![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > orngogrp | Structured version Visualization version GIF version |
Description: An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
Ref | Expression |
---|---|
orngogrp | β’ (π β oRing β π β oGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
2 | eqid 2726 | . . 3 β’ (0gβπ ) = (0gβπ ) | |
3 | eqid 2726 | . . 3 β’ (.rβπ ) = (.rβπ ) | |
4 | eqid 2726 | . . 3 β’ (leβπ ) = (leβπ ) | |
5 | 1, 2, 3, 4 | isorng 32920 | . 2 β’ (π β oRing β (π β Ring β§ π β oGrp β§ βπ β (Baseβπ )βπ β (Baseβπ )(((0gβπ )(leβπ )π β§ (0gβπ )(leβπ )π) β (0gβπ )(leβπ )(π(.rβπ )π)))) |
6 | 5 | simp2bi 1143 | 1 β’ (π β oRing β π β oGrp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β wcel 2098 βwral 3055 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Basecbs 17153 .rcmulr 17207 lecple 17213 0gc0g 17394 Ringcrg 20138 oGrpcogrp 32722 oRingcorng 32916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 df-ov 7408 df-orng 32918 |
This theorem is referenced by: orngsqr 32925 ornglmulle 32926 orngrmulle 32927 ofldtos 32932 ofldchr 32935 suborng 32936 isarchiofld 32938 nn0omnd 32963 |
Copyright terms: Public domain | W3C validator |