HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hhssabloilem Structured version   Visualization version   GIF version

Theorem hhssabloilem 29037
Description: Lemma for hhssabloi 29038. Formerly part of proof for hhssabloi 29038 which was based on the deprecated definition "SubGrpOp" for subgroups. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 27-Aug-2021.) (New usage is discouraged.)
Hypothesis
Ref Expression
hhssabl.1 𝐻S
Assertion
Ref Expression
hhssabloilem ( + ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ⊆ + )

Proof of Theorem hhssabloilem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hilablo 28936 . . 3 + ∈ AbelOp
2 ablogrpo 28323 . . 3 ( + ∈ AbelOp → + ∈ GrpOp)
31, 2ax-mp 5 . 2 + ∈ GrpOp
4 hhssabl.1 . . . 4 𝐻S
54elexi 3513 . . 3 𝐻 ∈ V
6 eqid 2821 . . . . . . . 8 ran + = ran +
76grpofo 28275 . . . . . . 7 ( + ∈ GrpOp → + :(ran + × ran + )–onto→ran + )
8 fof 6589 . . . . . . 7 ( + :(ran + × ran + )–onto→ran + → + :(ran + × ran + )⟶ran + )
93, 7, 8mp2b 10 . . . . . 6 + :(ran + × ran + )⟶ran +
104shssii 28989 . . . . . . . 8 𝐻 ⊆ ℋ
11 df-hba 28745 . . . . . . . . 9 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
12 eqid 2821 . . . . . . . . . 10 ⟨⟨ + , · ⟩, norm⟩ = ⟨⟨ + , · ⟩, norm
1312hhva 28942 . . . . . . . . 9 + = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
1411, 13bafval 28380 . . . . . . . 8 ℋ = ran +
1510, 14sseqtri 4002 . . . . . . 7 𝐻 ⊆ ran +
16 xpss12 5569 . . . . . . 7 ((𝐻 ⊆ ran +𝐻 ⊆ ran + ) → (𝐻 × 𝐻) ⊆ (ran + × ran + ))
1715, 15, 16mp2an 690 . . . . . 6 (𝐻 × 𝐻) ⊆ (ran + × ran + )
18 fssres 6543 . . . . . 6 (( + :(ran + × ran + )⟶ran + ∧ (𝐻 × 𝐻) ⊆ (ran + × ran + )) → ( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶ran + )
199, 17, 18mp2an 690 . . . . 5 ( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶ran +
20 ffn 6513 . . . . 5 (( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶ran + → ( + ↾ (𝐻 × 𝐻)) Fn (𝐻 × 𝐻))
2119, 20ax-mp 5 . . . 4 ( + ↾ (𝐻 × 𝐻)) Fn (𝐻 × 𝐻)
22 ovres 7313 . . . . . 6 ((𝑥𝐻𝑦𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))𝑦) = (𝑥 + 𝑦))
23 shaddcl 28993 . . . . . . 7 ((𝐻S𝑥𝐻𝑦𝐻) → (𝑥 + 𝑦) ∈ 𝐻)
244, 23mp3an1 1444 . . . . . 6 ((𝑥𝐻𝑦𝐻) → (𝑥 + 𝑦) ∈ 𝐻)
2522, 24eqeltrd 2913 . . . . 5 ((𝑥𝐻𝑦𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))𝑦) ∈ 𝐻)
2625rgen2 3203 . . . 4 𝑥𝐻𝑦𝐻 (𝑥( + ↾ (𝐻 × 𝐻))𝑦) ∈ 𝐻
27 ffnov 7277 . . . 4 (( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻 ↔ (( + ↾ (𝐻 × 𝐻)) Fn (𝐻 × 𝐻) ∧ ∀𝑥𝐻𝑦𝐻 (𝑥( + ↾ (𝐻 × 𝐻))𝑦) ∈ 𝐻))
2821, 26, 27mpbir2an 709 . . 3 ( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻
2922oveq1d 7170 . . . . 5 ((𝑥𝐻𝑦𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦) + 𝑧) = ((𝑥 + 𝑦) + 𝑧))
