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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 2821 | . . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
5 | 1, 2, 3, 4 | isidl 35291 | . . . . 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 479 | . . . 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 1140 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))) |
8 | simpr 487 | . . . . . 6 ⊢ (((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → (𝑥𝐻𝑧) ∈ 𝐼) | |
9 | 8 | ralimi 3160 | . . . . 5 ⊢ (∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼) → ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼) |
10 | 9 | adantl 484 | . . . 4 ⊢ ((∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)) → ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼) |
11 | 10 | ralimi 3160 | . . 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 7162 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧)) | |
14 | 13 | eleq1d 2897 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝑧) ∈ 𝐼)) |
15 | oveq2 7163 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝐴𝐻𝑧) = (𝐴𝐻𝐵)) | |
16 | 15 | eleq1d 2897 | . . 3 ⊢ (𝑧 = 𝐵 → ((𝐴𝐻𝑧) ∈ 𝐼 ↔ (𝐴𝐻𝐵) ∈ 𝐼)) |
17 | 14, 16 | rspc2v 3632 | . 2 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝐼 ∀𝑧 ∈ 𝑋 (𝑥𝐻𝑧) ∈ 𝐼 → (𝐴𝐻𝐵) ∈ 𝐼)) |
18 | 12, 17 | mpan9 509 | 1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ran crn 5555 ‘cfv 6354 (class class class)co 7155 1st c1st 7686 2nd c2nd 7687 GIdcgi 28266 RingOpscrngo 35171 Idlcidl 35284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-idl 35287 |
This theorem is referenced by: 1idl 35303 intidl 35306 unichnidl 35308 |
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