 62syl2anc 565 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6453oveq2d 6808 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩))
651, 2, 3, 4, 10, 8, 11dvhopvadd 36896 . . . . 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 1484 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩)
67 f1ococnv1 6306 . . . . . 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 17675 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7045, 47, 69syl2anc 565 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7168, 70opeq12d 4545 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩ = ⟨( I ↾ 𝐵), 0 ⟩)
7266, 71eqtrd 2804 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨( I ↾ 𝐵), 0 ⟩)
7364, 72eqtrd 2804 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = ⟨( I ↾ 𝐵), 0 ⟩)
747, 9, 13, 14, 30, 55, 63, 73isgrpd 17651 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  cop 4320   I cid 5156   × cxp 5247  ccnv 5248  cres 5251  ccom 5253  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  Basecbs 16063  +gcplusg 16148  Scalarcsca 16151  0gc0g 16307  Grpcgrp 17629  invgcminusg 17630  DivRingcdr 18956  HLchlt 35152  LHypclh 35785  LTrncltrn 35902  TEndoctendo 36554  EDRingcedring 36555  DVecHcdvh 36881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-riotaBAD 34754
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-tpos 7503  df-undef 7550  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-0g 16309  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-p1 17247  df-lat 17253  df-clat 17315  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-mgp 18697  df-ur 18709  df-ring 18756  df-oppr 18830  df-dvdsr 18848  df-unit 18849  df-invr 18879  df-dvr 18890  df-drng 18958  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300  df-lvols 35301  df-lines 35302  df-psubsp 35304  df-pmap 35305  df-padd 35597  df-lhyp 35789  df-laut 35790  df-ldil 35905  df-ltrn 35906  df-trl 35961  df-tendo 36557  df-edring 36559  df-dvech 36882
This theorem is referenced by:  dvhlveclem  36911
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