Theorem phlsrng 21850

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 2762	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		phlsrng.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3		eqid 2762	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
4		eqid 2762	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2762	. . 3 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹)
6		eqid 2762	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 21847	. 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 1164	1 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  *_𝑟cstv 17350  Scalarcsca 17351  ·_𝑖cip 17353  0_gc0g 17530  *-Ringcsr 21010   LMHom clmhm 21209  LVecclvec 21292  ringLModcrglmod 21362  PreHilcphl 21843

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-iota 6493  df-fv 6545  df-ov 7420  df-phl 21845

This theorem is used by:  iporthcom  21854  ip0r  21856  ipdi  21859  ip2di  21860  ipassr  21865  ipassr2  21866  phlssphl  21878  cphcjcl  25417