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Mirrors > Home > MPE Home > Th. List > Mathboxes > idladdcl | Structured version Visualization version GIF version |
Description: An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
idladdcl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
Ref | Expression |
---|---|
idladdcl | ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idladdcl.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | eqid 2821 | . . . . . 6 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | eqid 2821 | . . . . . 6 ⊢ ran 𝐺 = ran 𝐺 | |
4 | eqid 2821 | . . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
5 | 1, 2, 3, 4 | isidl 35307 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ ran 𝐺 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼))))) |
6 | 5 | biimpa 479 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ ran 𝐺 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)))) |
7 | 6 | simp3d 1140 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼))) |
8 | simpl 485 | . . . 4 ⊢ ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)) → ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼) | |
9 | 8 | ralimi 3160 | . . 3 ⊢ (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ ran 𝐺((𝑧(2nd ‘𝑅)𝑥) ∈ 𝐼 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝐼)) → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼) |
10 | 7, 9 | syl 17 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼) |
11 | oveq1 7163 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦)) | |
12 | 11 | eleq1d 2897 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝑦) ∈ 𝐼)) |
13 | oveq2 7164 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵)) | |
14 | 13 | eleq1d 2897 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦) ∈ 𝐼 ↔ (𝐴𝐺𝐵) ∈ 𝐼)) |
15 | 12, 14 | rspc2v 3633 | . 2 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼) → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 → (𝐴𝐺𝐵) ∈ 𝐼)) |
16 | 10, 15 | mpan9 509 | 1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 ran crn 5556 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 GIdcgi 28267 RingOpscrngo 35187 Idlcidl 35300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-idl 35303 |
This theorem is referenced by: idlsubcl 35316 intidl 35322 unichnidl 35324 |
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