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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orngring | Structured version Visualization version GIF version |
Description: An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
Ref | Expression |
---|---|
orngring | β’ (π β oRing β π β Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
2 | eqid 2733 | . . 3 β’ (0gβπ ) = (0gβπ ) | |
3 | eqid 2733 | . . 3 β’ (.rβπ ) = (.rβπ ) | |
4 | eqid 2733 | . . 3 β’ (leβπ ) = (leβπ ) | |
5 | 1, 2, 3, 4 | isorng 32417 | . 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 | simp1bi 1146 | 1 β’ (π β oRing β π β Ring) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 βwral 3062 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Basecbs 17144 .rcmulr 17198 lecple 17204 0gc0g 17385 Ringcrg 20056 oGrpcogrp 32216 oRingcorng 32413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 df-orng 32415 |
This theorem is referenced by: orngsqr 32422 ornglmulle 32423 orngrmulle 32424 ornglmullt 32425 orngrmullt 32426 orngmullt 32427 orng0le1 32430 suborng 32433 isarchiofld 32435 |
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