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Theorem 0rngo 37398
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 = (1st𝑅)
0ring.2 𝐻 = (2nd𝑅)
0ring.3 𝑋 = ran 𝐺
0ring.4 𝑍 = (GId‘𝐺)
0ring.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
0rngo (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))

Proof of Theorem 0rngo
StepHypRef Expression
1 0ring.4 . . . . . . 7 𝑍 = (GId‘𝐺)
21fvexi 6896 . . . . . 6 𝑍 ∈ V
32snid 4657 . . . . 5 𝑍 ∈ {𝑍}
4 eleq1 2813 . . . . 5 (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
53, 4mpbii 232 . . . 4 (𝑍 = 𝑈𝑈 ∈ {𝑍})
6 0ring.1 . . . . . 6 𝐺 = (1st𝑅)
76, 10idl 37396 . . . . 5 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
8 0ring.2 . . . . . 6 𝐻 = (2nd𝑅)
9 0ring.3 . . . . . 6 𝑋 = ran 𝐺
10 0ring.5 . . . . . 6 𝑈 = (GId‘𝐻)
116, 8, 9, 101idl 37397 . . . . 5 ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
127, 11mpdan 684 . . . 4 (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
135, 12imbitrid 243 . . 3 (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
14 eqcom 2731 . . 3 ({𝑍} = 𝑋𝑋 = {𝑍})
1513, 14imbitrdi 250 . 2 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
166rneqi 5927 . . . . 5 ran 𝐺 = ran (1st𝑅)
179, 16eqtri 2752 . . . 4 𝑋 = ran (1st𝑅)
1817, 8, 10rngo1cl 37310 . . 3 (𝑅 ∈ RingOps → 𝑈𝑋)
19 eleq2 2814 . . . 4 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
20 elsni 4638 . . . . 5 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2120eqcomd 2730 . . . 4 (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
2219, 21syl6bi 253 . . 3 (𝑋 = {𝑍} → (𝑈𝑋𝑍 = 𝑈))
2318, 22syl5com 31 . 2 (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
2415, 23impbid 211 1 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {csn 4621  ran crn 5668  cfv 6534  1st c1st 7967  2nd c2nd 7968  GIdcgi 30237  RingOpscrngo 37265  Idlcidl 37378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-1st 7969  df-2nd 7970  df-grpo 30240  df-gid 30241  df-ginv 30242  df-ablo 30292  df-ass 37214  df-exid 37216  df-mgmOLD 37220  df-sgrOLD 37232  df-mndo 37238  df-rngo 37266  df-idl 37381
This theorem is referenced by:  smprngopr  37423  isfldidl2  37440
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