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Theorem phlsrng 20703
Description: The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypothesis
Ref Expression
phlsrng.f 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
phlsrng (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)

Proof of Theorem phlsrng
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2818 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 phlsrng.f . . 3 𝐹 = (Scalar‘𝑊)
3 eqid 2818 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2818 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2818 . . 3 (*𝑟𝐹) = (*𝑟𝐹)
6 eqid 2818 . . 3 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 20700 . 2 (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥(·𝑖𝑊)𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟𝐹)‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))))
87simp2bi 1138 1 (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cmpt 5137  cfv 6348  (class class class)co 7145  Basecbs 16471  *𝑟cstv 16555  Scalarcsca 16556  ·𝑖cip 16558  0gc0g 16701  *-Ringcsr 19544   LMHom clmhm 19720  LVecclvec 19803  ringLModcrglmod 19870  PreHilcphl 20696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-iota 6307  df-fv 6356  df-ov 7148  df-phl 20698
This theorem is referenced by:  iporthcom  20707  ip0r  20709  ipdi  20712  ip2di  20713  ipassr  20718  ipassr2  20719  phlssphl  20731  cphcjcl  23714
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