Theorem 0idl 39162

Description: Obsolete theorem, use 2idl0 21587 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 2763	. . . 4 ⊢ ran 𝐺 = ran 𝐺
3		0idl.2	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
4	1, 2, 3	rngo0cl 39056	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ ran 𝐺)
5	4	snssd 4747	. 2 ⊢ (𝑅 ∈ RingOps → {𝑍} ⊆ ran 𝐺)
6	3	fvexi 6925	. . . 4 ⊢ 𝑍 ∈ V
7	6	snid 4623	. . 3 ⊢ 𝑍 ∈ {𝑍}
8	7	a1i 11	. 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ {𝑍})
9		velsn 4600	. . . 4 ⊢ (𝑥 ∈ {𝑍} ↔ 𝑥 = 𝑍)
10		velsn 4600	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑍} ↔ 𝑦 = 𝑍)
11	1, 2, 3	rngo0rid 39057	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
12	4, 11	mpdan 700	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
13		ovex 7479	. . . . . . . . . . 11 ⊢ (𝑍𝐺𝑍) ∈ V
14	13	elsn 4599	. . . . . . . . . 10 ⊢ ((𝑍𝐺𝑍) ∈ {𝑍} ↔ (𝑍𝐺𝑍) = 𝑍)
15	12, 14	sylibr 237	. . . . . . . . 9 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) ∈ {𝑍})
16		oveq2 7454	. . . . . . . . . 10 ⊢ (𝑦 = 𝑍 → (𝑍𝐺𝑦) = (𝑍𝐺𝑍))
17	16	eleq1d 2849	. . . . . . . . 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 3157	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍})
21		eqid 2763	. . . . . . . . . 10 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
22	3, 2, 1, 21	rngorz 39060	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑧(2nd ‘𝑅)𝑍) = 𝑍)
23		ovex 7479	. . . . . . . . . 10 ⊢ (𝑧(2nd ‘𝑅)𝑍) ∈ V
24	23	elsn 4599	. . . . . . . . 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 39059	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → (𝑍(2nd ‘𝑅)𝑧) = 𝑍)
27		ovex 7479	. . . . . . . . . 10 ⊢ (𝑍(2nd ‘𝑅)𝑧) ∈ V
28	27	elsn 4599	. . . . . . . . 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 521	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran 𝐺) → ((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
31	30	ralrimiva 3158	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
32	20, 31	jca 521	. . . . 5 ⊢ (𝑅 ∈ RingOps → (∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍} ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍} ∧ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍})))
33		oveq1 7453	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → (𝑥𝐺𝑦) = (𝑍𝐺𝑦))
34	33	eleq1d 2849	. . . . . . 7 ⊢ (𝑥 = 𝑍 → ((𝑥𝐺𝑦) ∈ {𝑍} ↔ (𝑍𝐺𝑦) ∈ {𝑍}))
35	34	ralbidv 3189	. . . . . 6 ⊢ (𝑥 = 𝑍 → (∀𝑦 ∈ {𝑍} (𝑥𝐺𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ {𝑍} (𝑍𝐺𝑦) ∈ {𝑍}))
36		oveq2 7454	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑧(2nd ‘𝑅)𝑥) = (𝑧(2nd ‘𝑅)𝑍))
37	36	eleq1d 2849	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑧(2nd ‘𝑅)𝑥) ∈ {𝑍} ↔ (𝑧(2nd ‘𝑅)𝑍) ∈ {𝑍}))
38		oveq1 7453	. . . . . . . . 9 ⊢ (𝑥 = 𝑍 → (𝑥(2nd ‘𝑅)𝑧) = (𝑍(2nd ‘𝑅)𝑧))
39	38	eleq1d 2849	. . . . . . . 8 ⊢ (𝑥 = 𝑍 → ((𝑥(2nd ‘𝑅)𝑧) ∈ {𝑍} ↔ (𝑍(2nd ‘𝑅)𝑧) ∈ {𝑍}))
40	37, 39	anbi12d 644	. . . . . . 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 3189	. . . . . 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 644	. . . . 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 3157	. 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 39151	. 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 1361	1 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  {csn 4588  ran crn 5657  ‘cfv 6561  (class class class)co 7446  1^st c1st 8026  2^nd c2nd 8027  GIdcgi 31125  RingOpscrngo 39031  Idlcidl 39144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-1st 8028  df-2nd 8029  df-grpo 31128  df-gid 31129  df-ginv 31130  df-ablo 31180  df-rngo 39032  df-idl 39147

This theorem is used by:  0rngo  39164  divrngidl  39165  smprngopr  39189  isdmn3  39211