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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoidl | Structured version Visualization version GIF version |
Description: A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
rngidl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
rngidl.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
rngoidl | ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3587 | . . 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 2610 | . . 3 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
6 | 3, 4, 5 | rngo0cl 32888 | . 2 ⊢ (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋) |
7 | 3, 4 | rngogcl 32881 | . . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) ∈ 𝑋) |
8 | 7 | 3expa 1257 | . . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) ∈ 𝑋) |
9 | 8 | ralrimiva 2949 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋) |
10 | eqid 2610 | . . . . . . . . 9 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
11 | 3, 10, 4 | rngocl 32870 | . . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋) |
12 | 11 | 3com23 1263 | . . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋) |
13 | 3, 10, 4 | rngocl 32870 | . . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋) |
14 | 12, 13 | jca 553 | . . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)) |
15 | 14 | 3expa 1257 | . . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)) |
16 | 15 | ralrimiva 2949 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)) |
17 | 9, 16 | jca 553 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))) |
18 | 17 | ralrimiva 2949 | . 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 32983 | . 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 1238 | 1 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ran crn 5039 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 GIdcgi 26728 RingOpscrngo 32863 Idlcidl 32976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-riota 6511 df-ov 6552 df-1st 7059 df-2nd 7060 df-grpo 26731 df-gid 26732 df-ablo 26783 df-rngo 32864 df-idl 32979 |
This theorem is referenced by: divrngidl 32997 igenval 33030 igenidl 33032 |
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