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Theorem cphnm 18645
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 18644 . . 3  |-  ( W  e.  CPreHil  ->  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )
54fveq1d 5543 . 2  |-  ( W  e.  CPreHil  ->  ( N `  A )  =  ( ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) `  A
) )
6 oveq12 5883 . . . . 5  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .,  x
)  =  ( A 
.,  A ) )
76anidms 626 . . . 4  |-  ( x  =  A  ->  (
x  .,  x )  =  ( A  .,  A ) )
87fveq2d 5545 . . 3  |-  ( x  =  A  ->  ( sqr `  ( x  .,  x ) )  =  ( sqr `  ( A  .,  A ) ) )
9 eqid 2296 . . 3  |-  ( x  e.  V  |->  ( sqr `  ( x  .,  x
) ) )  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) )
10 fvex 5555 . . 3  |-  ( sqr `  ( A  .,  A
) )  e.  _V
118, 9, 10fvmpt 5618 . 2  |-  ( A  e.  V  ->  (
( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) `  A
)  =  ( sqr `  ( A  .,  A
) ) )
125, 11sylan9eq 2348 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 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   sqrcsqr 11734   Basecbs 13164   .icip 13229   normcnm 18115   CPreHilccph 18618
This theorem is referenced by:  nmsq  18646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-cph 18620
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