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Theorem hilid 24498
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 24497 . . . 4  |-  +h  e.  AbelOp
2 ablogrpo 23706 . . . 4  |-  (  +h  e.  AbelOp  ->  +h  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +h  e.  GrpOp
4 ax-hfvadd 24337 . . . . . 6  |-  +h  :
( ~H  X.  ~H )
--> ~H
54fdmi 5561 . . . . 5  |-  dom  +h  =  ( ~H  X.  ~H )
63, 5grporn 23634 . . . 4  |-  ~H  =  ran  +h
7 eqid 2441 . . . 4  |-  (GId `  +h  )  =  (GId ` 
+h  )
86, 7grpoidval 23638 . . 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 24360 . . . 4  |-  ( x  e.  ~H  ->  ( 0h  +h  x )  =  x )
1110rgen 2779 . . 3  |-  A. x  e.  ~H  ( 0h  +h  x )  =  x
12 ax-hv0cl 24340 . . . 4  |-  0h  e.  ~H
136grpoideu 23631 . . . . 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 6097 . . . . . . 7  |-  ( y  =  0h  ->  (
y  +h  x )  =  ( 0h  +h  x ) )
1615eqeq1d 2449 . . . . . 6  |-  ( y  =  0h  ->  (
( y  +h  x
)  =  x  <->  ( 0h  +h  x )  =  x ) )
1716ralbidv 2733 . . . . 5  |-  ( y  =  0h  ->  ( A. x  e.  ~H  ( y  +h  x
)  =  x  <->  A. x  e.  ~H  ( 0h  +h  x )  =  x ) )
1817riota2 6073 . . . 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 667 . . 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 2461 1  |-  (GId `  +h  )  =  0h
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1364    e. wcel 1761   A.wral 2713   E!wreu 2715    X. cxp 4834   ` cfv 5415   iota_crio 6048  (class class class)co 6090   GrpOpcgr 23608  GIdcgi 23609   AbelOpcablo 23703   ~Hchil 24256    +h cva 24257   0hc0v 24261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-hilex 24336  ax-hfvadd 24337  ax-hvcom 24338  ax-hvass 24339  ax-hv0cl 24340  ax-hvaddid 24341  ax-hfvmul 24342  ax-hvmulid 24343  ax-hvdistr2 24346  ax-hvmul0 24347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-sub 9593  df-neg 9594  df-grpo 23613  df-gid 23614  df-ablo 23704  df-hvsub 24308
This theorem is referenced by:  hhnv  24502  hh0v  24505  hhssabloi  24598
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