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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlss | Structured version Visualization version GIF version |
Description: An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
idlss.1 | ⊢ 𝐺 = (1st ‘𝑅) |
idlss.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
idlss | ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlss.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | eqid 2770 | . . . 4 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | idlss.2 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
4 | eqid 2770 | . . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
5 | 1, 2, 3, 4 | isidl 34138 | . . 3 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼))))) |
6 | 5 | biimpa 462 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)))) |
7 | 6 | simp1d 1135 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1070 = wceq 1630 ∈ wcel 2144 ∀wral 3060 ⊆ wss 3721 ran crn 5250 ‘cfv 6031 (class class class)co 6792 1st c1st 7312 2nd c2nd 7313 GIdcgi 27678 RingOpscrngo 34018 Idlcidl 34131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-iota 5994 df-fun 6033 df-fv 6039 df-ov 6795 df-idl 34134 |
This theorem is referenced by: idlcl 34141 idlnegcl 34146 1idl 34150 divrngidl 34152 intidl 34153 unichnidl 34155 ispridl2 34162 igenmin 34188 igenidl2 34189 |
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