Theorem isfieldidl

Description: Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010) (Revised by AV, 30-Jun-2026)

	Ref	Expression
Hypotheses	isfieldidl.b	$$⊢B={\mathrm{Base}}_R$$
	isfieldidl.0	$$⊢\underset{\dot{}}{0}=0_R$$
	isfieldidl.i	$$⊢I=\mathrm{LIdeal}\left(R\right)$$
	isfieldidl.1	$$⊢\underset{\dot{}}{1}=1_R$$
Assertion	isfieldidl	$$⊢R\in \mathrm{Field}\leftrightarrow \left(R\in \mathrm{CRing}\wedge \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		isfieldidl.b	$$⊢B={\mathrm{Base}}_R$$
2		isfieldidl.0	$$⊢\underset{\dot{}}{0}=0_R$$
3		isfieldidl.i	$$⊢I=\mathrm{LIdeal}\left(R\right)$$
4		isfieldidl.1	$$⊢\underset{\dot{}}{1}=1_R$$
5		isfld	$$⊢R\in \mathrm{Field}\leftrightarrow \left(R\in \mathrm{DivRing}\wedge R\in \mathrm{CRing}\right)$$
6	1 2 3	drngidl	$$⊢R\in \mathrm{NzRing}\to \left(R\in \mathrm{DivRing}\leftrightarrow I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$
7	6	adantr	$$⊢\left(R\in \mathrm{NzRing}\wedge R\in \mathrm{CRing}\right)\to \left(R\in \mathrm{DivRing}\leftrightarrow I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$
8	4 2	nzrnz	$$⊢R\in \mathrm{NzRing}\to \underset{\dot{}}{1}\ne \underset{\dot{}}{0}$$
9	8	necomd	$$⊢R\in \mathrm{NzRing}\to \underset{\dot{}}{0}\ne \underset{\dot{}}{1}$$
10	9	a1d	$$⊢R\in \mathrm{NzRing}\to \left(I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\to \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\right)$$
11	10	adantr	$$⊢\left(R\in \mathrm{NzRing}\wedge R\in \mathrm{CRing}\right)\to \left(I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\to \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\right)$$
12	11	pm4.71rd	$$⊢\left(R\in \mathrm{NzRing}\wedge R\in \mathrm{CRing}\right)\to \left(I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\leftrightarrow \left(\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
13	7 12	bitrd	$$⊢\left(R\in \mathrm{NzRing}\wedge R\in \mathrm{CRing}\right)\to \left(R\in \mathrm{DivRing}\leftrightarrow \left(\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
14		drngnzr	$$⊢R\in \mathrm{DivRing}\to R\in \mathrm{NzRing}$$
15	14	con3i	$$⊢\neg R\in \mathrm{NzRing}\to \neg R\in \mathrm{DivRing}$$
16	15	adantr	$$⊢\left(\neg R\in \mathrm{NzRing}\wedge R\in \mathrm{CRing}\right)\to \neg R\in \mathrm{DivRing}$$
17		ianor	$$⊢\neg \left(R\in \mathrm{Ring}\wedge \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\right)\leftrightarrow \left(\neg R\in \mathrm{Ring}\vee \neg \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\right)$$
18	4 2	isnzr	$$⊢R\in \mathrm{NzRing}\leftrightarrow \left(R\in \mathrm{Ring}\wedge \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\right)$$
19	17 18	xchnxbir	$$⊢\neg R\in \mathrm{NzRing}\leftrightarrow \left(\neg R\in \mathrm{Ring}\vee \neg \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\right)$$
20		pm2.24	$$⊢R\in \mathrm{Ring}\to \left(\neg R\in \mathrm{Ring}\to \left(\neg \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\vee \neg I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
21		crngring	$$⊢R\in \mathrm{CRing}\to R\in \mathrm{Ring}$$
22	20 21	syl11	$$⊢\neg R\in \mathrm{Ring}\to \left(R\in \mathrm{CRing}\to \left(\neg \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\vee \neg I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
23		id	$$⊢\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\to \underset{\dot{}}{0}\ne \underset{\dot{}}{1}$$
24	23	necomd	$$⊢\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\to \underset{\dot{}}{1}\ne \underset{\dot{}}{0}$$
25	24	con3i	$$⊢\neg \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\to \neg \underset{\dot{}}{0}\ne \underset{\dot{}}{1}$$
26	25	orcd	$$⊢\neg \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\to \left(\neg \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\vee \neg I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$
27	26	a1d	$$⊢\neg \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\to \left(R\in \mathrm{CRing}\to \left(\neg \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\vee \neg I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
28	22 27	jaoi	$$⊢\left(\neg R\in \mathrm{Ring}\vee \neg \underset{\dot{}}{1}\ne \underset{\dot{}}{0}\right)\to \left(R\in \mathrm{CRing}\to \left(\neg \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\vee \neg I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
29	19 28	sylbi	$$⊢\neg R\in \mathrm{NzRing}\to \left(R\in \mathrm{CRing}\to \left(\neg \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\vee \neg I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
30	29	imp	$$⊢\left(\neg R\in \mathrm{NzRing}\wedge R\in \mathrm{CRing}\right)\to \left(\neg \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\vee \neg I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$
31		ianor	$$⊢\neg \left(\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\leftrightarrow \left(\neg \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\vee \neg I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$
32	30 31	sylibr	$$⊢\left(\neg R\in \mathrm{NzRing}\wedge R\in \mathrm{CRing}\right)\to \neg \left(\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$
33	16 32	2falsed	$$⊢\left(\neg R\in \mathrm{NzRing}\wedge R\in \mathrm{CRing}\right)\to \left(R\in \mathrm{DivRing}\leftrightarrow \left(\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
34	13 33	pm2.61ian	$$⊢R\in \mathrm{CRing}\to \left(R\in \mathrm{DivRing}\leftrightarrow \left(\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
35	34	pm5.32i	$$⊢\left(R\in \mathrm{CRing}\wedge R\in \mathrm{DivRing}\right)\leftrightarrow \left(R\in \mathrm{CRing}\wedge \left(\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
36		ancom	$$⊢\left(R\in \mathrm{DivRing}\wedge R\in \mathrm{CRing}\right)\leftrightarrow \left(R\in \mathrm{CRing}\wedge R\in \mathrm{DivRing}\right)$$
37		3anass	$$⊢\left(R\in \mathrm{CRing}\wedge \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\leftrightarrow \left(R\in \mathrm{CRing}\wedge \left(\underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\right)$$
38	35 36 37	3bitr4i	$$⊢\left(R\in \mathrm{DivRing}\wedge R\in \mathrm{CRing}\right)\leftrightarrow \left(R\in \mathrm{CRing}\wedge \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$
39	5 38	bitri	$$⊢R\in \mathrm{Field}\leftrightarrow \left(R\in \mathrm{CRing}\wedge \underset{\dot{}}{0}\ne \underset{\dot{}}{1}\wedge I=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$