Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 30873 | . 2 ⊢ oField = (Field ∩ oRing) | |
2 | 1 | elin2 4176 | 1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Fieldcfield 19505 oRingcorng 30870 oFieldcofld 30871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-in 3945 df-ofld 30873 |
This theorem is referenced by: ofldfld 30885 ofldtos 30886 ofldlt1 30888 ofldchr 30889 subofld 30891 isarchiofld 30892 reofld 30915 nn0omnd 30916 |
Copyright terms: Public domain | W3C validator |