Theorem iscsrg 42796

Description: A commutative semiring is a semiring whose multiplication is a commutative monoid. (Contributed by metakunt, 4-Apr-2025.)

Hypothesis

Ref	Expression
iscsrg.g	⊢ 𝐺 = (mulGrp‘𝑅)

Assertion

Ref	Expression
iscsrg	⊢ (𝑅 ∈ CSRing ↔ (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd))

Proof of Theorem iscsrg

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2		iscsrg.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
3	1, 2	eqtr4di 2818	. . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
4	3	eleq1d 2850	. 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5		df-csring 42795	. 2 ⊢ CSRing = {𝑟 ∈ SRing ∣ (mulGrp‘𝑟) ∈ CMnd}
6	4, 5	elrab2 3656	1 ⊢ (𝑅 ∈ CSRing ↔ (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  CMndccmn 19894  mulGrpcmgp 20260  SRingcsrg 20312   CSRing ccsrg 42794

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 2737

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 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-csring 42795

This theorem is used by: (None)