![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > srgmnd | GIF version |
Description: A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
Ref | Expression |
---|---|
srgmnd | ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgcmn 13102 | . 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) | |
2 | cmnmnd 13057 | . 2 ⊢ (𝑅 ∈ CMnd → 𝑅 ∈ Mnd) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Mndcmnd 12771 CMndccmn 13041 SRingcsrg 13099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-riota 5830 df-ov 5877 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-plusg 12543 df-mulr 12544 df-0g 12697 df-cmn 13043 df-srg 13100 |
This theorem is referenced by: srg0cl 13113 srgacl 13118 srg1zr 13123 srgmulgass 13125 srgpcomppsc 13128 srglmhm 13129 srgrmhm 13130 |
Copyright terms: Public domain | W3C validator |