Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0rngo Structured version   Visualization version   GIF version

Theorem 0rngo 38014
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 6921 . . . . . 6 𝑍 ∈ V
32snid 4667 . . . . 5 𝑍 ∈ {𝑍}
4 eleq1 2827 . . . . 5 (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
53, 4mpbii 233 . . . 4 (𝑍 = 𝑈𝑈 ∈ {𝑍})
6 0ring.1 . . . . . 6 𝐺 = (1st𝑅)
76, 10idl 38012 . . . . 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 38013 . . . . 5 ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
127, 11mpdan 687 . . . 4 (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋))
135, 12imbitrid 244 . . 3 (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋))
14 eqcom 2742 . . 3 ({𝑍} = 𝑋𝑋 = {𝑍})
1513, 14imbitrdi 251 . 2 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
166rneqi 5951 . . . . 5 ran 𝐺 = ran (1st𝑅)
179, 16eqtri 2763 . . . 4 𝑋 = ran (1st𝑅)
1817, 8, 10rngo1cl 37926 . . 3 (𝑅 ∈ RingOps → 𝑈𝑋)
19 eleq2 2828 . . . 4 (𝑋 = {𝑍} → (𝑈𝑋𝑈 ∈ {𝑍}))
20 elsni 4648 . . . . 5 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2120eqcomd 2741 . . . 4 (𝑈 ∈ {𝑍} → 𝑍 = 𝑈)
2219, 21biimtrdi 253 . . 3 (𝑋 = {𝑍} → (𝑈𝑋𝑍 = 𝑈))
2318, 22syl5com 31 . 2 (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈))
2415, 23impbid 212 1 (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  {csn 4631  ran crn 5690  cfv 6563  1st c1st 8011  2nd c2nd 8012  GIdcgi 30519  RingOpscrngo 37881  Idlcidl 37994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-1st 8013  df-2nd 8014  df-grpo 30522  df-gid 30523  df-ginv 30524  df-ablo 30574  df-ass 37830  df-exid 37832  df-mgmOLD 37836  df-sgrOLD 37848  df-mndo 37854  df-rngo 37882  df-idl 37997
This theorem is referenced by:  smprngopr  38039  isfldidl2  38056
  Copyright terms: Public domain W3C validator