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Theorem hhssabloi 22764
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 22664 . . . . . 6  |-  +h  e.  AbelOp
2 ablogrpo 21874 . . . . . 6  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
31, 2ax-mp 8 . . . . 5  |-  +h  e.  GrpOp
4 df-hba 22474 . . . . . 6  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
5 eqid 2438 . . . . . . 7  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
65hhva 22670 . . . . . 6  |-  +h  =  ( +v `  <. <.  +h  ,  .h  >. ,  normh >. )
74, 6bafval 22085 . . . . 5  |-  ~H  =  ran  +h
8 hilid 22665 . . . . . 6  |-  (GId `  +h  )  =  0h
98eqcomi 2442 . . . . 5  |-  0h  =  (GId `  +h  )
105hhnv 22669 . . . . . 6  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
115hhsm 22673 . . . . . . 7  |-  .h  =  ( .s OLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
12 eqid 2438 . . . . . . 7  |-  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )  =  (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )
136, 11, 12nvinvfval 22123 . . . . . 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 22717 . . . . 5  |-  H  C_  ~H
17 eqid 2438 . . . . 5  |-  (  +h  |`  ( H  X.  H
) )  =  (  +h  |`  ( H  X.  H ) )
18 shaddcl 22721 . . . . . 6  |-  ( ( H  e.  SH  /\  x  e.  H  /\  y  e.  H )  ->  ( x  +h  y
)  e.  H )
1915, 18mp3an1 1267 . . . . 5  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x  +h  y
)  e.  H )
20 sh0 22720 . . . . . 6  |-  ( H  e.  SH  ->  0h  e.  H )
2115, 20ax-mp 8 . . . . 5  |-  0h  e.  H
22 ax-hfvmul 22510 . . . . . . . 8  |-  .h  :
( CC  X.  ~H )
--> ~H
23 ffn 5593 . . . . . . . 8  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  .h  Fn  ( CC  X.  ~H )
)
2422, 23ax-mp 8 . . . . . . 7  |-  .h  Fn  ( CC  X.  ~H )
25 neg1cn 10069 . . . . . . 7  |-  -u 1  e.  CC
2612curry1val 6441 . . . . . . 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 655 . . . . . 6  |-  ( (  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) ) `  x )  =  (
-u 1  .h  x
)
28 shmulcl 22722 . . . . . . 7  |-  ( ( H  e.  SH  /\  -u 1  e.  CC  /\  x  e.  H )  ->  ( -u 1  .h  x )  e.  H
)
2915, 25, 28mp3an12 1270 . . . . . 6  |-  ( x  e.  H  ->  ( -u 1  .h  x )  e.  H )
3027, 29syl5eqel 2522 . . . . 5  |-  ( x  e.  H  ->  (
(  .h  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) ) `  x )  e.  H
)
313, 7, 9, 14, 16, 17, 19, 21, 30issubgoi 21900 . . . 4  |-  (  +h  |`  ( H  X.  H
) )  e.  (
SubGrpOp `  +h  )
32 issubgo 21893 . . . 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 201 . . 3  |-  (  +h  e.  GrpOp  /\  (  +h  |`  ( H  X.  H
) )  e.  GrpOp  /\  (  +h  |`  ( H  X.  H ) ) 
C_  +h  )
3433simp2i 968 . 2  |-  (  +h  |`  ( H  X.  H
) )  e.  GrpOp
35 xpss12 4983 . . . . 5  |-  ( ( H  C_  ~H  /\  H  C_ 
~H )  ->  ( H  X.  H )  C_  ( ~H  X.  ~H )
)
3616, 16, 35mp2an 655 . . . 4  |-  ( H  X.  H )  C_  ( ~H  X.  ~H )
37 ax-hfvadd 22505 . . . . 5  |-  +h  :
( ~H  X.  ~H )
--> ~H
3837fdmi 5598 . . . 4  |-  dom  +h  =  ( ~H  X.  ~H )
3936, 38sseqtr4i 3383 . . 3  |-  ( H  X.  H )  C_  dom  +h
40 ssdmres 5170 . . 3  |-  ( ( H  X.  H ) 
C_  dom  +h  <->  dom  (  +h  |`  ( H  X.  H
) )  =  ( H  X.  H ) )
4139, 40mpbi 201 . 2  |-  dom  (  +h  |`  ( H  X.  H ) )  =  ( H  X.  H
)
4215sheli 22718 . . . 4  |-  ( x  e.  H  ->  x  e.  ~H )
4315sheli 22718 . . . 4  |-  ( y  e.  H  ->  y  e.  ~H )
44 ax-hvcom 22506 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
4542, 43, 44syl2an 465 . . 3  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
46 ovres 6215 . . 3  |-  ( ( x  e.  H  /\  y  e.  H )  ->  ( x (  +h  |`  ( H  X.  H
) ) y )  =  ( x  +h  y ) )
47 ovres 6215 . . . 4  |-  ( ( y  e.  H  /\  x  e.  H )  ->  ( y (  +h  |`  ( H  X.  H
) ) x )  =  ( y  +h  x ) )
4847ancoms 441 . . 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 21878 1  |-  (  +h  |`  ( H  X.  H
) )  e.  AbelOp
Colors of variables: wff set class
Syntax hints:    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   {csn 3816   <.cop 3819    X. cxp 4878   `'ccnv 4879   dom cdm 4880    |` cres 4882    o. ccom 4884    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   2ndc2nd 6350   CCcc 8990   1c1 8993   -ucneg 9294   GrpOpcgr 21776  GIdcgi 21777   invcgn 21778   AbelOpcablo 21871   SubGrpOpcsubgo 21891   NrmCVeccnv 22065   ~Hchil 22424    +h cva 22425    .h csm 22426   normhcno 22428   0hc0v 22429   SHcsh 22433
This theorem is referenced by:  hhssablo  22765  hhssnv  22766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-hilex 22504  ax-hfvadd 22505  ax-hvcom 22506  ax-hvass 22507  ax-hv0cl 22508  ax-hvaddid 22509  ax-hfvmul 22510  ax-hvmulid 22511  ax-hvmulass 22512  ax-hvdistr1 22513  ax-hvdistr2 22514  ax-hvmul0 22515  ax-hfi 22583  ax-his1 22586  ax-his2 22587  ax-his3 22588  ax-his4 22589
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-grpo 21781  df-gid 21782  df-ginv 21783  df-ablo 21872  df-subgo 21892  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-nmcv 22081  df-hnorm 22473  df-hba 22474  df-hvsub 22476  df-sh 22711
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