Theorem iscsrg 41912

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 6902	. . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2		iscsrg.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
3	1, 2	eqtr4di 2791	. . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
4	3	eleq1d 2822	. 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5		df-csring 41911	. 2 ⊢ CSRing = {𝑟 ∈ SRing ∣ (mulGrp‘𝑟) ∈ CMnd}
6	4, 5	elrab2 3698	1 ⊢ (𝑅 ∈ CSRing ↔ (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1535   ∈ wcel 2104  ‘cfv 6559  CMndccmn 19799  mulGrpcmgp 20138  SRingcsrg 20190   CSRing ccsrg 41910

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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4916  df-br 5151  df-iota 6511  df-fv 6567  df-csring 41911

This theorem is referenced by: (None)