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Theorem phlsrng 20593
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 2737 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 phlsrng.f . . 3 𝐹 = (Scalar‘𝑊)
3 eqid 2737 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2737 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2737 . . 3 (*𝑟𝐹) = (*𝑟𝐹)
6 eqid 2737 . . 3 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 20590 . 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 1148 1 (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2110  wral 3061  cmpt 5135  cfv 6380  (class class class)co 7213  Basecbs 16760  *𝑟cstv 16804  Scalarcsca 16805  ·𝑖cip 16807  0gc0g 16944  *-Ringcsr 19880   LMHom clmhm 20056  LVecclvec 20139  ringLModcrglmod 20206  PreHilcphl 20586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-iota 6338  df-fv 6388  df-ov 7216  df-phl 20588
This theorem is referenced by:  iporthcom  20597  ip0r  20599  ipdi  20602  ip2di  20603  ipassr  20608  ipassr2  20609  phlssphl  20621  cphcjcl  24080
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