Theorem phlsrng 21547

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 21544	. 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 1146	1 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  *_𝑟cstv 17229  Scalarcsca 17230  ·_𝑖cip 17232  0_gc0g 17409  *-Ringcsr 20754   LMHom clmhm 20933  LVecclvec 21016  ringLModcrglmod 21086  PreHilcphl 21540

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-iota 6467  df-fv 6522  df-ov 7393  df-phl 21542

This theorem is referenced by:  iporthcom  21551  ip0r  21553  ipdi  21556  ip2di  21557  ipassr  21562  ipassr2  21563  phlssphl  21575  cphcjcl  25090