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Theorem tchval 22134
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 5825 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 tchval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2480 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
6 fveq2 5825 . . . . . . . . 9  |-  ( w  =  W  ->  ( .i `  w )  =  ( .i `  W
) )
7 tchval.h . . . . . . . . 9  |-  .,  =  ( .i `  W )
86, 7syl6eqr 2480 . . . . . . . 8  |-  ( w  =  W  ->  ( .i `  w )  = 
.,  )
98oveqd 6266 . . . . . . 7  |-  ( w  =  W  ->  (
x ( .i `  w ) x )  =  ( x  .,  x ) )
109fveq2d 5829 . . . . . 6  |-  ( w  =  W  ->  ( sqr `  ( x ( .i `  w ) x ) )  =  ( sqr `  (
x  .,  x )
) )
115, 10mpteq12dv 4445 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( sqr `  ( x ( .i
`  w ) x ) ) )  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
122, 11oveq12d 6267 . . . 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 22089 . . . 4  |- toCHil  =  ( w  e.  _V  |->  ( w toNrmGrp  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) ) )
14 ovex 6277 . . . 4  |-  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  e. 
_V
1512, 13, 14fvmpt 5908 . . 3  |-  ( W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
16 fvprc 5819 . . . 4  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  (/) )
17 reldmtng 21588 . . . . 5  |-  Rel  dom toNrmGrp
1817ovprc1 6280 . . . 4  |-  ( -.  W  e.  _V  ->  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  =  (/) )
1916, 18eqtr4d 2465 . . 3  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
2015, 19pm2.61i 167 . 2  |-  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
211, 20eqtri 2450 1  |-  G  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437    e. wcel 1872   _Vcvv 3022   (/)c0 3704    |-> cmpt 4425   ` cfv 5544  (class class class)co 6249   sqrcsqrt 13240   Basecbs 15064   .icip 15138   toNrmGrp ctng 21535  toCHilctch 22087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-iota 5508  df-fun 5546  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-tng 21541  df-tch 22089
This theorem is referenced by:  tchbas  22135  tchplusg  22136  tchmulr  22138  tchsca  22139  tchvsca  22140  tchip  22141  tchtopn  22142  tchnmfval  22144  tchds  22147  tchcph  22153
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