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Mathbox for Jeff Madsen |
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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 | 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)) |
Colors of variables: wff setvar class |
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 1st c1st 7425 2nd 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 |
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