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Theorem hilid 24563
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 24562 . . . 4  |-  +h  e.  AbelOp
2 ablogrpo 23771 . . . 4  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +h  e.  GrpOp
4 ax-hfvadd 24402 . . . . . 6  |-  +h  :
( ~H  X.  ~H )
--> ~H
54fdmi 5564 . . . . 5  |-  dom  +h  =  ( ~H  X.  ~H )
63, 5grporn 23699 . . . 4  |-  ~H  =  ran  +h
7 eqid 2443 . . . 4  |-  (GId `  +h  )  =  (GId ` 
+h  )
86, 7grpoidval 23703 . . 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 24425 . . . 4  |-  ( x  e.  ~H  ->  ( 0h  +h  x )  =  x )
1110rgen 2781 . . 3  |-  A. x  e.  ~H  ( 0h  +h  x )  =  x
12 ax-hv0cl 24405 . . . 4  |-  0h  e.  ~H
136grpoideu 23696 . . . . 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 6098 . . . . . . 7  |-  ( y  =  0h  ->  (
y  +h  x )  =  ( 0h  +h  x ) )
1615eqeq1d 2451 . . . . . 6  |-  ( y  =  0h  ->  (
( y  +h  x
)  =  x  <->  ( 0h  +h  x )  =  x ) )
1716ralbidv 2735 . . . . 5  |-  ( y  =  0h  ->  ( A. x  e.  ~H  ( y  +h  x
)  =  x  <->  A. x  e.  ~H  ( 0h  +h  x )  =  x ) )
1817riota2 6075 . . . 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 2463 1  |-  (GId `  +h  )  =  0h
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756   A.wral 2715   E!wreu 2717    X. cxp 4838   ` cfv 5418   iota_crio 6051  (class class class)co 6091   GrpOpcgr 23673  GIdcgi 23674   AbelOpcablo 23768   ~Hchil 24321    +h cva 24322   0hc0v 24326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-hilex 24401  ax-hfvadd 24402  ax-hvcom 24403  ax-hvass 24404  ax-hv0cl 24405  ax-hvaddid 24406  ax-hfvmul 24407  ax-hvmulid 24408  ax-hvdistr2 24411  ax-hvmul0 24412
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423  df-sub 9597  df-neg 9598  df-grpo 23678  df-gid 23679  df-ablo 23769  df-hvsub 24373
This theorem is referenced by:  hhnv  24567  hh0v  24570  hhssabloi  24663
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