Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhgrp Structured version   Visualization version   GIF version

Theorem dvhgrp 35414
 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 2610 . . . 4 (Base‘𝑈) = (Base‘𝑈)
61, 2, 3, 4, 5dvhvbase 35394 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
76eqcomd 2616 . 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 35402 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13123impb 1252 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
141, 2, 3, 4, 10, 11, 8dvhvaddass 35404 . 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 34454 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17 eqid 2610 . . . . . . . 8 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
181, 17, 4, 10dvhsca 35389 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
191, 17erngdv 35299 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
2018, 19eqeltrd 2688 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
21 drnggrp 18578 . . . . . 6 (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
2220, 21syl 17 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
23 eqid 2610 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
24 dvhgrp.o . . . . . 6 0 = (0g𝐷)
2523, 24grpidcl 17273 . . . . 5 (𝐷 ∈ Grp → 0 ∈ (Base‘𝐷))
2622, 25syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 ∈ (Base‘𝐷))
271, 3, 4, 10, 23dvhbase 35390 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘𝐷) = 𝐸)
2826, 27eleqtrd 2690 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0𝐸)
29 opelxpi 5072 . . 3 ((( I ↾ 𝐵) ∈ 𝑇0𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
3016, 28, 29syl2anc 691 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31 simpl 472 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3216adantr 480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
3328adantr 480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0𝐸)
34 xp1st 7089 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (1st𝑓) ∈ 𝑇)
3534adantl 481 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
36 xp2nd 7090 . . . . . 6 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
3736adantl 481 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ 𝐸)
381, 2, 3, 4, 10, 8, 11dvhopvadd 35400 . . . . 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 1327 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩)
4015, 1, 2ltrn1o 34428 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓):𝐵1-1-onto𝐵)
4135, 40syldan 486 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓):𝐵1-1-onto𝐵)
42 f1of 6050 . . . . . 6 ((1st𝑓):𝐵1-1-onto𝐵 → (1st𝑓):𝐵𝐵)
43 fcoi2 5992 . . . . . 6 ((1st𝑓):𝐵𝐵 → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4441, 42, 433syl 18 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝐵) ∘ (1st𝑓)) = (1st𝑓))
4522adantr 480 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
4627adantr 480 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
4737, 46eleqtrrd 2691 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd𝑓) ∈ (Base‘𝐷))
4823, 11, 24grplid 17275 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ( 0 (2nd𝑓)) = (2nd𝑓))
4945, 47, 48syl2anc 691 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 (2nd𝑓)) = (2nd𝑓))
5044, 49opeq12d 4348 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝐵) ∘ (1st𝑓)), ( 0 (2nd𝑓))⟩ = ⟨(1st𝑓), (2nd𝑓)⟩)
5139, 50eqtrd 2644 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨(1st𝑓), (2nd𝑓)⟩)
52 1st2nd2 7096 . . . . 5 (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5352adantl 481 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
5453oveq2d 6565 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = (⟨( I ↾ 𝐵), 0+ ⟨(1st𝑓), (2nd𝑓)⟩))
5551, 54, 533eqtr4d 2654 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0+ 𝑓) = 𝑓)
561, 2ltrncnv 34450 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (1st𝑓) ∈ 𝑇) → (1st𝑓) ∈ 𝑇)
5735, 56syldan 486 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st𝑓) ∈ 𝑇)
58 dvhgrp.i . . . . . 6 𝐼 = (invg𝐷)
5923, 58grpinvcl 17290 . . . . 5 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6045, 47, 59syl2anc 691 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ (Base‘𝐷))
6160, 46eleqtrd 2690 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd𝑓)) ∈ 𝐸)
62 opelxpi 5072 . . 3 (((1st𝑓) ∈ 𝑇 ∧ (𝐼‘(2nd𝑓)) ∈ 𝐸) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6357, 61, 62syl2anc 691 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ ∈ (𝑇 × 𝐸))
6453oveq2d 6565 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩))
651, 2, 3, 4, 10, 8, 11dvhopvadd 35400 . . . . 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 1327 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩)
67 f1ococnv1 6078 . . . . . 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 17291 . . . . . 6 ((𝐷 ∈ Grp ∧ (2nd𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7045, 47, 69syl2anc 691 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2nd𝑓)) (2nd𝑓)) = 0 )
7168, 70opeq12d 4348 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨((1st𝑓) ∘ (1st𝑓)), ((𝐼‘(2nd𝑓)) (2nd𝑓))⟩ = ⟨( I ↾ 𝐵), 0 ⟩)
7266, 71eqtrd 2644 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + ⟨(1st𝑓), (2nd𝑓)⟩) = ⟨( I ↾ 𝐵), 0 ⟩)
7364, 72eqtrd 2644 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨(1st𝑓), (𝐼‘(2nd𝑓))⟩ + 𝑓) = ⟨( I ↾ 𝐵), 0 ⟩)
747, 9, 13, 14, 30, 55, 63, 73isgrpd 17267 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   I cid 4948   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246  DivRingcdr 18570  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  TEndoctendo 35058  EDRingcedring 35059  DVecHcdvh 35385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-undef 7286  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-0g 15925  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464  df-tendo 35061  df-edring 35063  df-dvech 35386 This theorem is referenced by:  dvhlveclem  35415
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