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Theorem 0rngo 37500
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 6911 . . . . . 6 𝑍 ∈ V
32snid 4665 . . . . 5 𝑍 ∈ {𝑍}
4 eleq1 2817 . . . . 5 (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
53, 4mpbii 232 . . . 4 (𝑍 = 𝑈𝑈 ∈ {𝑍})
6 0ring.1 . . . . . 6 𝐺 = (1st𝑅)
76, 10idl 37498 . . . . 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 37499 . . . . 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 2735 . . 3 ({𝑍} = 𝑋𝑋 = {𝑍})
1513, 14imbitrdi 250 . 2 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
166rneqi 5939 . . . . 5 ran 𝐺 = ran (1st𝑅)
179, 16eqtri 2756 . . . 4 𝑋 = ran (1st𝑅)
1817, 8, 10rngo1cl 37412 . . 3 (𝑅 ∈ RingOps → 𝑈𝑋)
19 eleq2 2818 . . . 4 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
20 elsni 4646 . . . . 5 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2120eqcomd 2734 . . . 4 (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
2219, 21biimtrdi 252 . . 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 1534  wcel 2099  {csn 4629  ran crn 5679  cfv 6548  1st c1st 7991  2nd c2nd 7992  GIdcgi 30299  RingOpscrngo 37367  Idlcidl 37480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-1st 7993  df-2nd 7994  df-grpo 30302  df-gid 30303  df-ginv 30304  df-ablo 30354  df-ass 37316  df-exid 37318  df-mgmOLD 37322  df-sgrOLD 37334  df-mndo 37340  df-rngo 37368  df-idl 37483
This theorem is referenced by:  smprngopr  37525  isfldidl2  37542
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