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Theorem hhssabloi 26748
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 26648 . . . . . 6  |-  +h  e.  AbelOp
2 ablogrpo 25857 . . . . . 6  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
31, 2ax-mp 5 . . . . 5  |-  +h  e.  GrpOp
4 df-hba 26457 . . . . . 6  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
5 eqid 2429 . . . . . . 7  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
65hhva 26654 . . . . . 6  |-  +h  =  ( +v `  <. <.  +h  ,  .h  >. ,  normh >. )
74, 6bafval 26068 . . . . 5  |-  ~H  =  ran  +h
8 hilid 26649 . . . . . 6  |-  (GId `  +h  )  =  0h
98eqcomi 2442 . . . . 5  |-  0h  =  (GId `  +h  )
105hhnv 26653 . . . . . 6  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
115hhsm 26657 . . . . . . 7  |-  .h  =  ( .sOLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
12 eqid 2429 . . . . . . 7  |-  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )  =  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )
136, 11, 12nvinvfval 26106 . . . . . 6  |-  ( <. <.  +h  ,  .h  >. , 
normh >.  e.  NrmCVec  ->  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )  =  ( inv `  +h  ) )
1410, 13ax-mp 5 . . . . 5  |-  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )  =  ( inv `  +h  )
15 hhssabl.1 . . . . . 6  |-  H  e.  SH
1615shssii 26701 . . . . 5  |-  H  C_  ~H
17 eqid 2429 . . . . 5  |-  (  +h  |`  ( H  X.  H
) )  =  (  +h  |`  ( H  X.  H ) )
18 shaddcl 26705 . . . . . 6  |-  ( ( H  e.  SH  /\  x  e.  H  /\  y  e.  H )  ->  ( x  +h  y
)  e.  H )
1915, 18mp3an1 1347 . . . . 5  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x  +h  y
)  e.  H )
20 sh0 26704 . . . . . 6  |-  ( H  e.  SH  ->  0h  e.  H )
2115, 20ax-mp 5 . . . . 5  |-  0h  e.  H
22 ax-hfvmul 26493 . . . . . . . 8  |-  .h  :
( CC  X.  ~H )
--> ~H
23 ffn 5746 . . . . . . . 8  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  .h  Fn  ( CC  X.  ~H )
)
2422, 23ax-mp 5 . . . . . . 7  |-  .h  Fn  ( CC  X.  ~H )
25 neg1cn 10713 . . . . . . 7  |-  -u 1  e.  CC
2612curry1val 6900 . . . . . . 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 676 . . . . . 6  |-  ( (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) ) `  x )  =  (
-u 1  .h  x
)
28 shmulcl 26706 . . . . . . 7  |-  ( ( H  e.  SH  /\  -u 1  e.  CC  /\  x  e.  H )  ->  ( -u 1  .h  x )  e.  H
)
2915, 25, 28mp3an12 1350 . . . . . 6  |-  ( x  e.  H  ->  ( -u 1  .h  x )  e.  H )
3027, 29syl5eqel 2521 . . . . 5  |-  ( x  e.  H  ->  (
(  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) ) `  x )  e.  H
)
313, 7, 9, 14, 16, 17, 19, 21, 30issubgoi 25883 . . . 4  |-  (  +h  |`  ( H  X.  H
) )  e.  (
SubGrpOp `  +h  )
32 issubgo 25876 . . . 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 211 . . 3  |-  (  +h  e.  GrpOp  /\  (  +h  |`  ( H  X.  H
) )  e.  GrpOp  /\  (  +h  |`  ( H  X.  H ) ) 
C_  +h  )
3433simp2i 1015 . 2  |-  (  +h  |`  ( H  X.  H
) )  e.  GrpOp
35 xpss12 4960 . . . . 5  |-  ( ( H  C_  ~H  /\  H  C_ 
~H )  ->  ( H  X.  H )  C_  ( ~H  X.  ~H )
)
3616, 16, 35mp2an 676 . . . 4  |-  ( H  X.  H )  C_  ( ~H  X.  ~H )
37 ax-hfvadd 26488 . . . . 5  |-  +h  :
( ~H  X.  ~H )
--> ~H
3837fdmi 5751 . . . 4  |-  dom  +h  =  ( ~H  X.  ~H )
3936, 38sseqtr4i 3503 . . 3  |-  ( H  X.  H )  C_  dom  +h
40 ssdmres 5146 . . 3  |-  ( ( H  X.  H ) 
C_  dom  +h  <->  dom  (  +h  |`  ( H  X.  H
) )  =  ( H  X.  H ) )
4139, 40mpbi 211 . 2  |-  dom  (  +h  |`  ( H  X.  H ) )  =  ( H  X.  H
)
4215sheli 26702 . . . 4  |-  ( x  e.  H  ->  x  e.  ~H )
4315sheli 26702 . . . 4  |-  ( y  e.  H  ->  y  e.  ~H )
44 ax-hvcom 26489 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
4542, 43, 44syl2an 479 . . 3  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
46 ovres 6450 . . 3  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x (  +h  |`  ( H  X.  H
) ) y )  =  ( x  +h  y ) )
47 ovres 6450 . . . 4  |-  ( ( y  e.  H  /\  x  e.  H )  ->  ( y (  +h  |`  ( H  X.  H
) ) x )  =  ( y  +h  x ) )
4847ancoms 454 . . 3  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( y (  +h  |`  ( H  X.  H
) ) x )  =  ( y  +h  x ) )
4945, 46, 483eqtr4d 2480 . 2  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x (  +h  |`  ( H  X.  H
) ) y )  =  ( y (  +h  |`  ( H  X.  H ) ) x ) )
5034, 41, 49isabloi 25861 1  |-  (  +h  |`  ( H  X.  H
) )  e.  AbelOp
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   _Vcvv 3087    C_ wss 3442   {csn 4002   <.cop 4008    X. cxp 4852   `'ccnv 4853   dom cdm 4854    |` cres 4856    o. ccom 4858    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   2ndc2nd 6806   CCcc 9536   1c1 9539   -ucneg 9860   GrpOpcgr 25759  GIdcgi 25760   invcgn 25761   AbelOpcablo 25854   SubGrpOpcsubgo 25874   NrmCVeccnv 26048   ~Hchil 26407    +h cva 26408    .h csm 26409   normhcno 26411   0hc0v 26412   SHcsh 26416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-hilex 26487  ax-hfvadd 26488  ax-hvcom 26489  ax-hvass 26490  ax-hv0cl 26491  ax-hvaddid 26492  ax-hfvmul 26493  ax-hvmulid 26494  ax-hvmulass 26495  ax-hvdistr1 26496  ax-hvdistr2 26497  ax-hvmul0 26498  ax-hfi 26567  ax-his1 26570  ax-his2 26571  ax-his3 26572  ax-his4 26573
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-grpo 25764  df-gid 25765  df-ginv 25766  df-ablo 25855  df-subgo 25875  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-nmcv 26064  df-hnorm 26456  df-hba 26457  df-hvsub 26459  df-sh 26695
This theorem is referenced by:  hhssablo  26749  hhssnv  26750
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