isfld2 - Mathbox for Jeff Madsen

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Theorem isfld2 35282

Description: The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)

**Assertion**
Ref	Expression
isfld2	⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))

**Proof of Theorem isfld2**
Step	Hyp	Ref	Expression
1		flddivrng 35276	. . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
2		fldcrng 35281	. . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
3	1, 2	jca 514	. 2 ⊢ (𝐾 ∈ Fld → (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
4		iscrngo 35273	. . . 4 ⊢ (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
5	4	simprbi 499	. . 3 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ Com2)
6		elin 4168	. . . . 5 ⊢ (𝐾 ∈ (DivRingOps ∩ Com2) ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
7	6	biimpri 230	. . . 4 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ (DivRingOps ∩ Com2))
8		df-fld 35269	. . . 4 ⊢ Fld = (DivRingOps ∩ Com2)
9	7, 8	eleqtrrdi 2924	. . 3 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ Fld)
10	5, 9	sylan2 594	. 2 ⊢ ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → 𝐾 ∈ Fld)
11	3, 10	impbii 211	1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∩ cin 3934 RingOpscrngo 35171 DivRingOpscdrng 35225 Com2ccm2 35266 Fldcfld 35268 CRingOpsccring 35270

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-iota 6313 df-fun 6356 df-fv 6362 df-1st 7688 df-2nd 7689 df-drngo 35226 df-fld 35269 df-crngo 35271

This theorem is referenced by: flddmn 35335 isfldidl 35345