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Theorem hhssabloilem 28965
Description: Lemma for hhssabloi 28966. Formerly part of proof for hhssabloi 28966 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 28864 . . 3 + ∈ AbelOp
2 ablogrpo 28251 . . 3 ( + ∈ AbelOp → + ∈ GrpOp)
31, 2ax-mp 5 . 2 + ∈ GrpOp
4 hhssabl.1 . . . 4 𝐻S
54elexi 3511 . . 3 𝐻 ∈ V
6 eqid 2818 . . . . . . . 8 ran + = ran +
76grpofo 28203 . . . . . . 7 ( + ∈ GrpOp → + :(ran + × ran + )–onto→ran + )
8 fof 6583 . . . . . . 7 ( + :(ran + × ran + )–onto→ran + → + :(ran + × ran + )⟶ran + )
93, 7, 8mp2b 10 . . . . . 6 + :(ran + × ran + )⟶ran +
104shssii 28917 . . . . . . . 8 𝐻 ⊆ ℋ
11 df-hba 28673 . . . . . . . . 9 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
12 eqid 2818 . . . . . . . . . 10 ⟨⟨ + , · ⟩, norm⟩ = ⟨⟨ + , · ⟩, norm
1312hhva 28870 . . . . . . . . 9 + = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
1411, 13bafval 28308 . . . . . . . 8 ℋ = ran +
1510, 14sseqtri 4000 . . . . . . 7 𝐻 ⊆ ran +
16 xpss12 5563 . . . . . . 7 ((𝐻 ⊆ ran +𝐻 ⊆ ran + ) → (𝐻 × 𝐻) ⊆ (ran + × ran + ))
1715, 15, 16mp2an 688 . . . . . 6 (𝐻 × 𝐻) ⊆ (ran + × ran + )
18 fssres 6537 . . . . . 6 (( + :(ran + × ran + )⟶ran + ∧ (𝐻 × 𝐻) ⊆ (ran + × ran + )) → ( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶ran + )
199, 17, 18mp2an 688 . . . . 5 ( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶ran +
20 ffn 6507 . . . . 5 (( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶ran + → ( + ↾ (𝐻 × 𝐻)) Fn (𝐻 × 𝐻))
2119, 20ax-mp 5 . . . 4 ( + ↾ (𝐻 × 𝐻)) Fn (𝐻 × 𝐻)
22 ovres 7303 . . . . . 6 ((𝑥𝐻𝑦𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))𝑦) = (𝑥 + 𝑦))
23 shaddcl 28921 . . . . . . 7 ((𝐻S𝑥𝐻𝑦𝐻) → (𝑥 + 𝑦) ∈ 𝐻)
244, 23mp3an1 1439 . . . . . 6 ((𝑥𝐻𝑦𝐻) → (𝑥 + 𝑦) ∈ 𝐻)
2522, 24eqeltrd 2910 . . . . 5 ((𝑥𝐻𝑦𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))𝑦) ∈ 𝐻)
2625rgen2 3200 . . . 4 𝑥𝐻𝑦𝐻 (𝑥( + ↾ (𝐻 × 𝐻))𝑦) ∈ 𝐻
27 ffnov 7267 . . . 4 (( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻 ↔ (( + ↾ (𝐻 × 𝐻)) Fn (𝐻 × 𝐻) ∧ ∀𝑥𝐻𝑦𝐻 (𝑥( + ↾ (𝐻 × 𝐻))𝑦) ∈ 𝐻))
2821, 26, 27mpbir2an 707 . . 3 ( + ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻
2922oveq1d 7160 . . . . 5 ((𝑥𝐻𝑦𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦) + 𝑧) = ((𝑥 + 𝑦) + 𝑧))
30293adant3 1124 . . . 4 ((𝑥𝐻𝑦𝐻𝑧𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦) + 𝑧) = ((𝑥 + 𝑦) + 𝑧))
31 ovres 7303 . . . . 5 (((𝑥( + ↾ (𝐻 × 𝐻))𝑦) ∈ 𝐻𝑧𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦)( + ↾ (𝐻 × 𝐻))𝑧) = ((𝑥( + ↾ (𝐻 × 𝐻))𝑦) + 𝑧))
