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| Mirrors > Home > MPE Home > Th. List > Mathboxes > idlsubcl | Structured version Visualization version GIF version | ||
| Description: An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.) | 
| Ref | Expression | 
|---|---|
| idlsubcl.1 | ⊢ 𝐺 = (1st ‘𝑅) | 
| idlsubcl.2 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) | 
| Ref | Expression | 
|---|---|
| idlsubcl | ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idlsubcl.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
| 2 | eqid 2737 | . . . . 5 ⊢ ran 𝐺 = ran 𝐺 | |
| 3 | 1, 2 | idlcl 38024 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ran 𝐺) | 
| 4 | 1, 2 | idlcl 38024 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵 ∈ 𝐼) → 𝐵 ∈ ran 𝐺) | 
| 5 | 3, 4 | anim12dan 619 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴 ∈ ran 𝐺 ∧ 𝐵 ∈ ran 𝐺)) | 
| 6 | eqid 2737 | . . . . . 6 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
| 7 | idlsubcl.2 | . . . . . 6 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
| 8 | 1, 2, 6, 7 | rngosub 37937 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ ran 𝐺 ∧ 𝐵 ∈ ran 𝐺) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵))) | 
| 9 | 8 | 3expb 1121 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ ran 𝐺 ∧ 𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵))) | 
| 10 | 9 | adantlr 715 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ ran 𝐺 ∧ 𝐵 ∈ ran 𝐺)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵))) | 
| 11 | 5, 10 | syldan 591 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵))) | 
| 12 | simprl 771 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → 𝐴 ∈ 𝐼) | |
| 13 | 1, 6 | idlnegcl 38029 | . . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐵 ∈ 𝐼) → ((inv‘𝐺)‘𝐵) ∈ 𝐼) | 
| 14 | 13 | adantrl 716 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → ((inv‘𝐺)‘𝐵) ∈ 𝐼) | 
| 15 | 12, 14 | jca 511 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴 ∈ 𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼)) | 
| 16 | 1 | idladdcl 38026 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼) | 
| 17 | 15, 16 | syldan 591 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺((inv‘𝐺)‘𝐵)) ∈ 𝐼) | 
| 18 | 11, 17 | eqeltrd 2841 | 1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ran crn 5686 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 invcgn 30510 /𝑔 cgs 30511 RingOpscrngo 37901 Idlcidl 38014 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-ass 37850 df-exid 37852 df-mgmOLD 37856 df-sgrOLD 37868 df-mndo 37874 df-rngo 37902 df-idl 38017 | 
| This theorem is referenced by: (None) | 
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