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Theorem 0idl 36487
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 2737 . . . 4 ran 𝐺 = ran 𝐺
3 0idl.2 . . . 4 𝑍 = (GId‘𝐺)
41, 2, 3rngo0cl 36381 . . 3 (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
54snssd 4770 . 2 (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
63fvexi 6857 . . . 4 𝑍 ∈ V
76snid 4623 . . 3 𝑍 ∈ {𝑍}
87a1i 11 . 2 (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
9 velsn 4603 . . . 4 (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
10 velsn 4603 . . . . . . . 8 (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
111, 2, 3rngo0rid 36382 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
124, 11mpdan 686 . . . . . . . . . 10 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
13 ovex 7391 . . . . . . . . . . 11 (𝑍𝐺𝑍) ∈ V
1413elsn 4602 . . . . . . . . . 10 ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
1512, 14sylibr 233 . . . . . . . . 9 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
16 oveq2 7366 . . . . . . . . . 10 (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
1716eleq1d 2823 . . . . . . . . 9 (𝑦 = 𝑍 → ((𝑍𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑍) ∈ {𝑍}))
1815, 17syl5ibrcom 247 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑦 = 𝑍 → (𝑍𝐺𝑦) ∈ {𝑍}))
1910, 18biimtrid 241 . . . . . . 7 (𝑅 ∈ RingOps → (𝑦 ∈ {𝑍} → (𝑍𝐺𝑦) ∈ {𝑍}))
2019ralrimiv 3143 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
21 eqid 2737 . . . . . . . . . 10 (2nd𝑅) = (2nd𝑅)
223, 2, 1, 21rngorz 36385 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) = 𝑍)
23 ovex 7391 . . . . . . . . . 10 (𝑧(2nd𝑅)𝑍) ∈ V
2423elsn 4602 . . . . . . . . 9 ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) = 𝑍)
2522, 24sylibr 233 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) ∈ {𝑍})
263, 2, 1, 21rngolz 36384 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) = 𝑍)
27 ovex 7391 . . . . . . . . . 10 (𝑍(2nd𝑅)𝑧) ∈ V
2827elsn 4602 . . . . . . . . 9 ((𝑍(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) = 𝑍)
2926, 28sylibr 233 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) ∈ {𝑍})
3025, 29jca 513 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3130ralrimiva 3144 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3220, 31jca 513 . . . . 5 (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
33 oveq1 7365 . . . . . . . 8 (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
3433eleq1d 2823 . . . . . . 7 (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
3534ralbidv 3175 . . . . . 6 (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
36 oveq2 7366 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑧(2nd𝑅)𝑥) = (𝑧(2nd𝑅)𝑍))
3736eleq1d 2823 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) ∈ {𝑍}))
38 oveq1 7365 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑥(2nd𝑅)𝑧) = (𝑍(2nd𝑅)𝑧))
3938eleq1d 2823 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑥(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
4037, 39anbi12d 632 . . . . . . 7 (𝑥 = 𝑍 → (((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}) ↔ ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
4140ralbidv 3175 . . . . . 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 632 . . . . 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 247 . . . 4 (𝑅 ∈ RingOps → (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
449, 43biimtrid 241 . . 3 (𝑅 ∈ RingOps → (𝑥 ∈ {𝑍} → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
4544ralrimiv 3143 . 2 (𝑅 ∈ RingOps → ∀𝑥 ∈ {𝑍} (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍})))
461, 21, 2, 3isidl 36476 . 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 1343 1 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  wss 3911  {csn 4587  ran crn 5635  cfv 6497  (class class class)co 7358  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-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-rngo 36357  df-idl 36472
This theorem is referenced by:  0rngo  36489  divrngidl  36490  smprngopr  36514  isdmn3  36536
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