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Mirrors > Home > MPE Home > Th. List > isfld | Structured version Visualization version GIF version |
Description: A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.) |
Ref | Expression |
---|---|
isfld | ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-field 18573 | . 2 ⊢ Field = (DivRing ∩ CRing) | |
2 | 1 | elin2 3763 | 1 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 CRingccrg 18371 DivRingcdr 18570 Fieldcfield 18571 |
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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-field 18573 |
This theorem is referenced by: fldpropd 18598 rng1nfld 19099 fldidom 19126 fiidomfld 19129 refld 19784 recrng 19786 frlmphllem 19938 frlmphl 19939 rrxcph 22988 ply1pid 23743 lgseisenlem3 24902 lgseisenlem4 24903 ofldlt1 29144 ofldchr 29145 subofld 29147 isarchiofld 29148 reofld 29171 rearchi 29173 qqhrhm 29361 matunitlindflem1 32575 matunitlindflem2 32576 matunitlindf 32577 fldcat 41874 fldcatALTV 41893 |
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