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Mirrors > Home > MPE Home > Th. List > Mathboxes > crngorngo | Structured version Visualization version GIF version |
Description: A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
crngorngo | ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscrngo 33795 | . 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 RingOpscrngo 33693 Com2ccm2 33788 CRingOpsccring 33792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-crngo 33793 |
This theorem is referenced by: crngm23 33801 crngm4 33802 crngohomfo 33805 isidlc 33814 dmnrngo 33856 prnc 33866 isfldidl 33867 isfldidl2 33868 ispridlc 33869 pridlc3 33872 isdmn3 33873 |
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