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Theorem 0idl 35171
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 2825 . . . 4 ran 𝐺 = ran 𝐺
3 0idl.2 . . . 4 𝑍 = (GId‘𝐺)
41, 2, 3rngo0cl 35065 . . 3 (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
54snssd 4740 . 2 (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
63fvexi 6680 . . . 4 𝑍 ∈ V
76snid 4597 . . 3 𝑍 ∈ {𝑍}
87a1i 11 . 2 (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
9 velsn 4579 . . . 4 (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
10 velsn 4579 . . . . . . . 8 (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
111, 2, 3rngo0rid 35066 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
124, 11mpdan 683 . . . . . . . . . 10 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
13 ovex 7184 . . . . . . . . . . 11 (𝑍𝐺𝑍) ∈ V
1413elsn 4578 . . . . . . . . . 10 ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
1512, 14sylibr 235 . . . . . . . . 9 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
16 oveq2 7159 . . . . . . . . . 10 (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
1716eleq1d 2901 . . . . . . . . 9 (𝑦 = 𝑍 → ((𝑍𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑍) ∈ {𝑍}))
1815, 17syl5ibrcom 248 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑦 = 𝑍 → (𝑍𝐺𝑦) ∈ {𝑍}))
1910, 18syl5bi 243 . . . . . . 7 (𝑅 ∈ RingOps → (𝑦 ∈ {𝑍} → (𝑍𝐺𝑦) ∈ {𝑍}))
2019ralrimiv 3185 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
21 eqid 2825 . . . . . . . . . 10 (2nd𝑅) = (2nd𝑅)
223, 2, 1, 21rngorz 35069 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) = 𝑍)
23 ovex 7184 . . . . . . . . . 10 (𝑧(2nd𝑅)𝑍) ∈ V
2423elsn 4578 . . . . . . . . 9 ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) = 𝑍)
2522, 24sylibr 235 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) ∈ {𝑍})
263, 2, 1, 21rngolz 35068 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) = 𝑍)
27 ovex 7184 . . . . . . . . . 10 (𝑍(2nd𝑅)𝑧) ∈ V
2827elsn 4578 . . . . . . . . 9 ((𝑍(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) = 𝑍)
2926, 28sylibr 235 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) ∈ {𝑍})
3025, 29jca 512 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3130ralrimiva 3186 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3220, 31jca 512 . . . . 5 (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
33 oveq1 7158 . . . . . . . 8 (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
3433eleq1d 2901 . . . . . . 7 (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
3534ralbidv 3201 . . . . . 6 (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
36 oveq2 7159 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑧(2nd𝑅)𝑥) = (𝑧(2nd𝑅)𝑍))
3736eleq1d 2901 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) ∈ {𝑍}))
38 oveq1 7158 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑥(2nd𝑅)𝑧) = (𝑍(2nd𝑅)𝑧))
3938eleq1d 2901 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑥(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
4037, 39anbi12d 630 . . . . . . 7 (𝑥 = 𝑍 → (((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}) ↔ ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
4140ralbidv 3201 . . . . . 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})))
4235, 41anbi12d 630 . . . . 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}))))
4332, 42syl5ibrcom 248 . . . 4 (𝑅 ∈ RingOps → (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
449, 43syl5bi 243 . . 3 (𝑅 ∈ RingOps → (𝑥 ∈ {𝑍} → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
4544ralrimiv 3185 . 2 (𝑅 ∈ RingOps → ∀𝑥 ∈ {𝑍} (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍})))
461, 21, 2, 3isidl 35160 . 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})))))
475, 8, 45, 46mpbir3and 1336 1 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3142  wss 3939  {csn 4563  ran crn 5554  cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682  GIdcgi 28181  RingOpscrngo 35040  Idlcidl 35153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-1st 7683  df-2nd 7684  df-grpo 28184  df-gid 28185  df-ginv 28186  df-ablo 28236  df-rngo 35041  df-idl 35156
This theorem is referenced by:  0rngo  35173  divrngidl  35174  smprngopr  35198  isdmn3  35220
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