Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ogrpgrp | Structured version Visualization version GIF version |
Description: An left ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.) |
Ref | Expression |
---|---|
ogrpgrp | ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isogrp 29702 | . 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd)) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 Grpcgrp 17422 oMndcomnd 29697 oGrpcogrp 29698 |
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-ogrp 29700 |
This theorem is referenced by: ogrpinv0le 29716 ogrpsub 29717 ogrpaddlt 29718 ogrpaddltbi 29719 ogrpaddltrbid 29721 ogrpsublt 29722 ogrpinv0lt 29723 ogrpinvlt 29724 isarchi3 29741 archirng 29742 archirngz 29743 archiabllem1a 29745 archiabllem1b 29746 archiabllem1 29747 archiabllem2a 29748 archiabllem2c 29749 archiabllem2b 29750 archiabllem2 29751 |
Copyright terms: Public domain | W3C validator |