Theorem 0idl 38405

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 2741	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3		0idl.2	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
4	1, 2, 3	rngo0cl 38299	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
5	4	snssd 4720	. 2 ⊢ (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
6	3	fvexi 6844	. . . 4 ⊢ 𝑍 ∈ V
7	6	snid 4596	. . 3 ⊢ 𝑍 ∈ {𝑍}
8	7	a1i 11	. 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
9		velsn 4573	. . . 4 ⊢ (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
10		velsn 4573	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
11	1, 2, 3	rngo0rid 38300	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
12	4, 11	mpdan 694	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
13		ovex 7392	. . . . . . . . . . 11 ⊢ (𝑍𝐺𝑍) ∈ V
14	13	elsn 4572	. . . . . . . . . 10 ⊢ ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
15	12, 14	sylibr 236	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
16		oveq2 7367	. . . . . . . . . 10 ⊢ (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
17	16	eleq1d 2826	. . . . . . . . 9 ⊢ (𝑦 = 𝑍 → ((𝑍𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑍) ∈ {𝑍}))
18	15, 17	syl5ibrcom 249	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑦 = 𝑍 → (𝑍𝐺𝑦) ∈ {𝑍}))
19	10, 18	biimtrid 244	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑦 ∈ {𝑍} → (𝑍𝐺𝑦) ∈ {𝑍}))
20	19	ralrimiv 3132	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
21		eqid 2741	. . . . . . . . . 10 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
22	3, 2, 1, 21	rngorz 38303	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd ‘𝑅)𝑍) = 𝑍)
23		ovex 7392	. . . . . . . . . 10 ⊢ (𝑧(2nd ‘𝑅)𝑍) ∈ V
24	23	elsn 4572	. . . . . . . . 9 ⊢ ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ↔ (𝑧(2nd ‘𝑅)𝑍) = 𝑍)
25	22, 24	sylibr 236	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍})
26	3, 2, 1, 21	rngolz 38302	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd ‘𝑅)𝑧) = 𝑍)
27		ovex 7392	. . . . . . . . . 10 ⊢ (𝑍(2nd ‘𝑅)𝑧) ∈ V
28	27	elsn 4572	. . . . . . . . 9 ⊢ ((𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd ‘𝑅)𝑧) = 𝑍)
29	26, 28	sylibr 236	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})
30	25, 29	jca 517	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
31	30	ralrimiva 3133	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
32	20, 31	jca 517	. . . . 5 ⊢ (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})))
33		oveq1 7366	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
34	33	eleq1d 2826	. . . . . . 7 ⊢ (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
35	34	ralbidv 3164	. . . . . 6 ⊢ (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
36		oveq2 7367	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑧(2nd ‘𝑅)𝑥) = (𝑧(2nd ‘𝑅)𝑍))
37	36	eleq1d 2826	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍}))
38		oveq1 7366	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑥(2nd ‘𝑅)𝑧) = (𝑍(2nd ‘𝑅)𝑧))
39	38	eleq1d 2826	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
40	37, 39	anbi12d 639	. . . . . . 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 3164	. . . . . 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 639	. . . . 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 249	. . . 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 244	. . 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 3132	. 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 38394	. 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 1350	1 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055   ⊆ wss 3884  {csn 4557  ran crn 5621  ‘cfv 6488  (class class class)co 7359  1^st c1st 7931  2^nd c2nd 7932  GIdcgi 30581  RingOpscrngo 38274  Idlcidl 38387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-1st 7933  df-2nd 7934  df-grpo 30584  df-gid 30585  df-ginv 30586  df-ablo 30636  df-rngo 38275  df-idl 38390

This theorem is referenced by:  0rngo  38407  divrngidl  38408  smprngopr  38432  isdmn3  38454