Theorem phlsrng 21403

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.

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		phlsrng.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3		eqid 2730	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
4		eqid 2730	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2730	. . 3 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹)
6		eqid 2730	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 21400	. 2 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘𝐹) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘𝐹)‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥))))
8	7	simp2bi 1144	1 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059   ↦ cmpt 5230  ‘cfv 6542  (class class class)co 7411  Basecbs 17148  *_𝑟cstv 17203  Scalarcsca 17204  ·_𝑖cip 17206  0_gc0g 17389  *-Ringcsr 20595   LMHom clmhm 20774  LVecclvec 20857  ringLModcrglmod 20927  PreHilcphl 21396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-iota 6494  df-fv 6550  df-ov 7414  df-phl 21398

This theorem is referenced by:  iporthcom  21407  ip0r  21409  ipdi  21412  ip2di  21413  ipassr  21418  ipassr2  21419  phlssphl  21431  cphcjcl  24931