Theorem phlsrng 20777

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 2823	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		phlsrng.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3		eqid 2823	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
4		eqid 2823	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2823	. . 3 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹)
6		eqid 2823	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 20774	. 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 1142	1 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ↦ cmpt 5148  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  *_𝑟cstv 16569  Scalarcsca 16570  ·_𝑖cip 16572  0_gc0g 16715  *-Ringcsr 19617   LMHom clmhm 19793  LVecclvec 19876  ringLModcrglmod 19943  PreHilcphl 20770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-iota 6316  df-fv 6365  df-ov 7161  df-phl 20772

This theorem is referenced by:  iporthcom  20781  ip0r  20783  ipdi  20786  ip2di  20787  ipassr  20792  ipassr2  20793  phlssphl  20805  cphcjcl  23789