MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cphnmfval Structured version   Visualization version   GIF version

Theorem cphnmfval 24462
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 2736 . . 3 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2736 . . 3 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
61, 2, 3, 4, 5iscph 24440 . 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 1146 1 (π‘Š ∈ β„‚PreHil β†’ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ∩ cin 3897   βŠ† wss 3898   ↦ cmpt 5175   β€œ cima 5623  β€˜cfv 6479  (class class class)co 7337  0cc0 10972  +∞cpnf 11107  [,)cico 13182  βˆšcsqrt 15043  Basecbs 17009   β†Ύs cress 17038  Scalarcsca 17062  Β·π‘–cip 17064  β„‚fldccnfld 20703  PreHilcphl 20935  normcnm 23838  NrmModcnlm 23842  β„‚PreHilccph 24436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-nul 5250
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-xp 5626  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fv 6487  df-ov 7340  df-cph 24438
This theorem is referenced by:  cphnm  24463  cphnmf  24465  cphtcphnm  24500  cphsscph  24521
  Copyright terms: Public domain W3C validator