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

Theorem hilid 25770
Description: The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
hilid  |-  (GId `  +h  )  =  0h

Proof of Theorem hilid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hilablo 25769 . . . 4  |-  +h  e.  AbelOp
2 ablogrpo 24978 . . . 4  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +h  e.  GrpOp
4 ax-hfvadd 25609 . . . . . 6  |-  +h  :
( ~H  X.  ~H )
--> ~H
54fdmi 5735 . . . . 5  |-  dom  +h  =  ( ~H  X.  ~H )
63, 5grporn 24906 . . . 4  |-  ~H  =  ran  +h
7 eqid 2467 . . . 4  |-  (GId `  +h  )  =  (GId ` 
+h  )
86, 7grpoidval 24910 . . 3  |-  (  +h  e.  GrpOp  ->  (GId `  +h  )  =  ( iota_ y  e.  ~H  A. x  e.  ~H  ( y  +h  x )  =  x ) )
93, 8ax-mp 5 . 2  |-  (GId `  +h  )  =  ( iota_ y  e.  ~H  A. x  e.  ~H  (
y  +h  x )  =  x )
10 hvaddid2 25632 . . . 4  |-  ( x  e.  ~H  ->  ( 0h  +h  x )  =  x )
1110rgen 2824 . . 3  |-  A. x  e.  ~H  ( 0h  +h  x )  =  x
12 ax-hv0cl 25612 . . . 4  |-  0h  e.  ~H
136grpoideu 24903 . . . . 5  |-  (  +h  e.  GrpOp  ->  E! y  e.  ~H  A. x  e. 
~H  ( y  +h  x )  =  x )
143, 13ax-mp 5 . . . 4  |-  E! y  e.  ~H  A. x  e.  ~H  ( y  +h  x )  =  x
15 oveq1 6290 . . . . . . 7  |-  ( y  =  0h  ->  (
y  +h  x )  =  ( 0h  +h  x ) )
1615eqeq1d 2469 . . . . . 6  |-  ( y  =  0h  ->  (
( y  +h  x
)  =  x  <->  ( 0h  +h  x )  =  x ) )
1716ralbidv 2903 . . . . 5  |-  ( y  =  0h  ->  ( A. x  e.  ~H  ( y  +h  x
)  =  x  <->  A. x  e.  ~H  ( 0h  +h  x )  =  x ) )
1817riota2 6267 . . . 4  |-  ( ( 0h  e.  ~H  /\  E! y  e.  ~H  A. x  e.  ~H  (
y  +h  x )  =  x )  -> 
( A. x  e. 
~H  ( 0h  +h  x )  =  x  <-> 
( iota_ y  e.  ~H  A. x  e.  ~H  (
y  +h  x )  =  x )  =  0h ) )
1912, 14, 18mp2an 672 . . 3  |-  ( A. x  e.  ~H  ( 0h  +h  x )  =  x  <->  ( iota_ y  e. 
~H  A. x  e.  ~H  ( y  +h  x
)  =  x )  =  0h )
2011, 19mpbi 208 . 2  |-  ( iota_ y  e.  ~H  A. x  e.  ~H  ( y  +h  x )  =  x )  =  0h
219, 20eqtri 2496 1  |-  (GId `  +h  )  =  0h
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   A.wral 2814   E!wreu 2816    X. cxp 4997   ` cfv 5587   iota_crio 6243  (class class class)co 6283   GrpOpcgr 24880  GIdcgi 24881   AbelOpcablo 24975   ~Hchil 25528    +h cva 25529   0hc0v 25533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-hilex 25608  ax-hfvadd 25609  ax-hvcom 25610  ax-hvass 25611  ax-hv0cl 25612  ax-hvaddid 25613  ax-hfvmul 25614  ax-hvmulid 25615  ax-hvdistr2 25618  ax-hvmul0 25619
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-ltxr 9632  df-sub 9806  df-neg 9807  df-grpo 24885  df-gid 24886  df-ablo 24976  df-hvsub 25580
This theorem is referenced by:  hhnv  25774  hh0v  25777  hhssabloi  25870
  Copyright terms: Public domain W3C validator