HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hhssabloi Unicode version

Theorem hhssabloi 22715
Description: Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
hhssabl.1  |-  H  e.  SH
Assertion
Ref Expression
hhssabloi  |-  (  +h  |`  ( H  X.  H
) )  e.  AbelOp

Proof of Theorem hhssabloi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hilablo 22615 . . . . . 6  |-  +h  e.  AbelOp
2 ablogrpo 21825 . . . . . 6  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
31, 2ax-mp 8 . . . . 5  |-  +h  e.  GrpOp
4 df-hba 22425 . . . . . 6  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
5 eqid 2404 . . . . . . 7  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
65hhva 22621 . . . . . 6  |-  +h  =  ( +v `  <. <.  +h  ,  .h  >. ,  normh >. )
74, 6bafval 22036 . . . . 5  |-  ~H  =  ran  +h
8 hilid 22616 . . . . . 6  |-  (GId `  +h  )  =  0h
98eqcomi 2408 . . . . 5  |-  0h  =  (GId `  +h  )
105hhnv 22620 . . . . . 6  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
115hhsm 22624 . . . . . . 7  |-  .h  =  ( .s OLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
12 eqid 2404 . . . . . . 7  |-  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )  =  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )
136, 11, 12nvinvfval 22074 . . . . . 6  |-  ( <. <.  +h  ,  .h  >. , 
normh >.  e.  NrmCVec  ->  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )  =  ( inv `  +h  ) )
1410, 13ax-mp 8 . . . . 5  |-  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )  =  ( inv `  +h  )
15 hhssabl.1 . . . . . 6  |-  H  e.  SH
1615shssii 22668 . . . . 5  |-  H  C_  ~H
17 eqid 2404 . . . . 5  |-  (  +h  |`  ( H  X.  H
) )  =  (  +h  |`  ( H  X.  H ) )
18 shaddcl 22672 . . . . . 6  |-  ( ( H  e.  SH  /\  x  e.  H  /\  y  e.  H )  ->  ( x  +h  y
)  e.  H )
1915, 18mp3an1 1266 . . . . 5  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x  +h  y
)  e.  H )
20 sh0 22671 . . . . . 6  |-  ( H  e.  SH  ->  0h  e.  H )
2115, 20ax-mp 8 . . . . 5  |-  0h  e.  H
22 ax-hfvmul 22461 . . . . . . . 8  |-  .h  :
( CC  X.  ~H )
--> ~H
23 ffn 5550 . . . . . . . 8  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  .h  Fn  ( CC  X.  ~H )
)
2422, 23ax-mp 8 . . . . . . 7  |-  .h  Fn  ( CC  X.  ~H )
25 neg1cn 10023 . . . . . . 7  |-  -u 1  e.  CC
2612curry1val 6398 . . . . . . 7  |-  ( (  .h  Fn  ( CC 
X.  ~H )  /\  -u 1  e.  CC )  ->  (
(  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) ) `  x )  =  (
-u 1  .h  x
) )
2724, 25, 26mp2an 654 . . . . . 6  |-  ( (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) ) `  x )  =  (
-u 1  .h  x
)
28 shmulcl 22673 . . . . . . 7  |-  ( ( H  e.  SH  /\  -u 1  e.  CC  /\  x  e.  H )  ->  ( -u 1  .h  x )  e.  H
)
2915, 25, 28mp3an12 1269 . . . . . 6  |-  ( x  e.  H  ->  ( -u 1  .h  x )  e.  H )
3027, 29syl5eqel 2488 . . . . 5  |-  ( x  e.  H  ->  (
(  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) ) `  x )  e.  H
)
313, 7, 9, 14, 16, 17, 19, 21, 30issubgoi 21851 . . . 4  |-  (  +h  |`  ( H  X.  H
) )  e.  (
SubGrpOp `  +h  )
32 issubgo 21844 . . . 4  |-  ( (  +h  |`  ( H  X.  H ) )  e.  ( SubGrpOp `  +h  )  <->  (  +h  e.  GrpOp  /\  (  +h  |`  ( H  X.  H
) )  e.  GrpOp  /\  (  +h  |`  ( H  X.  H ) ) 
C_  +h  ) )
3331, 32mpbi 200 . . 3  |-  (  +h  e.  GrpOp  /\  (  +h  |`  ( H  X.  H
) )  e.  GrpOp  /\  (  +h  |`  ( H  X.  H ) ) 
C_  +h  )
3433simp2i 967 . 2  |-  (  +h  |`  ( H  X.  H
) )  e.  GrpOp
35 xpss12 4940 . . . . 5  |-  ( ( H  C_  ~H  /\  H  C_ 
~H )  ->  ( H  X.  H )  C_  ( ~H  X.  ~H )
)
3616, 16, 35mp2an 654 . . . 4  |-  ( H  X.  H )  C_  ( ~H  X.  ~H )
37 ax-hfvadd 22456 . . . . 5  |-  +h  :
( ~H  X.  ~H )
--> ~H
3837fdmi 5555 . . . 4  |-  dom  +h  =  ( ~H  X.  ~H )
3936, 38sseqtr4i 3341 . . 3  |-  ( H  X.  H )  C_  dom  +h
40 ssdmres 5127 . . 3  |-  ( ( H  X.  H ) 
C_  dom  +h  <->  dom  (  +h  |`  ( H  X.  H
) )  =  ( H  X.  H ) )
4139, 40mpbi 200 . 2  |-  dom  (  +h  |`  ( H  X.  H ) )  =  ( H  X.  H
)
4215sheli 22669 . . . 4  |-  ( x  e.  H  ->  x  e.  ~H )
4315sheli 22669 . . . 4  |-  ( y  e.  H  ->  y  e.  ~H )
44 ax-hvcom 22457 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
4542, 43, 44syl2an 464 . . 3  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
46 ovres 6172 . . 3  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x (  +h  |`  ( H  X.  H
) ) y )  =  ( x  +h  y ) )
47 ovres 6172 . . . 4  |-  ( ( y  e.  H  /\  x  e.  H )  ->  ( y (  +h  |`  ( H  X.  H
) ) x )  =  ( y  +h  x ) )
4847ancoms 440 . . 3  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( y (  +h  |`  ( H  X.  H
) ) x )  =  ( y  +h  x ) )
4945, 46, 483eqtr4d 2446 . 2  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x (  +h  |`  ( H  X.  H
) ) y )  =  ( y (  +h  |`  ( H  X.  H ) ) x ) )
5034, 41, 49isabloi 21829 1  |-  (  +h  |`  ( H  X.  H
) )  e.  AbelOp
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   {csn 3774   <.cop 3777    X. cxp 4835   `'ccnv 4836   dom cdm 4837    |` cres 4839    o. ccom 4841    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   2ndc2nd 6307   CCcc 8944   1c1 8947   -ucneg 9248   GrpOpcgr 21727  GIdcgi 21728   invcgn 21729   AbelOpcablo 21822   SubGrpOpcsubgo 21842   NrmCVeccnv 22016   ~Hchil 22375    +h cva 22376    .h csm 22377   normhcno 22379   0hc0v 22380   SHcsh 22384
This theorem is referenced by:  hhssablo  22716  hhssnv  22717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-grpo 21732  df-gid 21733  df-ginv 21734  df-ablo 21823  df-subgo 21843  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-nmcv 22032  df-hnorm 22424  df-hba 22425  df-hvsub 22427  df-sh 22662
  Copyright terms: Public domain W3C validator