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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlrmulcl | Structured version Visualization version GIF version |
Description: An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
idllmulcl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
idllmulcl.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
idllmulcl.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
idlrmulcl | ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idllmulcl.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | idllmulcl.2 | . . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | idllmulcl.3 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
4 | eqid 2736 | . . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
5 | 1, 2, 3, 4 | isidl 35858 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))) |
6 | 5 | biimpa 480 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))) |
7 | 6 | simp3d 1146 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))) |
8 | simpr 488 | . . . . . 6 ⊢ (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → (𝑥𝐻𝑧) ∈ 𝐼) | |
9 | 8 | ralimi 3073 | . . . . 5 ⊢ (∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼) |
10 | 9 | adantl 485 | . . . 4 ⊢ ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼) |
11 | 10 | ralimi 3073 | . . 3 ⊢ (∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼) |
12 | 7, 11 | syl 17 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼) |
13 | oveq1 7198 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧)) | |
14 | 13 | eleq1d 2815 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝑧) ∈ 𝐼)) |
15 | oveq2 7199 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝐴𝐻𝑧) = (𝐴𝐻𝐵)) | |
16 | 15 | eleq1d 2815 | . . 3 ⊢ (𝑧 = 𝐵 → ((𝐴𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝐵) ∈ 𝐼)) |
17 | 14, 16 | rspc2v 3537 | . 2 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼 → (𝐴𝐻𝐵) ∈ 𝐼)) |
18 | 12, 17 | mpan9 510 | 1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ⊆ wss 3853 ran crn 5537 ‘cfv 6358 (class class class)co 7191 1st c1st 7737 2nd c2nd 7738 GIdcgi 28525 RingOpscrngo 35738 Idlcidl 35851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-idl 35854 |
This theorem is referenced by: 1idl 35870 intidl 35873 unichnidl 35875 |
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