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Mathbox for Thierry Arnoux |
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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 2728 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
2 | eqid 2728 | . . 3 β’ (0gβπ ) = (0gβπ ) | |
3 | eqid 2728 | . . 3 β’ (.rβπ ) = (.rβπ ) | |
4 | eqid 2728 | . . 3 β’ (leβπ ) = (leβπ ) | |
5 | 1, 2, 3, 4 | isorng 33046 | . 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 394 β wcel 2098 βwral 3058 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Basecbs 17189 .rcmulr 17243 lecple 17249 0gc0g 17430 Ringcrg 20187 oGrpcogrp 32807 oRingcorng 33042 |
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 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-orng 33044 |
This theorem is referenced by: orngsqr 33051 ornglmulle 33052 orngrmulle 33053 ofldtos 33058 ofldchr 33061 suborng 33062 isarchiofld 33064 nn0omnd 33089 |
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