Theorem 0rngo 33956

Description: In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.)

Hypotheses

Ref	Expression
0ring.1	⊢ 𝐺 = (1st ‘𝑅)
0ring.2	⊢ 𝐻 = (2nd ‘𝑅)
0ring.3	⊢ 𝑋 = ran 𝐺
0ring.4	⊢ 𝑍 = (GId‘𝐺)
0ring.5	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
0rngo	⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))

Proof of Theorem 0rngo

Step	Hyp	Ref	Expression
1		0ring.4	. . . . . . 7 ⊢ 𝑍 = (GId‘𝐺)
2		fvex 6239	. . . . . . 7 ⊢ (GId‘𝐺) ∈ V
3	1, 2	eqeltri 2726	. . . . . 6 ⊢ 𝑍 ∈ V
4	3	snid 4241	. . . . 5 ⊢ 𝑍 ∈ {𝑍}
5		eleq1 2718	. . . . 5 ⊢ (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
6	4, 5	mpbii 223	. . . 4 ⊢ (𝑍 = 𝑈 → 𝑈 ∈ {𝑍})
7		0ring.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
8	7, 1	0idl 33954	. . . . 5 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
9		0ring.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
10		0ring.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
11		0ring.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
12	7, 9, 10, 11	1idl 33955	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
13	8, 12	mpdan 703	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
14	6, 13	syl5ib 234	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
15		eqcom 2658	. . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
16	14, 15	syl6ib 241	. 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍}))
17	7	rneqi 5384	. . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅)
18	10, 17	eqtri 2673	. . . 4 ⊢ 𝑋 = ran (1st ‘𝑅)
19	18, 9, 11	rngo1cl 33868	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
20		eleq2 2719	. . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
21		elsni 4227	. . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
22	21	eqcomd 2657	. . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
23	20, 22	syl6bi 243	. . 3 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑍 = 𝑈))
24	19, 23	syl5com 31	. 2 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
25	16, 24	impbid 202	1 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523   ∈ wcel 2030  Vcvv 3231  {csn 4210  ran crn 5144  ‘cfv 5926  1^st c1st 7208  2^nd c2nd 7209  GIdcgi 27472  RingOpscrngo 33823  Idlcidl 33936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-1st 7210  df-2nd 7211  df-grpo 27475  df-gid 27476  df-ginv 27477  df-ablo 27527  df-ass 33772  df-exid 33774  df-mgmOLD 33778  df-sgrOLD 33790  df-mndo 33796  df-rngo 33824  df-idl 33939

This theorem is referenced by:  smprngopr  33981  isfldidl2  33998