30293adant3 1128 . . . 4 ((𝑥𝐻𝑦𝐻𝑧𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦) + 𝑧) = ((𝑥 + 𝑦) + 𝑧))
31 ovres 7313 . . . . 5 (((𝑥( + ↾ (𝐻 × 𝐻))𝑦) ∈ 𝐻𝑧𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦)( + ↾ (𝐻 × 𝐻))𝑧) = ((𝑥( + ↾ (𝐻 × 𝐻))𝑦) + 𝑧))
3225, 31stoic3 1773 . . . 4 ((𝑥𝐻𝑦𝐻𝑧𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦)( + ↾ (𝐻 × 𝐻))𝑧) = ((𝑥( + ↾ (𝐻 × 𝐻))𝑦) + 𝑧))
33 ovres 7313 . . . . . . 7 ((𝑦𝐻𝑧𝐻) → (𝑦( + ↾ (𝐻 × 𝐻))𝑧) = (𝑦 + 𝑧))
3433oveq2d 7171 . . . . . 6 ((𝑦𝐻𝑧𝐻) → (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦 + 𝑧)))
35343adant1 1126 . . . . 5 ((𝑥𝐻𝑦𝐻𝑧𝐻) → (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦 + 𝑧)))
3628fovcl 7278 . . . . . . 7 ((𝑦𝐻𝑧𝐻) → (𝑦( + ↾ (𝐻 × 𝐻))𝑧) ∈ 𝐻)
37 ovres 7313 . . . . . . 7 ((𝑥𝐻 ∧ (𝑦( + ↾ (𝐻 × 𝐻))𝑧) ∈ 𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)))
3836, 37sylan2 594 . . . . . 6 ((𝑥𝐻 ∧ (𝑦𝐻𝑧𝐻)) → (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)))
39383impb 1111 . . . . 5 ((𝑥𝐻𝑦𝐻𝑧𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)))
4015sseli 3962 . . . . . 6 (𝑥𝐻𝑥 ∈ ran + )
4115sseli 3962 . . . . . 6 (𝑦𝐻𝑦 ∈ ran + )
4215sseli 3962 . . . . . 6 (𝑧𝐻𝑧 ∈ ran + )
436grpoass 28279 . . . . . . 7 (( + ∈ GrpOp ∧ (𝑥 ∈ ran +𝑦 ∈ ran +𝑧 ∈ ran + )) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
443, 43mpan 688 . . . . . 6 ((𝑥 ∈ ran +𝑦 ∈ ran +𝑧 ∈ ran + ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
4540, 41, 42, 44syl3an 1156 . . . . 5 ((𝑥𝐻𝑦𝐻𝑧𝐻) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
4635, 39, 453eqtr4d 2866 . . . 4 ((𝑥𝐻𝑦𝐻𝑧𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = ((𝑥 + 𝑦) + 𝑧))
4730, 32, 463eqtr4d 2866 . . 3 ((𝑥𝐻𝑦𝐻𝑧𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦)( + ↾ (𝐻 × 𝐻))𝑧) = (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)))
48 hilid 28937 . . . 4 (GId‘ + ) = 0
49 sh0 28992 . . . . 5 (𝐻S → 0𝐻)
504, 49ax-mp 5 . . . 4 0𝐻
5148, 50eqeltri 2909 . . 3 (GId‘ + ) ∈ 𝐻
52 ovres 7313 . . . . 5 (((GId‘ + ) ∈ 𝐻𝑥𝐻) → ((GId‘ + )( + ↾ (𝐻 × 𝐻))𝑥) = ((GId‘ + ) + 𝑥))
5351, 52mpan 688 . . . 4 (𝑥𝐻 → ((GId‘ + )( + ↾ (𝐻 × 𝐻))𝑥) = ((GId‘ + ) + 𝑥))
54 eqid 2821 . . . . . 6 (GId‘ + ) = (GId‘ + )
556, 54grpolid 28292 . . . . 5 (( + ∈ GrpOp ∧ 𝑥 ∈ ran + ) → ((GId‘ + ) + 𝑥) = 𝑥)
563, 40, 55sylancr 589 . . . 4 (𝑥𝐻 → ((GId‘ + ) + 𝑥) = 𝑥)
5753, 56eqtrd 2856 . . 3 (𝑥𝐻 → ((GId‘ + )( + ↾ (𝐻 × 𝐻))𝑥) = 𝑥)
5812hhnv 28941 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
5912hhsm 28945 . . . . . . . 8 · = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
60 eqid 2821 . . . . . . . 8 ( ·(2nd ↾ ({-1} × V))) = ( ·(2nd ↾ ({-1} × V)))
6113, 59, 60nvinvfval 28416 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( ·(2nd ↾ ({-1} × V))) = (inv‘ + ))
6258, 61ax-mp 5 . . . . . 6 ( ·(2nd ↾ ({-1} × V))) = (inv‘ + )
