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Theorem tchval 18748
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 19 . . . . 5  |-  ( w  =  W  ->  w  =  W )
3 fveq2 5605 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 tchval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2408 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  V )
6 fveq2 5605 . . . . . . . . 9  |-  ( w  =  W  ->  ( .i `  w )  =  ( .i `  W
) )
7 tchval.h . . . . . . . . 9  |-  .,  =  ( .i `  W )
86, 7syl6eqr 2408 . . . . . . . 8  |-  ( w  =  W  ->  ( .i `  w )  = 
.,  )
98oveqd 5959 . . . . . . 7  |-  ( w  =  W  ->  (
x ( .i `  w ) x )  =  ( x  .,  x ) )
109fveq2d 5609 . . . . . 6  |-  ( w  =  W  ->  ( sqr `  ( x ( .i `  w ) x ) )  =  ( sqr `  (
x  .,  x )
) )
115, 10mpteq12dv 4177 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( sqr `  ( x ( .i
`  w ) x ) ) )  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
122, 11oveq12d 5960 . . . 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 18703 . . . 4  |- toCHil  =  ( w  e.  _V  |->  ( w toNrmGrp  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) ) )
14 ovex 5967 . . . 4  |-  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  e. 
_V
1512, 13, 14fvmpt 5682 . . 3  |-  ( W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
16 fvprc 5599 . . . 4  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  (/) )
17 reldmtng 18250 . . . . 5  |-  Rel  dom toNrmGrp
1817ovprc1 5970 . . . 4  |-  ( -.  W  e.  _V  ->  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )  =  (/) )
1916, 18eqtr4d 2393 . . 3  |-  ( -.  W  e.  _V  ->  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
2015, 19pm2.61i 156 . 2  |-  (toCHil `  W )  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) )
211, 20eqtri 2378 1  |-  G  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531    e. cmpt 4156   ` cfv 5334  (class class class)co 5942   sqrcsqr 11808   Basecbs 13239   .icip 13304   toNrmGrp ctng 18197  toCHilctch 18701
This theorem is referenced by:  tchbas  18749  tchplusg  18750  tchmulr  18751  tchsca  18752  tchvsca  18753  tchip  18754  tchtopn  18755  tchnmfval  18757  tchcph  18765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-tng 18203  df-tch 18703
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