Theorem rngoidl 33823

Description: A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
rngidl.1	⊢ 𝐺 = (1st ‘𝑅)
rngidl.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
rngoidl	⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))

Proof of Theorem rngoidl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3624	. . 3 ⊢ 𝑋 ⊆ 𝑋
2	1	a1i 11	. 2 ⊢ (𝑅 ∈ RingOps → 𝑋 ⊆ 𝑋)
3		rngidl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
4		rngidl.2	. . 3 ⊢ 𝑋 = ran 𝐺
5		eqid 2622	. . 3 ⊢ (GId‘𝐺) = (GId‘𝐺)
6	3, 4, 5	rngo0cl 33718	. 2 ⊢ (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋)
7	3, 4	rngogcl 33711	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
8	7	3expa 1265	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
9	8	ralrimiva 2966	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋)
10		eqid 2622	. . . . . . . . 9 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
11	3, 10, 4	rngocl 33700	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋)
12	11	3com23 1271	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋)
13	3, 10, 4	rngocl 33700	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)
14	12, 13	jca 554	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))
15	14	3expa 1265	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))
16	15	ralrimiva 2966	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))
17	9, 16	jca 554	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)))
18	17	ralrimiva 2966	. 2 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)))
19	3, 10, 4, 5	isidl 33813	. 2 ⊢ (𝑅 ∈ RingOps → (𝑋 ∈ (Idl‘𝑅) ↔ (𝑋 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)))))
20	2, 6, 18, 19	mpbir3and 1245	1 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  ran crn 5115  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  GIdcgi 27344  RingOpscrngo 33693  Idlcidl 33806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ablo 27399  df-rngo 33694  df-idl 33809

This theorem is referenced by:  divrngidl  33827  igenval  33860  igenidl  33862