Theorem assasca 22049

Description: The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by SN, 2-Mar-2025.)

Hypothesis

Ref	Expression
assasca.f	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
assasca	⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)

Proof of Theorem assasca

Step	Hyp	Ref	Expression
1		assalmod 22047	. 2 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
2		assasca.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 21026	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 18	1 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ‘cfv 6543  Scalarcsca 17338  Ringcrg 20346  LModclmod 21018  AssAlgcasa 22037

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 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

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 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-lmod 21020  df-assa 22040

This theorem is used by:  assa2ass  22050  assa2ass2  22051  ascldimul  22075  asclrhm  22077  assamulgscmlem2  22087  asclmulg  22089