Theorem rngoidl 34364

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		ssidd 3848	. 2 ⊢ (𝑅 ∈ RingOps → 𝑋 ⊆ 𝑋)
2		rngidl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
3		rngidl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		eqid 2824	. . 3 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	2, 3, 4	rngo0cl 34259	. 2 ⊢ (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋)
6	2, 3	rngogcl 34252	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
7	6	3expa 1153	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) ∈ 𝑋)
8	7	ralrimiva 3174	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋)
9		eqid 2824	. . . . . . . . 9 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
10	2, 9, 3	rngocl 34241	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋)
11	10	3com23 1162	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋)
12	2, 9, 3	rngocl 34241	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)
13	11, 12	jca 509	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))
14	13	3expa 1153	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))
15	14	ralrimiva 3174	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))
16	8, 15	jca 509	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)))
17	16	ralrimiva 3174	. 2 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)))
18	2, 9, 3, 4	isidl 34354	. 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)))))
19	1, 5, 17, 18	mpbir3and 1448	1 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3116   ⊆ wss 3797  ran crn 5342  ‘cfv 6122  (class class class)co 6904  1^st c1st 7425  2^nd c2nd 7426  GIdcgi 27899  RingOpscrngo 34234  Idlcidl 34347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-fo 6128  df-fv 6130  df-riota 6865  df-ov 6907  df-1st 7427  df-2nd 7428  df-grpo 27902  df-gid 27903  df-ablo 27954  df-rngo 34235  df-idl 34350

This theorem is referenced by:  divrngidl  34368  igenval  34401  igenidl  34403