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Theorem hilid 26649
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 26648 . . . 4  |-  +h  e.  AbelOp
2 ablogrpo 25857 . . . 4  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +h  e.  GrpOp
4 ax-hfvadd 26488 . . . . . 6  |-  +h  :
( ~H  X.  ~H )
--> ~H
54fdmi 5751 . . . . 5  |-  dom  +h  =  ( ~H  X.  ~H )
63, 5grporn 25785 . . . 4  |-  ~H  =  ran  +h
7 eqid 2429 . . . 4  |-  (GId `  +h  )  =  (GId ` 
+h  )
86, 7grpoidval 25789 . . 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 26511 . . . 4  |-  ( x  e.  ~H  ->  ( 0h  +h  x )  =  x )
1110rgen 2792 . . 3  |-  A. x  e.  ~H  ( 0h  +h  x )  =  x
12 ax-hv0cl 26491 . . . 4  |-  0h  e.  ~H
136grpoideu 25782 . . . . 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 6312 . . . . . . 7  |-  ( y  =  0h  ->  (
y  +h  x )  =  ( 0h  +h  x ) )
1615eqeq1d 2431 . . . . . 6  |-  ( y  =  0h  ->  (
( y  +h  x
)  =  x  <->  ( 0h  +h  x )  =  x ) )
1716ralbidv 2871 . . . . 5  |-  ( y  =  0h  ->  ( A. x  e.  ~H  ( y  +h  x
)  =  x  <->  A. x  e.  ~H  ( 0h  +h  x )  =  x ) )
1817riota2 6289 . . . 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 676 . . 3  |-  ( A. x  e.  ~H  ( 0h  +h  x )  =  x  <->  ( iota_ y  e. 
~H  A. x  e.  ~H  ( y  +h  x
)  =  x )  =  0h )
2011, 19mpbi 211 . 2  |-  ( iota_ y  e.  ~H  A. x  e.  ~H  ( y  +h  x )  =  x )  =  0h
219, 20eqtri 2458 1  |-  (GId `  +h  )  =  0h
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    e. wcel 1870   A.wral 2782   E!wreu 2784    X. cxp 4852   ` cfv 5601   iota_crio 6266  (class class class)co 6305   GrpOpcgr 25759  GIdcgi 25760   AbelOpcablo 25854   ~Hchil 26407    +h cva 26408   0hc0v 26412
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-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-hilex 26487  ax-hfvadd 26488  ax-hvcom 26489  ax-hvass 26490  ax-hv0cl 26491  ax-hvaddid 26492  ax-hfvmul 26493  ax-hvmulid 26494  ax-hvdistr2 26497  ax-hvmul0 26498
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-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  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-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-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-ltxr 9679  df-sub 9861  df-neg 9862  df-grpo 25764  df-gid 25765  df-ablo 25855  df-hvsub 26459
This theorem is referenced by:  hhnv  26653  hh0v  26656  hhssabloi  26748
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