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Theorem tchval 20733
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 22 . . . . 5  |-  ( w  =  W  ->  w  =  W )
3 fveq2 5691 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 tchval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2493 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
6 fveq2 5691 . . . . . . . . 9  |-  ( w  =  W  ->  ( .i `  w )  =  ( .i `  W
) )
7 tchval.h . . . . . . . . 9  |-  .,  =  ( .i `  W )
86, 7syl6eqr 2493 . . . . . . . 8  |-  ( w  =  W  ->  ( .i `  w )  = 
.,  )
98oveqd 6108 . . . . . . 7  |-  ( w  =  W  ->  (
x ( .i `  w ) x )  =  ( x  .,  x ) )
109fveq2d 5695 . . . . . 6  |-  ( w  =  W  ->  ( sqr `  ( x ( .i `  w ) x ) )  =  ( sqr `  (
x  .,  x )
) )
115, 10mpteq12dv 4370 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( sqr `  ( x ( .i
`  w ) x ) ) )  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
122, 11oveq12d 6109 . . . 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 20688 . . . 4  |- toCHil  =  ( w  e.  _V  |->  ( w toNrmGrp  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) ) )
14 ovex 6116 . . . 4  |-  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  e. 
_V
1512, 13, 14fvmpt 5774 . . 3  |-  ( W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
16 fvprc 5685 . . . 4  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  (/) )
17 reldmtng 20224 . . . . 5  |-  Rel  dom toNrmGrp
1817ovprc1 6119 . . . 4  |-  ( -.  W  e.  _V  ->  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  =  (/) )
1916, 18eqtr4d 2478 . . 3  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
2015, 19pm2.61i 164 . 2  |-  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
211, 20eqtri 2463 1  |-  G  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2972   (/)c0 3637    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   sqrcsqr 12722   Basecbs 14174   .icip 14243   toNrmGrp ctng 20171  toCHilctch 20686
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-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-tng 20177  df-tch 20688
This theorem is referenced by:  tchbas  20734  tchplusg  20735  tchmulr  20737  tchsca  20738  tchvsca  20739  tchip  20740  tchtopn  20741  tchnmfval  20743  tchds  20746  tchcph  20752
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