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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 = (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 6113 . . . . . . 7 (GId‘𝐺) ∈ V
31, 2eqeltri 2684 . . . . . 6 𝑍 ∈ V
43snid 4155 . . . . 5 𝑍 ∈ {𝑍}
5 eleq1 2676 . . . . 5 (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
64, 5mpbii 222 . . . 4 (𝑍 = 𝑈𝑈 ∈ {𝑍})
7 0ring.1 . . . . . 6 𝐺 = (1st𝑅)
87, 10idl 32994 . . . . 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 32995 . . . . 5 ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
138, 12mpdan 699 . . . 4 (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
146, 13syl5ib 233 . . 3 (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
15 eqcom 2617 . . 3 ({𝑍} = 𝑋𝑋 = {𝑍})
1614, 15syl6ib 240 . 2 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
177rneqi 5273 . . . . 5 ran 𝐺 = ran (1st𝑅)
1810, 17eqtri 2632 . . . 4 𝑋 = ran (1st𝑅)
1918, 9, 11rngo1cl 32908 . . 3 (𝑅 ∈ RingOps → 𝑈𝑋)
20 eleq2 2677 . . . 4 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
21 elsni 4142 . . . . 5 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2221eqcomd 2616 . . . 4 (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
2320, 22syl6bi 242 . . 3 (𝑋 = {𝑍} → (𝑈𝑋𝑍 = 𝑈))
2419, 23syl5com 31 . 2 (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
2516, 24impbid 201 1 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ran crn 5039  ‘cfv 5804  1st c1st 7057  2nd 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
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