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Theorem 0rngo 33956
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)
2 fvex 6239 . . . . . . 7 (GId‘𝐺) ∈ V
31, 2eqeltri 2726 . . . . . 6 𝑍 ∈ V
43snid 4241 . . . . 5 𝑍 ∈ {𝑍}
5 eleq1 2718 . . . . 5 (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
64, 5mpbii 223 . . . 4 (𝑍 = 𝑈𝑈 ∈ {𝑍})
7 0ring.1 . . . . . 6 𝐺 = (1st𝑅)
87, 10idl 33954 . . . . 5 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
9 0ring.2 . . . . . 6 𝐻 = (2nd𝑅)
10 0ring.3 . . . . . 6 𝑋 = ran 𝐺
11 0ring.5 . . . . . 6 𝑈 = (GId‘𝐻)
127, 9, 10, 111idl 33955 . . . . 5 ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
138, 12mpdan 703 . . . 4 (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
146, 13syl5ib 234 . . 3 (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
15 eqcom 2658 . . 3 ({𝑍} = 𝑋𝑋 = {𝑍})
1614, 15syl6ib 241 . 2 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
177rneqi 5384 . . . . 5 ran 𝐺 = ran (1st𝑅)
1810, 17eqtri 2673 . . . 4 𝑋 = ran (1st𝑅)
1918, 9, 11rngo1cl 33868 . . 3 (𝑅 ∈ RingOps → 𝑈𝑋)
20 eleq2 2719 . . . 4 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
21 elsni 4227 . . . . 5 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2221eqcomd 2657 . . . 4 (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
2320, 22syl6bi 243 . . 3 (𝑋 = {𝑍} → (𝑈𝑋𝑍 = 𝑈))
2419, 23syl5com 31 . 2 (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
2516, 24impbid 202 1 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  ran crn 5144  cfv 5926  1st c1st 7208  2nd c2nd 7209  GIdcgi 27472  RingOpscrngo 33823  Idlcidl 33936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-1st 7210  df-2nd 7211  df-grpo 27475  df-gid 27476  df-ginv 27477  df-ablo 27527  df-ass 33772  df-exid 33774  df-mgmOLD 33778  df-sgrOLD 33790  df-mndo 33796  df-rngo 33824  df-idl 33939
This theorem is referenced by:  smprngopr  33981  isfldidl2  33998
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