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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
nmsq.h
nmsq.n
Assertion
Ref Expression
cphnmfval
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem cphnmfval
StepHypRef Expression
1 nmsq.v . . 3
2 nmsq.h . . 3
3 nmsq.n . . 3
4 eqid 2436 . . 3 Scalar Scalar
5 eqid 2436 . . 3 Scalar Scalar
61, 2, 3, 4, 5iscph 19133 . 2 NrmMod Scalar flds Scalar Scalar Scalar
76simp3bi 974 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 936   wceq 1652   wcel 1725   cin 3319   wss 3320   cmpt 4266  cima 4881  cfv 5454  (class class class)co 6081  cc0 8990   cpnf 9117  cico 10918  csqr 12038  cbs 13469   ↾s cress 13470  Scalarcsca 13532  cip 13534  ℂfldccnfld 16703  cphl 16855  cnm 18624  NrmModcnlm 18628  ccph 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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