Theorem 0rngo 33826

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 6201	. . . . . . 7 ⊢ (GId‘𝐺) ∈ V
3	1, 2	eqeltri 2697	. . . . . 6 ⊢ 𝑍 ∈ V
4	3	snid 4208	. . . . 5 ⊢ 𝑍 ∈ {𝑍}
5		eleq1 2689	. . . . 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 33824	. . . . 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 33825	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
13	8, 12	mpdan 702	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
14	6, 13	syl5ib 234	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
15		eqcom 2629	. . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
16	14, 15	syl6ib 241	. 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍}))
17	7	rneqi 5352	. . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅)
18	10, 17	eqtri 2644	. . . 4 ⊢ 𝑋 = ran (1st ‘𝑅)
19	18, 9, 11	rngo1cl 33738	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
20		eleq2 2690	. . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍}))
21		elsni 4194	. . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
22	21	eqcomd 2628	. . . 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 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ran crn 5115  ‘cfv 5888  1^st c1st 7166  2^nd c2nd 7167  GIdcgi 27344  RingOpscrngo 33693  Idlcidl 33806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-idl 33809

This theorem is referenced by:  smprngopr  33851  isfldidl2  33868