![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 33041 | . . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | |
2 | 1 | simprbi 495 | . . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing) |
3 | orngogrp 33040 | . . 3 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp) | |
4 | isogrp 32803 | . . . 4 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd)) | |
5 | 4 | simprbi 495 | . . 3 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd) |
6 | 2, 3, 5 | 3syl 18 | . 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oMnd) |
7 | omndtos 32806 | . 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 18415 Grpcgrp 18897 Fieldcfield 20632 oMndcomnd 32798 oGrpcogrp 32799 oRingcorng 33034 oFieldcofld 33035 |
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-omnd 32800 df-ogrp 32801 df-orng 33036 df-ofld 33037 |
This theorem is referenced by: ofldchr 33053 |
Copyright terms: Public domain | W3C validator |