Theorem 0idl 38536

Description: Obsolete theorem, use 2idl0 21361 instead. The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

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 2765	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3		0idl.2	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
4	1, 2, 3	rngo0cl 38430	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
5	4	snssd 4748	. 2 ⊢ (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
6	3	fvexi 6885	. . . 4 ⊢ 𝑍 ∈ V
7	6	snid 4624	. . 3 ⊢ 𝑍 ∈ {𝑍}
8	7	a1i 11	. 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
9		velsn 4601	. . . 4 ⊢ (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
10		velsn 4601	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
11	1, 2, 3	rngo0rid 38431	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
12	4, 11	mpdan 699	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
13		ovex 7433	. . . . . . . . . . 11 ⊢ (𝑍𝐺𝑍) ∈ V
14	13	elsn 4600	. . . . . . . . . 10 ⊢ ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
15	12, 14	sylibr 237	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
16		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
17	16	eleq1d 2850	. . . . . . . . 9 ⊢ (𝑦 = 𝑍 → ((𝑍𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑍) ∈ {𝑍}))
18	15, 17	syl5ibrcom 250	. . . . . . . 8 ⊢ (𝑅 ∈ RingOps → (𝑦 = 𝑍 → (𝑍𝐺𝑦) ∈ {𝑍}))
19	10, 18	biimtrid 245	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑦 ∈ {𝑍} → (𝑍𝐺𝑦) ∈ {𝑍}))
20	19	ralrimiv 3156	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
21		eqid 2765	. . . . . . . . . 10 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
22	3, 2, 1, 21	rngorz 38434	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd ‘𝑅)𝑍) = 𝑍)
23		ovex 7433	. . . . . . . . . 10 ⊢ (𝑧(2nd ‘𝑅)𝑍) ∈ V
24	23	elsn 4600	. . . . . . . . 9 ⊢ ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ↔ (𝑧(2nd ‘𝑅)𝑍) = 𝑍)
25	22, 24	sylibr 237	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍})
26	3, 2, 1, 21	rngolz 38433	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd ‘𝑅)𝑧) = 𝑍)
27		ovex 7433	. . . . . . . . . 10 ⊢ (𝑍(2nd ‘𝑅)𝑧) ∈ V
28	27	elsn 4600	. . . . . . . . 9 ⊢ ((𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd ‘𝑅)𝑧) = 𝑍)
29	26, 28	sylibr 237	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})
30	25, 29	jca 520	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
31	30	ralrimiva 3157	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
32	20, 31	jca 520	. . . . 5 ⊢ (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})))
33		oveq1 7407	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
34	33	eleq1d 2850	. . . . . . 7 ⊢ (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
35	34	ralbidv 3188	. . . . . 6 ⊢ (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
36		oveq2 7408	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑧(2nd ‘𝑅)𝑥) = (𝑧(2nd ‘𝑅)𝑍))
37	36	eleq1d 2850	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍}))
38		oveq1 7407	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑥(2nd ‘𝑅)𝑧) = (𝑍(2nd ‘𝑅)𝑧))
39	38	eleq1d 2850	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
40	37, 39	anbi12d 643	. . . . . . 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 3188	. . . . . 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 643	. . . . 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 250	. . . 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 245	. . 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 3156	. 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 38525	. 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 1359	1 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ⊆ wss 3907  {csn 4585  ran crn 5653  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  GIdcgi 30751  RingOpscrngo 38405  Idlcidl 38518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-1st 7974  df-2nd 7975  df-grpo 30754  df-gid 30755  df-ginv 30756  df-ablo 30806  df-rngo 38406  df-idl 38521

This theorem is referenced by:  0rngo  38538  divrngidl  38539  smprngopr  38563  isdmn3  38585