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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofldtos | Structured version Visualization version GIF version |
Description: An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
Ref | Expression |
---|---|
ofldtos | ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isofld 32922 | . . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | |
2 | 1 | simprbi 496 | . . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing) |
3 | orngogrp 32921 | . . 3 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp) | |
4 | isogrp 32723 | . . . 4 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd)) | |
5 | 4 | simprbi 496 | . . 3 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd) |
6 | 2, 3, 5 | 3syl 18 | . 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oMnd) |
7 | omndtos 32726 | . 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset) | |
8 | 6, 7 | syl 17 | 1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Tosetctos 18378 Grpcgrp 18860 Fieldcfield 20585 oMndcomnd 32718 oGrpcogrp 32719 oRingcorng 32915 oFieldcofld 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 6488 df-fv 6544 df-ov 7407 df-omnd 32720 df-ogrp 32721 df-orng 32917 df-ofld 32918 |
This theorem is referenced by: ofldchr 32934 |
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