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Theorem 0idl 34311
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 2799 . . . 4 ran 𝐺 = ran 𝐺
3 0idl.2 . . . 4 𝑍 = (GId‘𝐺)
41, 2, 3rngo0cl 34205 . . 3 (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
54snssd 4528 . 2 (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
63fvexi 6425 . . . 4 𝑍 ∈ V
76snid 4400 . . 3 𝑍 ∈ {𝑍}
87a1i 11 . 2 (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
9 velsn 4384 . . . 4 (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
10 velsn 4384 . . . . . . . 8 (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
111, 2, 3rngo0rid 34206 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
124, 11mpdan 679 . . . . . . . . . 10 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
13 ovex 6910 . . . . . . . . . . 11 (𝑍𝐺𝑍) ∈ V
1413elsn 4383 . . . . . . . . . 10 ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
1512, 14sylibr 226 . . . . . . . . 9 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
16 oveq2 6886 . . . . . . . . . 10 (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
1716eleq1d 2863 . . . . . . . . 9 (𝑦 = 𝑍 → ((𝑍𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑍) ∈ {𝑍}))
1815, 17syl5ibrcom 239 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑦 = 𝑍 → (𝑍𝐺𝑦) ∈ {𝑍}))
1910, 18syl5bi 234 . . . . . . 7 (𝑅 ∈ RingOps → (𝑦 ∈ {𝑍} → (𝑍𝐺𝑦) ∈ {𝑍}))
2019ralrimiv 3146 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
21 eqid 2799 . . . . . . . . . 10 (2nd𝑅) = (2nd𝑅)
223, 2, 1, 21rngorz 34209 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) = 𝑍)
23 ovex 6910 . . . . . . . . . 10 (𝑧(2nd𝑅)𝑍) ∈ V
2423elsn 4383 . . . . . . . . 9 ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) = 𝑍)
2522, 24sylibr 226 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) ∈ {𝑍})
263, 2, 1, 21rngolz 34208 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) = 𝑍)
27 ovex 6910 . . . . . . . . . 10 (𝑍(2nd𝑅)𝑧) ∈ V
2827elsn 4383 . . . . . . . . 9 ((𝑍(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) = 𝑍)
2926, 28sylibr 226 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) ∈ {𝑍})
3025, 29jca 508 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3130ralrimiva 3147 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3220, 31jca 508 . . . . 5 (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
33 oveq1 6885 . . . . . . . 8 (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
3433eleq1d 2863 . . . . . . 7 (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
3534ralbidv 3167 . . . . . 6 (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
36 oveq2 6886 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑧(2nd𝑅)𝑥) = (𝑧(2nd𝑅)𝑍))
3736eleq1d 2863 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) ∈ {𝑍}))
38 oveq1 6885 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑥(2nd𝑅)𝑧) = (𝑍(2nd𝑅)𝑧))
3938eleq1d 2863 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑥(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
4037, 39anbi12d 625 . . . . . . 7 (𝑥 = 𝑍 → (((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}) ↔ ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
4140ralbidv 3167 . . . . . 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 625 . . . . 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 239 . . . 4 (𝑅 ∈ RingOps → (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
449, 43syl5bi 234 . . 3 (𝑅 ∈ RingOps → (𝑥 ∈ {𝑍} → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
4544ralrimiv 3146 . 2 (𝑅 ∈ RingOps → ∀𝑥 ∈ {𝑍} (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍})))
461, 21, 2, 3isidl 34300 . 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 1443 1 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  wss 3769  {csn 4368  ran crn 5313  cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  GIdcgi 27870  RingOpscrngo 34180  Idlcidl 34293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-1st 7401  df-2nd 7402  df-grpo 27873  df-gid 27874  df-ginv 27875  df-ablo 27925  df-rngo 34181  df-idl 34296
This theorem is referenced by:  0rngo  34313  divrngidl  34314  smprngopr  34338  isdmn3  34360
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