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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 29133 | . . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | |
2 | 1 | simprbi 479 | . . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing) |
3 | orngogrp 29132 | . . 3 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp) | |
4 | isogrp 29033 | . . . 4 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd)) | |
5 | 4 | simprbi 479 | . . 3 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd) |
6 | 2, 3, 5 | 3syl 18 | . 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oMnd) |
7 | omndtos 29036 | . 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 1977 Tosetctos 16856 Grpcgrp 17245 Fieldcfield 18571 oMndcomnd 29028 oGrpcogrp 29029 oRingcorng 29126 oFieldcofld 29127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-omnd 29030 df-ogrp 29031 df-orng 29128 df-ofld 29129 |
This theorem is referenced by: ofldchr 29145 |
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