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Theorem phlsrng 21031
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 2736 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 phlsrng.f . . 3 𝐹 = (Scalar‘𝑊)
3 eqid 2736 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2736 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2736 . . 3 (*𝑟𝐹) = (*𝑟𝐹)
6 eqid 2736 . . 3 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 21028 . 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 1146 1 (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  wral 3063  cmpt 5187  cfv 6494  (class class class)co 7354  Basecbs 17080  *𝑟cstv 17132  Scalarcsca 17133  ·𝑖cip 17135  0gc0g 17318  *-Ringcsr 20299   LMHom clmhm 20476  LVecclvec 20559  ringLModcrglmod 20626  PreHilcphl 21024
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 2707  ax-nul 5262
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 2714  df-cleq 2728  df-clel 2814  df-ne 2943  df-ral 3064  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-iota 6446  df-fv 6502  df-ov 7357  df-phl 21026
This theorem is referenced by:  iporthcom  21035  ip0r  21037  ipdi  21040  ip2di  21041  ipassr  21046  ipassr2  21047  phlssphl  21059  cphcjcl  24543
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