Theorem 0rngo 38021

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	1	fvexi 6872	. . . . . 6 ⊢ 𝑍 ∈ V
3	2	snid 4626	. . . . 5 ⊢ 𝑍 ∈ {𝑍}
4		eleq1 2816	. . . . 5 ⊢ (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
5	3, 4	mpbii 233	. . . 4 ⊢ (𝑍 = 𝑈 → 𝑈 ∈ {𝑍})
6		0ring.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
7	6, 1	0idl 38019	. . . . 5 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
8		0ring.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
9		0ring.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
10		0ring.5	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
11	6, 8, 9, 10	1idl 38020	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
12	7, 11	mpdan 687	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
13	5, 12	imbitrid 244	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
14		eqcom 2736	. . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
15	13, 14	imbitrdi 251	. 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍}))
16	6	rneqi 5901	. . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅)
17	9, 16	eqtri 2752	. . . 4 ⊢ 𝑋 = ran (1st ‘𝑅)
18	17, 8, 10	rngo1cl 37933	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
19		eleq2 2817	. . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
20		elsni 4606	. . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
21	20	eqcomd 2735	. . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
22	19, 21	biimtrdi 253	. . 3 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑍 = 𝑈))
23	18, 22	syl5com 31	. 2 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
24	15, 23	impbid 212	1 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {csn 4589  ran crn 5639  ‘cfv 6511  1^st c1st 7966  2^nd c2nd 7967  GIdcgi 30419  RingOpscrngo 37888  Idlcidl 38001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-1st 7968  df-2nd 7969  df-grpo 30422  df-gid 30423  df-ginv 30424  df-ablo 30474  df-ass 37837  df-exid 37839  df-mgmOLD 37843  df-sgrOLD 37855  df-mndo 37861  df-rngo 37889  df-idl 38004

This theorem is referenced by:  smprngopr  38046  isfldidl2  38063