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Theorem 0rngo 33479
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 6160 . . . . . . 7 (GId‘𝐺) ∈ V
31, 2eqeltri 2694 . . . . . 6 𝑍 ∈ V
43snid 4181 . . . . 5 𝑍 ∈ {𝑍}
5 eleq1 2686 . . . . 5 (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
64, 5mpbii 223 . . . 4 (𝑍 = 𝑈𝑈 ∈ {𝑍})
7 0ring.1 . . . . . 6 𝐺 = (1st𝑅)
87, 10idl 33477 . . . . 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 33478 . . . . 5 ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
138, 12mpdan 701 . . . 4 (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
146, 13syl5ib 234 . . 3 (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
15 eqcom 2628 . . 3 ({𝑍} = 𝑋𝑋 = {𝑍})
1614, 15syl6ib 241 . 2 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
177rneqi 5314 . . . . 5 ran 𝐺 = ran (1st𝑅)
1810, 17eqtri 2643 . . . 4 𝑋 = ran (1st𝑅)
1918, 9, 11rngo1cl 33391 . . 3 (𝑅 ∈ RingOps → 𝑈𝑋)
20 eleq2 2687 . . . 4 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
21 elsni 4167 . . . . 5 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2221eqcomd 2627 . . . 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 1480  wcel 1987  Vcvv 3186  {csn 4150  ran crn 5077  cfv 5849  1st c1st 7114  2nd c2nd 7115  GIdcgi 27205  RingOpscrngo 33346  Idlcidl 33459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-1st 7116  df-2nd 7117  df-grpo 27208  df-gid 27209  df-ginv 27210  df-ablo 27260  df-ass 33295  df-exid 33297  df-mgmOLD 33301  df-sgrOLD 33313  df-mndo 33319  df-rngo 33347  df-idl 33462
This theorem is referenced by:  smprngopr  33504  isfldidl2  33521
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