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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iscsrg | Structured version Visualization version GIF version | ||
| Description: A commutative semiring is a semiring whose multiplication is a commutative monoid. (Contributed by metakunt, 4-Apr-2025.) |
| Ref | Expression |
|---|---|
| iscsrg.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
| Ref | Expression |
|---|---|
| iscsrg | ⊢ (𝑅 ∈ CSRing ↔ (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
| 2 | iscsrg.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑅) | |
| 3 | 1, 2 | eqtr4di 2795 | . . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺) |
| 4 | 3 | eleq1d 2826 | . 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd)) |
| 5 | df-csring 41969 | . 2 ⊢ CSRing = {𝑟 ∈ SRing ∣ (mulGrp‘𝑟) ∈ CMnd} | |
| 6 | 4, 5 | elrab2 3695 | 1 ⊢ (𝑅 ∈ CSRing ↔ (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 CMndccmn 19798 mulGrpcmgp 20137 SRingcsrg 20183 CSRing ccsrg 41968 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-csring 41969 |
| This theorem is referenced by: (None) |
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