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Theorem cphnmfval 19155
Description: The value of the norm in a complex pre-Hilbert space is the square root of the inner product of a vector with itself. (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
cphnmfval  |-  ( W  e.  CPreHil  ->  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )
Distinct variable groups:    x,  .,    x, V   
x, W
Allowed substitution hint:    N( x)

Proof of Theorem cphnmfval
StepHypRef Expression
1 nmsq.v . . 3  |-  V  =  ( Base `  W
)
2 nmsq.h . . 3  |-  .,  =  ( .i `  W )
3 nmsq.n . . 3  |-  N  =  ( norm `  W
)
4 eqid 2436 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2436 . . 3  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 19133 . 2  |-  ( W  e.  CPreHil 
<->  ( ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) )  /\  ( sqr " ( (
Base `  (Scalar `  W
) )  i^i  (
0 [,)  +oo ) ) )  C_  ( Base `  (Scalar `  W )
)  /\  N  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
76simp3bi 974 1  |-  ( W  e.  CPreHil  ->  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320    e. cmpt 4266   "cima 4881   ` cfv 5454  (class class class)co 6081   0cc0 8990    +oocpnf 9117   [,)cico 10918   sqrcsqr 12038   Basecbs 13469   ↾s cress 13470  Scalarcsca 13532   .icip 13534  ℂfldccnfld 16703   PreHilcphl 16855   normcnm 18624  NrmModcnlm 18628   CPreHilccph 19129
This theorem is referenced by:  cphnm  19156  cphnmf  19158  cphtchnm  19188
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
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 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-ov 6084  df-cph 19131
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