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Theorem cphnm 18629
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 18628 . . 3  |-  ( W  e.  CPreHil  ->  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )
54fveq1d 5527 . 2  |-  ( W  e.  CPreHil  ->  ( N `  A )  =  ( ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) `  A
) )
6 oveq12 5867 . . . . 5  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .,  x
)  =  ( A 
.,  A ) )
76anidms 626 . . . 4  |-  ( x  =  A  ->  (
x  .,  x )  =  ( A  .,  A ) )
87fveq2d 5529 . . 3  |-  ( x  =  A  ->  ( sqr `  ( x  .,  x ) )  =  ( sqr `  ( A  .,  A ) ) )
9 eqid 2283 . . 3  |-  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) )  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) )
10 fvex 5539 . . 3  |-  ( sqr `  ( A  .,  A
) )  e.  _V
118, 9, 10fvmpt 5602 . 2  |-  ( A  e.  V  ->  (
( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) `  A
)  =  ( sqr `  ( A  .,  A
) ) )
125, 11sylan9eq 2335 1  |-  ( ( W  e.  CPreHil  /\  A  e.  V )  ->  ( N `  A )  =  ( sqr `  ( A  .,  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   sqrcsqr 11718   Basecbs 13148   .icip 13213   normcnm 18099   CPreHilccph 18602
This theorem is referenced by:  nmsq  18630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-cph 18604
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