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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 19436 | . 2 ⊢ Field = (DivRing ∩ CRing) | |
2 | 1 | elin2 4173 | 1 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∈ wcel 2105 CRingccrg 19229 DivRingcdr 19433 Fieldcfield 19434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-in 3942 df-field 19436 |
This theorem is referenced by: fldpropd 19461 primefld 19515 rng1nfld 19981 fldidom 20008 fiidomfld 20011 refld 20693 recrng 20695 frlmphllem 20854 frlmphl 20855 rrxcph 23924 rrx0 23929 ply1pid 24702 lgseisenlem3 25881 lgseisenlem4 25882 ofldlt1 30814 ofldchr 30815 subofld 30817 isarchiofld 30818 reofld 30841 rearchi 30843 srafldlvec 30891 rgmoddim 30908 ccfldextrr 30938 fldextsralvec 30945 extdgcl 30946 extdggt0 30947 fldextid 30949 extdgmul 30951 extdg1id 30953 ccfldsrarelvec 30956 qqhrhm 31130 matunitlindflem1 34770 matunitlindflem2 34771 matunitlindf 34772 fldcat 44251 fldcatALTV 44269 |
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