Theorem 0rngo 36185

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 6788	. . . . . 6 ⊢ 𝑍 ∈ V
3	2	snid 4597	. . . . 5 ⊢ 𝑍 ∈ {𝑍}
4		eleq1 2826	. . . . 5 ⊢ (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
5	3, 4	mpbii 232	. . . 4 ⊢ (𝑍 = 𝑈 → 𝑈 ∈ {𝑍})
6		0ring.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
7	6, 1	0idl 36183	. . . . 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 36184	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
12	7, 11	mpdan 684	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
13	5, 12	syl5ib 243	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
14		eqcom 2745	. . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
15	13, 14	syl6ib 250	. 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍}))
16	6	rneqi 5846	. . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅)
17	9, 16	eqtri 2766	. . . 4 ⊢ 𝑋 = ran (1st ‘𝑅)
18	17, 8, 10	rngo1cl 36097	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
19		eleq2 2827	. . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
20		elsni 4578	. . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
21	20	eqcomd 2744	. . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
22	19, 21	syl6bi 252	. . 3 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑍 = 𝑈))
23	18, 22	syl5com 31	. 2 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
24	15, 23	impbid 211	1 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {csn 4561  ran crn 5590  ‘cfv 6433  1^st c1st 7829  2^nd c2nd 7830  GIdcgi 28852  RingOpscrngo 36052  Idlcidl 36165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-1st 7831  df-2nd 7832  df-grpo 28855  df-gid 28856  df-ginv 28857  df-ablo 28907  df-ass 36001  df-exid 36003  df-mgmOLD 36007  df-sgrOLD 36019  df-mndo 36025  df-rngo 36053  df-idl 36168

This theorem is referenced by:  smprngopr  36210  isfldidl2  36227