Theorem 0rngo 38362

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 6848	. . . . . 6 ⊢ 𝑍 ∈ V
3	2	snid 4607	. . . . 5 ⊢ 𝑍 ∈ {𝑍}
4		eleq1 2825	. . . . 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 38360	. . . . 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 38361	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
12	7, 11	mpdan 688	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
13	5, 12	imbitrid 244	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
14		eqcom 2744	. . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
15	13, 14	imbitrdi 251	. 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍}))
16	6	rneqi 5886	. . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅)
17	9, 16	eqtri 2760	. . . 4 ⊢ 𝑋 = ran (1st ‘𝑅)
18	17, 8, 10	rngo1cl 38274	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
19		eleq2 2826	. . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
20		elsni 4585	. . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
21	20	eqcomd 2743	. . . 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 1542   ∈ wcel 2114  {csn 4568  ran crn 5625  ‘cfv 6492  1^st c1st 7933  2^nd c2nd 7934  GIdcgi 30576  RingOpscrngo 38229  Idlcidl 38342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-1st 7935  df-2nd 7936  df-grpo 30579  df-gid 30580  df-ginv 30581  df-ablo 30631  df-ass 38178  df-exid 38180  df-mgmOLD 38184  df-sgrOLD 38196  df-mndo 38202  df-rngo 38230  df-idl 38345

This theorem is referenced by:  smprngopr  38387  isfldidl2  38404