3225, 31stoic3 1768 . . . 4 ((𝑥𝐻𝑦𝐻𝑧𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦)( + ↾ (𝐻 × 𝐻))𝑧) = ((𝑥( + ↾ (𝐻 × 𝐻))𝑦) + 𝑧))
33 ovres 7303 . . . . . . 7 ((𝑦𝐻𝑧𝐻) → (𝑦( + ↾ (𝐻 × 𝐻))𝑧) = (𝑦 + 𝑧))
3433oveq2d 7161 . . . . . 6 ((𝑦𝐻𝑧𝐻) → (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦 + 𝑧)))
35343adant1 1122 . . . . 5 ((𝑥𝐻𝑦𝐻𝑧𝐻) → (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦 + 𝑧)))
3628fovcl 7268 . . . . . . 7 ((𝑦𝐻𝑧𝐻) → (𝑦( + ↾ (𝐻 × 𝐻))𝑧) ∈ 𝐻)
37 ovres 7303 . . . . . . 7 ((𝑥𝐻 ∧ (𝑦( + ↾ (𝐻 × 𝐻))𝑧) ∈ 𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)))
3836, 37sylan2 592 . . . . . 6 ((𝑥𝐻 ∧ (𝑦𝐻𝑧𝐻)) → (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)))
39383impb 1107 . . . . 5 ((𝑥𝐻𝑦𝐻𝑧𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = (𝑥 + (𝑦( + ↾ (𝐻 × 𝐻))𝑧)))
4015sseli 3960 . . . . . 6 (𝑥𝐻𝑥 ∈ ran + )
4115sseli 3960 . . . . . 6 (𝑦𝐻𝑦 ∈ ran + )
4215sseli 3960 . . . . . 6 (𝑧𝐻𝑧 ∈ ran + )
436grpoass 28207 . . . . . . 7 (( + ∈ GrpOp ∧ (𝑥 ∈ ran +𝑦 ∈ ran +𝑧 ∈ ran + )) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
443, 43mpan 686 . . . . . 6 ((𝑥 ∈ ran +𝑦 ∈ ran +𝑧 ∈ ran + ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
4540, 41, 42, 44syl3an 1152 . . . . 5 ((𝑥𝐻𝑦𝐻𝑧𝐻) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
4635, 39, 453eqtr4d 2863 . . . 4 ((𝑥𝐻𝑦𝐻𝑧𝐻) → (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)) = ((𝑥 + 𝑦) + 𝑧))
4730, 32, 463eqtr4d 2863 . . 3 ((𝑥𝐻𝑦𝐻𝑧𝐻) → ((𝑥( + ↾ (𝐻 × 𝐻))𝑦)( + ↾ (𝐻 × 𝐻))𝑧) = (𝑥( + ↾ (𝐻 × 𝐻))(𝑦( + ↾ (𝐻 × 𝐻))𝑧)))
48 hilid 28865 . . . 4 (GId‘ + ) = 0
49 sh0 28920 . . . . 5 (𝐻S → 0𝐻)
504, 49ax-mp 5 . . . 4 0𝐻
5148, 50eqeltri 2906 . . 3 (GId‘ + ) ∈ 𝐻
52 ovres 7303 . . . . 5 (((GId‘ + ) ∈ 𝐻𝑥𝐻) → ((GId‘ + )( + ↾ (𝐻 × 𝐻))𝑥) = ((GId‘ + ) + 𝑥))
5351, 52mpan 686 . . . 4 (𝑥𝐻 → ((GId‘ + )( + ↾ (𝐻 × 𝐻))𝑥) = ((GId‘ + ) + 𝑥))
54 eqid 2818 . . . . . 6 (GId‘ + ) = (GId‘ + )
556, 54grpolid 28220 . . . . 5 (( + ∈ GrpOp ∧ 𝑥 ∈ ran + ) → ((GId‘ + ) + 𝑥) = 𝑥)
563, 40, 55sylancr 587 . . . 4 (𝑥𝐻 → ((GId‘ + ) + 𝑥) = 𝑥)
5753, 56eqtrd 2853 . . 3 (𝑥𝐻 → ((GId‘ + )( + ↾ (𝐻 × 𝐻))𝑥) = 𝑥)
5812hhnv 28869 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
5912hhsm 28873 . . . . . . . 8 · = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
60 eqid 2818 . . . . . . . 8 ( ·(2nd ↾ ({-1} × V))) = ( ·(2nd ↾ ({-1} × V)))
6113, 59, 60nvinvfval 28344 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( ·(2nd ↾ ({-1} × V))) = (inv‘ + ))
6258, 61ax-mp 5 . . . . . 6 ( ·(2nd ↾ ({-1} × V))) = (inv‘ + )
