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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rngolz | Structured version Visualization version GIF version | ||
| Description: The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ringlz.1 | ⊢ 𝑍 = (GId‘𝐺) |
| ringlz.2 | ⊢ 𝑋 = ran 𝐺 |
| ringlz.3 | ⊢ 𝐺 = (1st ‘𝑅) |
| ringlz.4 | ⊢ 𝐻 = (2nd ‘𝑅) |
| Ref | Expression |
|---|---|
| rngolz | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = 𝑍) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ringlz.3 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
| 2 | 1 | rngogrpo 37917 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
| 3 | ringlz.2 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
| 4 | ringlz.1 | . . . . . . 7 ⊢ 𝑍 = (GId‘𝐺) | |
| 5 | 3, 4 | grpoidcl 30533 | . . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋) |
| 6 | 3, 4 | grpolid 30535 | . . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍) |
| 7 | 2, 5, 6 | syl2anc2 585 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍) |
| 8 | 7 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍) |
| 9 | 8 | oveq1d 7446 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = (𝑍𝐻𝐴)) |
| 10 | 1, 3, 4 | rngo0cl 37926 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋) |
| 11 | 10 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
| 12 | simpr 484 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
| 13 | 11, 11, 12 | 3jca 1129 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
| 14 | ringlz.4 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
| 15 | 1, 14, 3 | rngodir 37912 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴))) |
| 16 | 13, 15 | syldan 591 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑍𝐺𝑍)𝐻𝐴) = ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴))) |
| 17 | 2 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp) |
| 18 | simpl 482 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑅 ∈ RingOps) | |
| 19 | 1, 14, 3 | rngocl 37908 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) ∈ 𝑋) |
| 20 | 18, 11, 12, 19 | syl3anc 1373 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) ∈ 𝑋) |
| 21 | 3, 4 | grporid 30536 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → ((𝑍𝐻𝐴)𝐺𝑍) = (𝑍𝐻𝐴)) |
| 22 | 21 | eqcomd 2743 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑍𝐻𝐴) ∈ 𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍)) |
| 23 | 17, 20, 22 | syl2anc 584 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = ((𝑍𝐻𝐴)𝐺𝑍)) |
| 24 | 9, 16, 23 | 3eqtr3d 2785 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍)) |
| 25 | 3 | grpolcan 30549 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝑍𝐻𝐴) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ (𝑍𝐻𝐴) ∈ 𝑋)) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍)) |
| 26 | 17, 20, 11, 20, 25 | syl13anc 1374 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (((𝑍𝐻𝐴)𝐺(𝑍𝐻𝐴)) = ((𝑍𝐻𝐴)𝐺𝑍) ↔ (𝑍𝐻𝐴) = 𝑍)) |
| 27 | 24, 26 | mpbid 232 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐻𝐴) = 𝑍) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ran crn 5686 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 GrpOpcgr 30508 GIdcgi 30509 RingOpscrngo 37901 |
| 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-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-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-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-1st 8014 df-2nd 8015 df-grpo 30512 df-gid 30513 df-ginv 30514 df-ablo 30564 df-rngo 37902 |
| This theorem is referenced by: rngonegmn1l 37948 isdrngo3 37966 0idl 38032 keridl 38039 prnc 38074 |
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