Theorem 0idl 36557

Description: The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
0idl.1	⊢ 𝐺 = (1st ‘𝑅)
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.

Step	Hyp	Ref	Expression
1		0idl.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		eqid 2731	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3		0idl.2	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
4	1, 2, 3	rngo0cl 36451	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
5	4	snssd 4774	. 2 ⊢ (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
6	3	fvexi 6861	. . . 4 ⊢ 𝑍 ∈ V
7	6	snid 4627	. . 3 ⊢ 𝑍 ∈ {𝑍}
8	7	a1i 11	. 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
9		velsn 4607	. . . 4 ⊢ (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
10		velsn 4607	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
11	1, 2, 3	rngo0rid 36452	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
12	4, 11	mpdan 685	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
13		ovex 7395	. . . . . . . . . . 11 ⊢ (𝑍𝐺𝑍) ∈ V
14	13	elsn 4606	. . . . . . . . . 10 ⊢ ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
15	12, 14	sylibr 233	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
16		oveq2 7370	. . . . . . . . . 10 ⊢ (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
17	16	eleq1d 2817	. . . . . . . . 9 ⊢ (𝑦 = 𝑍 → ((𝑍𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑍) ∈ {𝑍}))
18	15, 17	syl5ibrcom 246	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑦 = 𝑍 → (𝑍𝐺𝑦) ∈ {𝑍}))
19	10, 18	biimtrid 241	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑦 ∈ {𝑍} → (𝑍𝐺𝑦) ∈ {𝑍}))
20	19	ralrimiv 3138	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
21		eqid 2731	. . . . . . . . . 10 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
22	3, 2, 1, 21	rngorz 36455	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd ‘𝑅)𝑍) = 𝑍)
23		ovex 7395	. . . . . . . . . 10 ⊢ (𝑧(2nd ‘𝑅)𝑍) ∈ V
24	23	elsn 4606	. . . . . . . . 9 ⊢ ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ↔ (𝑧(2nd ‘𝑅)𝑍) = 𝑍)
25	22, 24	sylibr 233	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍})
26	3, 2, 1, 21	rngolz 36454	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd ‘𝑅)𝑧) = 𝑍)
27		ovex 7395	. . . . . . . . . 10 ⊢ (𝑍(2nd ‘𝑅)𝑧) ∈ V
28	27	elsn 4606	. . . . . . . . 9 ⊢ ((𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd ‘𝑅)𝑧) = 𝑍)
29	26, 28	sylibr 233	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})
30	25, 29	jca 512	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
31	30	ralrimiva 3139	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
32	20, 31	jca 512	. . . . 5 ⊢ (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})))
33		oveq1 7369	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
34	33	eleq1d 2817	. . . . . . 7 ⊢ (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
35	34	ralbidv 3170	. . . . . 6 ⊢ (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
36		oveq2 7370	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑧(2nd ‘𝑅)𝑥) = (𝑧(2nd ‘𝑅)𝑍))
37	36	eleq1d 2817	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍}))
38		oveq1 7369	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑥(2nd ‘𝑅)𝑧) = (𝑍(2nd ‘𝑅)𝑧))
39	38	eleq1d 2817	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
40	37, 39	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝑍 → (((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍}) ↔ ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})))
41	40	ralbidv 3170	. . . . . 6 ⊢ (𝑥 = 𝑍 → (∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍}) ↔ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})))
42	35, 41	anbi12d 631	. . . . 5 ⊢ (𝑥 = 𝑍 → ((∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍})) ↔ (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))))
43	32, 42	syl5ibrcom 246	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍}))))
44	9, 43	biimtrid 241	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑥 ∈ {𝑍} → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍}))))
45	44	ralrimiv 3138	. 2 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ {𝑍} (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍})))
46	1, 21, 2, 3	isidl 36546	. 2 ⊢ (𝑅 ∈ RingOps → ({𝑍} ∈ (Idl‘𝑅) ↔ ({𝑍} ⊆ ran 𝐺 ∧ 𝑍 ∈ {𝑍} ∧ ∀𝑥 ∈ {𝑍} (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍})))))
47	5, 8, 45, 46	mpbir3and 1342	1 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060   ⊆ wss 3913  {csn 4591  ran crn 5639  ‘cfv 6501  (class class class)co 7362  1^st c1st 7924  2^nd c2nd 7925  GIdcgi 29495  RingOpscrngo 36426  Idlcidl 36539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-1st 7926  df-2nd 7927  df-grpo 29498  df-gid 29499  df-ginv 29500  df-ablo 29550  df-rngo 36427  df-idl 36542

This theorem is referenced by:  0rngo  36559  divrngidl  36560  smprngopr  36584  isdmn3  36606