Theorem 0idl 38064

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 37958	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
5	4	snssd 4761	. 2 ⊢ (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
6	3	fvexi 6836	. . . 4 ⊢ 𝑍 ∈ V
7	6	snid 4615	. . 3 ⊢ 𝑍 ∈ {𝑍}
8	7	a1i 11	. 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
9		velsn 4592	. . . 4 ⊢ (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
10		velsn 4592	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
11	1, 2, 3	rngo0rid 37959	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
12	4, 11	mpdan 687	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
13		ovex 7379	. . . . . . . . . . 11 ⊢ (𝑍𝐺𝑍) ∈ V
14	13	elsn 4591	. . . . . . . . . 10 ⊢ ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
15	12, 14	sylibr 234	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
16		oveq2 7354	. . . . . . . . . 10 ⊢ (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
17	16	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑦 = 𝑍 → ((𝑍𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑍) ∈ {𝑍}))
18	15, 17	syl5ibrcom 247	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑦 = 𝑍 → (𝑍𝐺𝑦) ∈ {𝑍}))
19	10, 18	biimtrid 242	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑦 ∈ {𝑍} → (𝑍𝐺𝑦) ∈ {𝑍}))
20	19	ralrimiv 3123	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
21		eqid 2731	. . . . . . . . . 10 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
22	3, 2, 1, 21	rngorz 37962	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd ‘𝑅)𝑍) = 𝑍)
23		ovex 7379	. . . . . . . . . 10 ⊢ (𝑧(2nd ‘𝑅)𝑍) ∈ V
24	23	elsn 4591	. . . . . . . . 9 ⊢ ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ↔ (𝑧(2nd ‘𝑅)𝑍) = 𝑍)
25	22, 24	sylibr 234	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍})
26	3, 2, 1, 21	rngolz 37961	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd ‘𝑅)𝑧) = 𝑍)
27		ovex 7379	. . . . . . . . . 10 ⊢ (𝑍(2nd ‘𝑅)𝑧) ∈ V
28	27	elsn 4591	. . . . . . . . 9 ⊢ ((𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd ‘𝑅)𝑧) = 𝑍)
29	26, 28	sylibr 234	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})
30	25, 29	jca 511	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
31	30	ralrimiva 3124	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
32	20, 31	jca 511	. . . . 5 ⊢ (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})))
33		oveq1 7353	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
34	33	eleq1d 2816	. . . . . . 7 ⊢ (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
35	34	ralbidv 3155	. . . . . 6 ⊢ (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
36		oveq2 7354	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑧(2nd ‘𝑅)𝑥) = (𝑧(2nd ‘𝑅)𝑍))
37	36	eleq1d 2816	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍}))
38		oveq1 7353	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑥(2nd ‘𝑅)𝑧) = (𝑍(2nd ‘𝑅)𝑧))
39	38	eleq1d 2816	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
40	37, 39	anbi12d 632	. . . . . . 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 3155	. . . . . 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 632	. . . . 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 247	. . . 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 242	. . 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 3123	. 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 38053	. 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 1343	1 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3902  {csn 4576  ran crn 5617  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  GIdcgi 30465  RingOpscrngo 37933  Idlcidl 38046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-1st 7921  df-2nd 7922  df-grpo 30468  df-gid 30469  df-ginv 30470  df-ablo 30520  df-rngo 37934  df-idl 38049

This theorem is referenced by:  0rngo  38066  divrngidl  38067  smprngopr  38091  isdmn3  38113