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Theorem cphnm 19027
Description: The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
nmsq.v  |-  V  =  ( Base `  W
)
nmsq.h  |-  .,  =  ( .i `  W )
nmsq.n  |-  N  =  ( norm `  W
)
Assertion
Ref Expression
cphnm  |-  ( ( W  e.  CPreHil  /\  A  e.  V )  ->  ( N `  A )  =  ( sqr `  ( A  .,  A ) ) )

Proof of Theorem cphnm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nmsq.v . . . 4  |-  V  =  ( Base `  W
)
2 nmsq.h . . . 4  |-  .,  =  ( .i `  W )
3 nmsq.n . . . 4  |-  N  =  ( norm `  W
)
41, 2, 3cphnmfval 19026 . . 3  |-  ( W  e.  CPreHil  ->  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )
54fveq1d 5670 . 2  |-  ( W  e.  CPreHil  ->  ( N `  A )  =  ( ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) `  A
) )
6 oveq12 6029 . . . . 5  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .,  x
)  =  ( A 
.,  A ) )
76anidms 627 . . . 4  |-  ( x  =  A  ->  (
x  .,  x )  =  ( A  .,  A ) )
87fveq2d 5672 . . 3  |-  ( x  =  A  ->  ( sqr `  ( x  .,  x ) )  =  ( sqr `  ( A  .,  A ) ) )
9 eqid 2387 . . 3  |-  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) )  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) )
10 fvex 5682 . . 3  |-  ( sqr `  ( A  .,  A
) )  e.  _V
118, 9, 10fvmpt 5745 . 2  |-  ( A  e.  V  ->  (
( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) `  A
)  =  ( sqr `  ( A  .,  A
) ) )
125, 11sylan9eq 2439 1  |-  ( ( W  e.  CPreHil  /\  A  e.  V )  ->  ( N `  A )  =  ( sqr `  ( A  .,  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   sqrcsqr 11965   Basecbs 13396   .icip 13461   normcnm 18495   CPreHilccph 19000
This theorem is referenced by:  nmsq  19028
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-cph 19002
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