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Theorem cphnmfval 18628
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 2283 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2283 . . 3  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 18606 . 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 972 1  |-  ( W  e.  CPreHil  ->  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152    e. cmpt 4077   "cima 4692   ` cfv 5255  (class class class)co 5858   0cc0 8737    +oocpnf 8864   [,)cico 10658   sqrcsqr 11718   Basecbs 13148   ↾s cress 13149  Scalarcsca 13211   .icip 13213  ℂfldccnfld 16377   PreHilcphl 16528   normcnm 18099  NrmModcnlm 18103   CPreHilccph 18602
This theorem is referenced by:  cphnm  18629  cphnmf  18631  cphtchnm  18661
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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-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-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-cph 18604
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