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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isofld | Structured version Visualization version GIF version |
Description: An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
Ref | Expression |
---|---|
isofld | ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ofld 33176 | . 2 ⊢ oField = (Field ∩ oRing) | |
2 | 1 | elin2 4198 | 1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2099 Fieldcfield 20708 oRingcorng 33173 oFieldcofld 33174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-in 3954 df-ofld 33176 |
This theorem is referenced by: ofldfld 33188 ofldtos 33189 ofldlt1 33191 ofldchr 33192 subofld 33194 isarchiofld 33195 reofld 33219 nn0omnd 33220 |
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