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Theorem cphsca 24696
Description: A subcomplex pre-Hilbert space is a vector space over a subfield of β„‚fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
cphsca.f 𝐹 = (Scalarβ€˜π‘Š)
cphsca.k 𝐾 = (Baseβ€˜πΉ)
Assertion
Ref Expression
cphsca (π‘Š ∈ β„‚PreHil β†’ 𝐹 = (β„‚fld β†Ύs 𝐾))

Proof of Theorem cphsca
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2733 . . . 4 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
3 eqid 2733 . . . 4 (normβ€˜π‘Š) = (normβ€˜π‘Š)
4 cphsca.f . . . 4 𝐹 = (Scalarβ€˜π‘Š)
5 cphsca.k . . . 4 𝐾 = (Baseβ€˜πΉ)
61, 2, 3, 4, 5iscph 24687 . . 3 (π‘Š ∈ β„‚PreHil ↔ ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ (√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ (normβ€˜π‘Š) = (π‘₯ ∈ (Baseβ€˜π‘Š) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘Š)π‘₯)))))
76simp1bi 1146 . 2 (π‘Š ∈ β„‚PreHil β†’ (π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)))
87simp3d 1145 1 (π‘Š ∈ β„‚PreHil β†’ 𝐹 = (β„‚fld β†Ύs 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3948   βŠ† wss 3949   ↦ cmpt 5232   β€œ cima 5680  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  +∞cpnf 11245  [,)cico 13326  βˆšcsqrt 15180  Basecbs 17144   β†Ύs cress 17173  Scalarcsca 17200  Β·π‘–cip 17202  β„‚fldccnfld 20944  PreHilcphl 21177  normcnm 24085  NrmModcnlm 24089  β„‚PreHilccph 24683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-cph 24685
This theorem is referenced by:  cphsubrg  24697  cphreccl  24698  cphcjcl  24700  cphqss  24705  cphclm  24706  ipcau  24755  cphsscph  24768  hlprlem  24884  ishl2  24887
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