6362eqcomi 2830 . . . . 5 (inv‘ + ) = ( ·(2nd ↾ ({-1} × V)))
6463fveq1i 6670 . . . 4 ((inv‘ + )‘𝑥) = (( ·(2nd ↾ ({-1} × V)))‘𝑥)
65 ax-hfvmul 28781 . . . . . . 7 · :(ℂ × ℋ)⟶ ℋ
66 ffn 6513 . . . . . . 7 ( · :(ℂ × ℋ)⟶ ℋ → · Fn (ℂ × ℋ))
6765, 66ax-mp 5 . . . . . 6 · Fn (ℂ × ℋ)
68 neg1cn 11750 . . . . . 6 -1 ∈ ℂ
6960curry1val 7799 . . . . . 6 (( · Fn (ℂ × ℋ) ∧ -1 ∈ ℂ) → (( ·(2nd ↾ ({-1} × V)))‘𝑥) = (-1 · 𝑥))
7067, 68, 69mp2an 690 . . . . 5 (( ·(2nd ↾ ({-1} × V)))‘𝑥) = (-1 · 𝑥)
71 shmulcl 28994 . . . . . 6 ((𝐻S ∧ -1 ∈ ℂ ∧ 𝑥𝐻) → (-1 · 𝑥) ∈ 𝐻)
724, 68, 71mp3an12 1447 . . . . 5 (𝑥𝐻 → (-1 · 𝑥) ∈ 𝐻)
7370, 72eqeltrid 2917 . . . 4 (𝑥𝐻 → (( ·(2nd ↾ ({-1} × V)))‘𝑥) ∈ 𝐻)
7464, 73eqeltrid 2917 . . 3 (𝑥𝐻 → ((inv‘ + )‘𝑥) ∈ 𝐻)
75 ovres 7313 . . . . 5 ((((inv‘ + )‘𝑥) ∈ 𝐻𝑥𝐻) → (((inv‘ + )‘𝑥)( + ↾ (𝐻 × 𝐻))𝑥) = (((inv‘ + )‘𝑥) + 𝑥))
7674, 75mpancom 686 . . . 4 (𝑥𝐻 → (((inv‘ + )‘𝑥)( + ↾ (𝐻 × 𝐻))𝑥) = (((inv‘ + )‘𝑥) + 𝑥))
77 eqid 2821 . . . . . 6 (inv‘ + ) = (inv‘ + )
786, 54, 77grpolinv 28302 . . . . 5 (( + ∈ GrpOp ∧ 𝑥 ∈ ran + ) → (((inv‘ + )‘𝑥) + 𝑥) = (GId‘ + ))
793, 40, 78sylancr 589 . . . 4 (𝑥𝐻 → (((inv‘ + )‘𝑥) + 𝑥) = (GId‘ + ))
8076, 79eqtrd 2856 . . 3 (𝑥𝐻 → (((inv‘ + )‘𝑥)( + ↾ (𝐻 × 𝐻))𝑥) = (GId‘ + ))
815, 28, 47, 51, 57, 74, 80isgrpoi 28274 . 2 ( + ↾ (𝐻 × 𝐻)) ∈ GrpOp
82 resss 5877 . 2 ( + ↾ (𝐻 × 𝐻)) ⊆ +
833, 81, 823pm3.2i 1335 1 ( + ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ⊆ + )
Colors of variables: wff setvar class
Syntax hints:  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  Vcvv 3494  wss 3935  {csn 4566  cop 4572   × cxp 5552  ccnv 5553  ran crn 5555  cres 5556  ccom 5558   Fn wfn 6349  wf 6350  ontowfo 6352  cfv 6354  (class class class)co 7155  2nd c2nd 7687  cc 10534  1c1 10537  -cneg 10870  GrpOpcgr 28265  GIdcgi 28266  invcgn 28267  AbelOpcablo 28320  NrmCVeccnv 28360  chba 28695   + cva 28696   · csm 28697  normcno 28699  0c0v 28700   S csh 28704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614  ax-hilex 28775  ax-hfvadd 28776  ax-hvcom 28777  ax-hvass 28778  ax-hv0cl 28779  ax-hvaddid 28780  ax-hfvmul 28781  ax-hvmulid 28782  ax-hvmulass 28783  ax-hvdistr1 28784  ax-hvdistr2 28785  ax-hvmul0 28786  ax-hfi 28855  ax-his1 28858  ax-his2 28859  ax-his3 28860  ax-his4 28861
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-sup 8905  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-seq 13369  df-exp 13429  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-grpo 28269  df-gid 28270  df-ginv 28271  df-ablo 28321  df-vc 28335  df-nv 28368  df-va 28371  df-ba 28372  df-sm 28373  df-0v 28374  df-nmcv 28376  df-hnorm 28744  df-hba 28745  df-hvsub 28747  df-sh 28983
This theorem is referenced by:  hhssabloi  29038
  Copyright terms: Public domain W3C validator