Theorem iscsrg 42623

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 6879	. . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2		iscsrg.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
3	1, 2	eqtr4di 2822	. . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
4	3	eleq1d 2854	. 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5		df-csring 42622	. 2 ⊢ CSRing = {𝑟 ∈ SRing ∣ (mulGrp‘𝑟) ∈ CMnd}
6	4, 5	elrab2 3663	1 ⊢ (𝑅 ∈ CSRing ↔ (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ‘cfv 6534  CMndccmn 19846  mulGrpcmgp 20212  SRingcsrg 20264   CSRing ccsrg 42621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-csring 42622

This theorem is referenced by: (None)