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Theorem 0rngo 36489
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 6857 . . . . . 6 𝑍 ∈ V
32snid 4623 . . . . 5 𝑍 ∈ {𝑍}
4 eleq1 2826 . . . . 5 (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
53, 4mpbii 232 . . . 4 (𝑍 = 𝑈𝑈 ∈ {𝑍})
6 0ring.1 . . . . . 6 𝐺 = (1st𝑅)
76, 10idl 36487 . . . . 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 36488 . . . . 5 ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
127, 11mpdan 686 . . . 4 (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
135, 12imbitrid 243 . . 3 (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
14 eqcom 2744 . . 3 ({𝑍} = 𝑋𝑋 = {𝑍})
1513, 14syl6ib 251 . 2 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
166rneqi 5893 . . . . 5 ran 𝐺 = ran (1st𝑅)
179, 16eqtri 2765 . . . 4 𝑋 = ran (1st𝑅)
1817, 8, 10rngo1cl 36401 . . 3 (𝑅 ∈ RingOps → 𝑈𝑋)
19 eleq2 2827 . . . 4 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
20 elsni 4604 . . . . 5 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2120eqcomd 2743 . . . 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 1542  wcel 2107  {csn 4587  ran crn 5635  cfv 6497  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  RingOpscrngo 36356  Idlcidl 36469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-1st 7922  df-2nd 7923  df-grpo 29438  df-gid 29439  df-ginv 29440  df-ablo 29490  df-ass 36305  df-exid 36307  df-mgmOLD 36311  df-sgrOLD 36323  df-mndo 36329  df-rngo 36357  df-idl 36472
This theorem is referenced by:  smprngopr  36514  isfldidl2  36531
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