6362eqcomi 2827 . . . . 5 (inv‘ + ) = ( ·(2nd ↾ ({-1} × V)))
6463fveq1i 6664 . . . 4 ((inv‘ + )‘𝑥) = (( ·(2nd ↾ ({-1} × V)))‘𝑥)
65 ax-hfvmul 28709 . . . . . . 7 · :(ℂ × ℋ)⟶ ℋ
66 ffn 6507 . . . . . . 7 ( · :(ℂ × ℋ)⟶ ℋ → · Fn (ℂ × ℋ))
6765, 66ax-mp 5 . . . . . 6 · Fn (ℂ × ℋ)
68 neg1cn 11739 . . . . . 6 -1 ∈ ℂ
6960curry1val 7789 . . . . . 6 (( · Fn (ℂ × ℋ) ∧ -1 ∈ ℂ) → (( ·(2nd ↾ ({-1} × V)))‘𝑥) = (-1 · 𝑥))
7067, 68, 69mp2an 688 . . . . 5 (( ·(2nd ↾ ({-1} × V)))‘𝑥) = (-1 · 𝑥)
71 shmulcl 28922 . . . . . 6 ((𝐻S ∧ -1 ∈ ℂ ∧ 𝑥𝐻) → (-1 · 𝑥) ∈ 𝐻)
724, 68, 71mp3an12 1442 . . . . 5 (𝑥𝐻 → (-1 · 𝑥) ∈ 𝐻)
7370, 72eqeltrid 2914 . . . 4 (𝑥𝐻 → (( ·(2nd ↾ ({-1} × V)))‘𝑥) ∈ 𝐻)
7464, 73eqeltrid 2914 . . 3 (𝑥𝐻 → ((inv‘ + )‘𝑥) ∈ 𝐻)
75 ovres 7303 . . . . 5 ((((inv‘ + )‘𝑥) ∈ 𝐻𝑥𝐻) → (((inv‘ + )‘𝑥)( + ↾ (𝐻 × 𝐻))𝑥) = (((inv‘ + )‘𝑥) + 𝑥))
7674, 75mpancom 684 . . . 4 (𝑥𝐻 → (((inv‘ + )‘𝑥)( + ↾ (𝐻 × 𝐻))𝑥) = (((inv‘ + )‘𝑥) + 𝑥))
77 eqid 2818 . . . . . 6 (inv‘ + ) = (inv‘ + )
786, 54, 77grpolinv 28230 . . . . 5 (( + ∈ GrpOp ∧ 𝑥 ∈ ran + ) → (((inv‘ + )‘𝑥) + 𝑥) = (GId‘ + ))
793, 40, 78sylancr 587 . . . 4 (𝑥𝐻 → (((inv‘ + )‘𝑥) + 𝑥) = (GId‘ + ))
8076, 79eqtrd 2853 . . 3 (𝑥𝐻 → (((inv‘ + )‘𝑥)( + ↾ (𝐻 × 𝐻))𝑥) = (GId‘ + ))
815, 28, 47, 51, 57, 74, 80isgrpoi 28202 . 2 ( + ↾ (𝐻 × 𝐻)) ∈ GrpOp
82 resss 5871 . 2 ( + ↾ (𝐻 × 𝐻)) ⊆ +
833, 81, 823pm3.2i 1331 1 ( + ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ⊆ + )
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933  {csn 4557  cop 4563   × cxp 5546  ccnv 5547  ran crn 5549  cres 5550  ccom 5552   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  (class class class)co 7145  2nd c2nd 7677  cc 10523  1c1 10526  -cneg 10859  GrpOpcgr 28193  GIdcgi 28194  invcgn 28195  AbelOpcablo 28248  NrmCVeccnv 28288  chba 28623   + cva 28624   · csm 28625  normcno 28627  0c0v 28628   S csh 28632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-hilex 28703  ax-hfvadd 28704  ax-hvcom 28705  ax-hvass 28706  ax-hv0cl 28707  ax-hvaddid 28708  ax-hfvmul 28709  ax-hvmulid 28710  ax-hvmulass 28711  ax-hvdistr1 28712  ax-hvdistr2 28713  ax-hvmul0 28714  ax-hfi 28783  ax-his1 28786  ax-his2 28787  ax-his3 28788  ax-his4 28789
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-grpo 28197  df-gid 28198  df-ginv 28199  df-ablo 28249  df-vc 28263  df-nv 28296  df-va 28299  df-ba 28300  df-sm 28301  df-0v 28302  df-nmcv 28304  df-hnorm 28672  df-hba 28673  df-hvsub 28675  df-sh 28911
This theorem is referenced by:  hhssabloi  28966
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