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Theorem tchval 19130
Description: Define a function to augment a pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
tchval.n  |-  G  =  (toCHil `  W )
tchval.v  |-  V  =  ( Base `  W
)
tchval.h  |-  .,  =  ( .i `  W )
Assertion
Ref Expression
tchval  |-  G  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) ) )
Distinct variable groups:    x,  .,    x, G   
x, V    x, W

Proof of Theorem tchval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 tchval.n . 2  |-  G  =  (toCHil `  W )
2 id 20 . . . . 5  |-  ( w  =  W  ->  w  =  W )
3 fveq2 5687 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 tchval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2454 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
6 fveq2 5687 . . . . . . . . 9  |-  ( w  =  W  ->  ( .i `  w )  =  ( .i `  W
) )
7 tchval.h . . . . . . . . 9  |-  .,  =  ( .i `  W )
86, 7syl6eqr 2454 . . . . . . . 8  |-  ( w  =  W  ->  ( .i `  w )  = 
.,  )
98oveqd 6057 . . . . . . 7  |-  ( w  =  W  ->  (
x ( .i `  w ) x )  =  ( x  .,  x ) )
109fveq2d 5691 . . . . . 6  |-  ( w  =  W  ->  ( sqr `  ( x ( .i `  w ) x ) )  =  ( sqr `  (
x  .,  x )
) )
115, 10mpteq12dv 4247 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( sqr `  ( x ( .i
`  w ) x ) ) )  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
122, 11oveq12d 6058 . . . 4  |-  ( w  =  W  ->  (
w toNrmGrp  ( x  e.  (
Base `  w )  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) ) ) )
13 df-tch 19085 . . . 4  |- toCHil  =  ( w  e.  _V  |->  ( w toNrmGrp  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) ) )
14 ovex 6065 . . . 4  |-  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  e. 
_V
1512, 13, 14fvmpt 5765 . . 3  |-  ( W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
16 fvprc 5681 . . . 4  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  (/) )
17 reldmtng 18632 . . . . 5  |-  Rel  dom toNrmGrp
1817ovprc1 6068 . . . 4  |-  ( -.  W  e.  _V  ->  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  =  (/) )
1916, 18eqtr4d 2439 . . 3  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
2015, 19pm2.61i 158 . 2  |-  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
211, 20eqtri 2424 1  |-  G  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   sqrcsqr 11993   Basecbs 13424   .icip 13489   toNrmGrp ctng 18579  toCHilctch 19083
This theorem is referenced by:  tchbas  19131  tchplusg  19132  tchmulr  19133  tchsca  19134  tchvsca  19135  tchip  19136  tchtopn  19137  tchnmfval  19139  tchcph  19147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-tng 18585  df-tch 19085
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