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Theorem cphnmfval 24716
Description: The value of the norm in a subcomplex 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 = (Baseβ€˜π‘Š)
nmsq.h , = (Β·π‘–β€˜π‘Š)
nmsq.n 𝑁 = (normβ€˜π‘Š)
Assertion
Ref Expression
cphnmfval (π‘Š ∈ β„‚PreHil β†’ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
Distinct variable groups:   π‘₯, ,   π‘₯,𝑉   π‘₯,π‘Š
Allowed substitution hint:   𝑁(π‘₯)

Proof of Theorem cphnmfval
StepHypRef Expression
1 nmsq.v . . 3 𝑉 = (Baseβ€˜π‘Š)
2 nmsq.h . . 3 , = (Β·π‘–β€˜π‘Š)
3 nmsq.n . . 3 𝑁 = (normβ€˜π‘Š)
4 eqid 2732 . . 3 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2732 . . 3 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
61, 2, 3, 4, 5iscph 24694 . 2 (π‘Š ∈ β„‚PreHil ↔ ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ (Scalarβ€˜π‘Š) = (β„‚fld β†Ύs (Baseβ€˜(Scalarβ€˜π‘Š)))) ∧ (√ β€œ ((Baseβ€˜(Scalarβ€˜π‘Š)) ∩ (0[,)+∞))) βŠ† (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))
76simp3bi 1147 1 (π‘Š ∈ β„‚PreHil β†’ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3947   βŠ† wss 3948   ↦ cmpt 5231   β€œ cima 5679  β€˜cfv 6543  (class class class)co 7411  0cc0 11112  +∞cpnf 11247  [,)cico 13328  βˆšcsqrt 15182  Basecbs 17146   β†Ύs cress 17175  Scalarcsca 17202  Β·π‘–cip 17204  β„‚fldccnfld 20950  PreHilcphl 21183  normcnm 24092  NrmModcnlm 24096  β„‚PreHilccph 24690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fv 6551  df-ov 7414  df-cph 24692
This theorem is referenced by:  cphnm  24717  cphnmf  24719  cphtcphnm  24754  cphsscph  24775
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