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Theorem 0idl 35733
 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 2759 . . . 4 ran 𝐺 = ran 𝐺
3 0idl.2 . . . 4 𝑍 = (GId‘𝐺)
41, 2, 3rngo0cl 35627 . . 3 (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
54snssd 4697 . 2 (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
63fvexi 6670 . . . 4 𝑍 ∈ V
76snid 4556 . . 3 𝑍 ∈ {𝑍}
87a1i 11 . 2 (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
9 velsn 4536 . . . 4 (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
10 velsn 4536 . . . . . . . 8 (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
111, 2, 3rngo0rid 35628 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
124, 11mpdan 687 . . . . . . . . . 10 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
13 ovex 7181 . . . . . . . . . . 11 (𝑍𝐺𝑍) ∈ V
1413elsn 4535 . . . . . . . . . 10 ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
1512, 14sylibr 237 . . . . . . . . 9 (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
16 oveq2 7156 . . . . . . . . . 10 (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
1716eleq1d 2837 . . . . . . . . 9 (𝑦 = 𝑍 → ((𝑍𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑍) ∈ {𝑍}))
1815, 17syl5ibrcom 250 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑦 = 𝑍 → (𝑍𝐺𝑦) ∈ {𝑍}))
1910, 18syl5bi 245 . . . . . . 7 (𝑅 ∈ RingOps → (𝑦 ∈ {𝑍} → (𝑍𝐺𝑦) ∈ {𝑍}))
2019ralrimiv 3113 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
21 eqid 2759 . . . . . . . . . 10 (2nd𝑅) = (2nd𝑅)
223, 2, 1, 21rngorz 35631 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) = 𝑍)
23 ovex 7181 . . . . . . . . . 10 (𝑧(2nd𝑅)𝑍) ∈ V
2423elsn 4535 . . . . . . . . 9 ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) = 𝑍)
2522, 24sylibr 237 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd𝑅)𝑍) ∈ {𝑍})
263, 2, 1, 21rngolz 35630 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) = 𝑍)
27 ovex 7181 . . . . . . . . . 10 (𝑍(2nd𝑅)𝑧) ∈ V
2827elsn 4535 . . . . . . . . 9 ((𝑍(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) = 𝑍)
2926, 28sylibr 237 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd𝑅)𝑧) ∈ {𝑍})
3025, 29jca 516 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3130ralrimiva 3114 . . . . . 6 (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
3220, 31jca 516 . . . . 5 (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
33 oveq1 7155 . . . . . . . 8 (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
3433eleq1d 2837 . . . . . . 7 (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
3534ralbidv 3127 . . . . . 6 (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
36 oveq2 7156 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑧(2nd𝑅)𝑥) = (𝑧(2nd𝑅)𝑍))
3736eleq1d 2837 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd𝑅)𝑍) ∈ {𝑍}))
38 oveq1 7155 . . . . . . . . 9 (𝑥 = 𝑍 → (𝑥(2nd𝑅)𝑧) = (𝑍(2nd𝑅)𝑧))
3938eleq1d 2837 . . . . . . . 8 (𝑥 = 𝑍 → ((𝑥(2nd𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd𝑅)𝑧) ∈ {𝑍}))
4037, 39anbi12d 634 . . . . . . 7 (𝑥 = 𝑍 → (((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}) ↔ ((𝑧(2nd𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd𝑅)𝑧) ∈ {𝑍})))
4140ralbidv 3127 . . . . . 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 634 . . . . 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 250 . . . 4 (𝑅 ∈ RingOps → (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
449, 43syl5bi 245 . . 3 (𝑅 ∈ RingOps → (𝑥 ∈ {𝑍} → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍}))))
4544ralrimiv 3113 . 2 (𝑅 ∈ RingOps → ∀𝑥 ∈ {𝑍} (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd𝑅)𝑧) ∈ {𝑍})))
461, 21, 2, 3isidl 35722 . 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 1340 1 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071   ⊆ wss 3859  {csn 4520  ran crn 5523  ‘cfv 6333  (class class class)co 7148  1st c1st 7689  2nd c2nd 7690  GIdcgi 28362  RingOpscrngo 35602  Idlcidl 35715 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7106  df-ov 7151  df-1st 7691  df-2nd 7692  df-grpo 28365  df-gid 28366  df-ginv 28367  df-ablo 28417  df-rngo 35603  df-idl 35718 This theorem is referenced by:  0rngo  35735  divrngidl  35736  smprngopr  35760  isdmn3  35782
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