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Theorem 0idl 32994
 Description: The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
0idl.1 𝐺 = (1st𝑅)
0idl.2 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
0idl (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))

Proof of Theorem 0idl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0idl.1 . . . 4 𝐺 = (1st𝑅)
2 eqid 2610 . . . 4 ran 𝐺 = ran 𝐺
3 0idl.2 . . . 4 𝑍 = (GId‘𝐺)
41, 2, 3rngo0cl 32888 . . 3 (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
54snssd 4281 . 2 (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
6 fvex 6113 . . . . 5 (GId‘𝐺) ∈ V
73, 6eqeltri 2684 . . . 4 𝑍 ∈ V
87snid 4155 . . 3 𝑍 ∈ {𝑍}
98a1i 11 . 2 (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
10 velsn 4141 . . . 4 (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
11 velsn 4141 . . . . . . . 8 (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
121, 2, 3rngo0rid 32889 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
134, 12mpdan 699 . . . . . . . . . 10 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
14 ovex 6577 . . . . . . . . . . 11 (𝑍𝐺𝑍) ∈ V
1514elsn 4140 . . . . . . . . . 10 ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
1613, 15sylibr 223 . . . . . . . . 9 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
17 oveq2 6557 . . . . . . . . . 10 (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
1817eleq1d 2672 . . . . . . . . 9 (𝑦 = 𝑍 → ((𝑍𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑍) ∈ {𝑍}))
1916, 18syl5ibrcom 236 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑦 = 𝑍 → (𝑍𝐺𝑦) ∈ {𝑍}))
2011, 19syl5bi 231 . . . . . . 7 (𝑅 ∈ RingOps → (𝑦 ∈ {𝑍} → (𝑍𝐺𝑦) ∈ {𝑍}))
2120ralrimiv 2948 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
22 eqid 2610 . . . . . . . . . 10 (2nd𝑅) = (2nd𝑅)
233, 2, 1, 22rngorz 32892 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) = 𝑍)
24 ovex 6577 . . . . . . . . . 10 (𝑧(2nd𝑅)𝑍) ∈ V
2524elsn 4140 . . . . . . . . 9 ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) = 𝑍)
2623, 25sylibr 223 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) ∈ {𝑍})
273, 2, 1, 22rngolz 32891 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) = 𝑍)
28 ovex 6577 . . . . . . . . . 10 (𝑍(2nd𝑅)𝑧) ∈ V
2928elsn 4140 . . . . . . . . 9 ((𝑍(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) = 𝑍)
3027, 29sylibr 223 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) ∈ {𝑍})
3126, 30jca 553 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3231ralrimiva 2949 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3321, 32jca 553 . . . . 5 (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
34 oveq1 6556 . . . . . . . 8 (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
3534eleq1d 2672 . . . . . . 7 (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
3635ralbidv 2969 . . . . . 6 (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
37 oveq2 6557 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑧(2nd𝑅)𝑥) = (𝑧(2nd𝑅)𝑍))
3837eleq1d 2672 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) ∈ {𝑍}))
39 oveq1 6556 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑥(2nd𝑅)𝑧) = (𝑍(2nd𝑅)𝑧))
4039eleq1d 2672 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑥(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
4138, 40anbi12d 743 . . . . . . 7 (𝑥 = 𝑍 → (((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}) ↔ ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
4241ralbidv 2969 . . . . . 6 (𝑥 = 𝑍 → (∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}) ↔ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
4336, 42anbi12d 743 . . . . 5 (𝑥 = 𝑍 → ((∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍})) ↔ (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))))
4433, 43syl5ibrcom 236 . . . 4 (𝑅 ∈ RingOps → (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
4510, 44syl5bi 231 . . 3 (𝑅 ∈ RingOps → (𝑥 ∈ {𝑍} → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
4645ralrimiv 2948 . 2 (𝑅 ∈ RingOps → ∀𝑥 ∈ {𝑍} (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍})))
471, 22, 2, 3isidl 32983 . 2 (𝑅 ∈ RingOps → ({𝑍} ∈ (Idl‘𝑅) ↔ ({𝑍} ⊆ ran 𝐺𝑍 ∈ {𝑍} ∧ ∀𝑥 ∈ {𝑍} (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍})))))
485, 9, 46, 47mpbir3and 1238 1 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  {csn 4125  ran crn 5039  ‘cfv 5804  (class class class)co 6549  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-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-rngo 32864  df-idl 32979 This theorem is referenced by:  0rngo  32996  divrngidl  32997  smprngopr  33021  isdmn3  33043
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