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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 30550 | . 2 ⊢ oField = (Field ∩ oRing) | |
2 | 1 | elin2 4064 | 1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∈ wcel 2050 Fieldcfield 19229 oRingcorng 30547 oFieldcofld 30548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-v 3417 df-in 3838 df-ofld 30550 |
This theorem is referenced by: ofldfld 30562 ofldtos 30563 ofldlt1 30565 ofldchr 30566 subofld 30568 isarchiofld 30569 reofld 30592 nn0omnd 30593 |
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