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Theorem tchval 22270
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 5879 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 tchval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2523 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
6 fveq2 5879 . . . . . . . . 9  |-  ( w  =  W  ->  ( .i `  w )  =  ( .i `  W
) )
7 tchval.h . . . . . . . . 9  |-  .,  =  ( .i `  W )
86, 7syl6eqr 2523 . . . . . . . 8  |-  ( w  =  W  ->  ( .i `  w )  = 
.,  )
98oveqd 6325 . . . . . . 7  |-  ( w  =  W  ->  (
x ( .i `  w ) x )  =  ( x  .,  x ) )
109fveq2d 5883 . . . . . 6  |-  ( w  =  W  ->  ( sqr `  ( x ( .i `  w ) x ) )  =  ( sqr `  (
x  .,  x )
) )
115, 10mpteq12dv 4474 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( sqr `  ( x ( .i
`  w ) x ) ) )  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
122, 11oveq12d 6326 . . . 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 22225 . . . 4  |- toCHil  =  ( w  e.  _V  |->  ( w toNrmGrp  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) ) )
14 ovex 6336 . . . 4  |-  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  e. 
_V
1512, 13, 14fvmpt 5963 . . 3  |-  ( W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
16 fvprc 5873 . . . 4  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  (/) )
17 reldmtng 21724 . . . . 5  |-  Rel  dom toNrmGrp
1817ovprc1 6339 . . . 4  |-  ( -.  W  e.  _V  ->  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  =  (/) )
1916, 18eqtr4d 2508 . . 3  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
2015, 19pm2.61i 169 . 2  |-  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
211, 20eqtri 2493 1  |-  G  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   sqrcsqrt 13373   Basecbs 15199   .icip 15273   toNrmGrp ctng 21671  toCHilctch 22223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-tng 21677  df-tch 22225
This theorem is referenced by:  tchbas  22271  tchplusg  22272  tchmulr  22274  tchsca  22275  tchvsca  22276  tchip  22277  tchtopn  22278  tchnmfval  22280  tchds  22283  tchcph  22289
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