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Theorem cphnm 19148
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 19147 . . 3  |-  ( W  e.  CPreHil  ->  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )
54fveq1d 5722 . 2  |-  ( W  e.  CPreHil  ->  ( N `  A )  =  ( ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) `  A
) )
6 oveq12 6082 . . . . 5  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .,  x
)  =  ( A 
.,  A ) )
76anidms 627 . . . 4  |-  ( x  =  A  ->  (
x  .,  x )  =  ( A  .,  A ) )
87fveq2d 5724 . . 3  |-  ( x  =  A  ->  ( sqr `  ( x  .,  x ) )  =  ( sqr `  ( A  .,  A ) ) )
9 eqid 2435 . . 3  |-  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) )  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) )
10 fvex 5734 . . 3  |-  ( sqr `  ( A  .,  A
) )  e.  _V
118, 9, 10fvmpt 5798 . 2  |-  ( A  e.  V  ->  (
( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) `  A
)  =  ( sqr `  ( A  .,  A
) ) )
125, 11sylan9eq 2487 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 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   sqrcsqr 12030   Basecbs 13461   .icip 13526   normcnm 18616   CPreHilccph 19121
This theorem is referenced by:  nmsq  19149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-cph 19123
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