![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > isfld2 | Structured version Visualization version GIF version |
Description: The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
isfld2 | ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flddivrng 34752 | . . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps) | |
2 | fldcrng 34757 | . . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps) | |
3 | 1, 2 | jca 504 | . 2 ⊢ (𝐾 ∈ Fld → (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) |
4 | iscrngo 34749 | . . . 4 ⊢ (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2)) | |
5 | 4 | simprbi 489 | . . 3 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ Com2) |
6 | elin 4052 | . . . . 5 ⊢ (𝐾 ∈ (DivRingOps ∩ Com2) ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2)) | |
7 | 6 | biimpri 220 | . . . 4 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ (DivRingOps ∩ Com2)) |
8 | df-fld 34745 | . . . 4 ⊢ Fld = (DivRingOps ∩ Com2) | |
9 | 7, 8 | syl6eleqr 2872 | . . 3 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ Fld) |
10 | 5, 9 | sylan2 584 | . 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → 𝐾 ∈ Fld) |
11 | 3, 10 | impbii 201 | 1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∈ wcel 2051 ∩ cin 3823 RingOpscrngo 34647 DivRingOpscdrng 34701 Com2ccm2 34742 Fldcfld 34744 CRingOpsccring 34746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-iota 6150 df-fun 6188 df-fv 6194 df-1st 7500 df-2nd 7501 df-drngo 34702 df-fld 34745 df-crngo 34747 |
This theorem is referenced by: flddmn 34811 isfldidl 34821 |
Copyright terms: Public domain | W3C validator |