Theorem iscsrg 41441

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 6897	. . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2		iscsrg.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
3	1, 2	eqtr4di 2786	. . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
4	3	eleq1d 2814	. 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5		df-csring 41440	. 2 ⊢ CSRing = {𝑟 ∈ SRing ∣ (mulGrp‘𝑟) ∈ CMnd}
6	4, 5	elrab2 3685	1 ⊢ (𝑅 ∈ CSRing ↔ (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ‘cfv 6548  CMndccmn 19734  mulGrpcmgp 20073  SRingcsrg 20125   CSRing ccsrg 41439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-csring 41440

This theorem is referenced by: (None)

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