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 30870 | . . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | |
2 | 1 | simprbi 499 | . . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing) |
3 | orngogrp 30869 | . . 3 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp) | |
4 | isogrp 30698 | . . . 4 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd)) | |
5 | 4 | simprbi 499 | . . 3 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd) |
6 | 2, 3, 5 | 3syl 18 | . 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oMnd) |
7 | omndtos 30701 | . 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 2110 Tosetctos 17637 Grpcgrp 18097 Fieldcfield 19497 oMndcomnd 30693 oGrpcogrp 30694 oRingcorng 30863 oFieldcofld 30864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-omnd 30695 df-ogrp 30696 df-orng 30865 df-ofld 30866 |
This theorem is referenced by: ofldchr 30882 |
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