Theorem 0rngo 32996

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 6113	. . . . . . 7 ⊢ (GId‘𝐺) ∈ V
3	1, 2	eqeltri 2684	. . . . . 6 ⊢ 𝑍 ∈ V
4	3	snid 4155	. . . . 5 ⊢ 𝑍 ∈ {𝑍}
5		eleq1 2676	. . . . 5 ⊢ (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
6	4, 5	mpbii 222	. . . 4 ⊢ (𝑍 = 𝑈 → 𝑈 ∈ {𝑍})
7		0ring.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
8	7, 1	0idl 32994	. . . . 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 32995	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
13	8, 12	mpdan 699	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
14	6, 13	syl5ib 233	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
15		eqcom 2617	. . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
16	14, 15	syl6ib 240	. 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍}))
17	7	rneqi 5273	. . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅)
18	10, 17	eqtri 2632	. . . 4 ⊢ 𝑋 = ran (1st ‘𝑅)
19	18, 9, 11	rngo1cl 32908	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
20		eleq2 2677	. . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
21		elsni 4142	. . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
22	21	eqcomd 2616	. . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
23	20, 22	syl6bi 242	. . 3 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑍 = 𝑈))
24	19, 23	syl5com 31	. 2 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
25	16, 24	impbid 201	1 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ran crn 5039  ‘cfv 5804  1^st c1st 7057  2^nd c2nd 7058  GIdcgi 26728  RingOpscrngo 32863  Idlcidl 32976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-1st 7059  df-2nd 7060  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-ass 32812  df-exid 32814  df-mgmOLD 32818  df-sgrOLD 32830  df-mndo 32836  df-rngo 32864  df-idl 32979

This theorem is referenced by:  smprngopr  33021  isfldidl2  33038