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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngorz | Structured version Visualization version GIF version |
Description: The zero of a unital ring is a right-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 |
---|---|
rngorz | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringlz.3 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 37289 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ringlz.2 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
4 | ringlz.1 | . . . . . . 7 ⊢ 𝑍 = (GId‘𝐺) | |
5 | 3, 4 | grpoidcl 30272 | . . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋) |
6 | 3, 4 | grpolid 30274 | . . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍) |
7 | 2, 5, 6 | syl2anc2 584 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍) |
8 | 7 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍) |
9 | 8 | oveq2d 7420 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = (𝐴𝐻𝑍)) |
10 | simpr 484 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
11 | 1, 3, 4 | rngo0cl 37298 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋) |
12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
13 | 10, 12, 12 | 3jca 1125 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋)) |
14 | ringlz.4 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
15 | 1, 14, 3 | rngodi 37283 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋)) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍))) |
16 | 13, 15 | syldan 590 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍))) |
17 | 2 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp) |
18 | 1, 14, 3 | rngocl 37280 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋) |
19 | 12, 18 | mpd3an3 1458 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋) |
20 | 3, 4 | grpolid 30274 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝑍𝐺(𝐴𝐻𝑍)) = (𝐴𝐻𝑍)) |
21 | 20 | eqcomd 2732 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍))) |
22 | 17, 19, 21 | syl2anc 583 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍))) |
23 | 9, 16, 22 | 3eqtr3d 2774 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍))) |
24 | 3 | grporcan 30276 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐻𝑍) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ (𝐴𝐻𝑍) ∈ 𝑋)) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍)) |
25 | 17, 19, 12, 19, 24 | syl13anc 1369 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍)) |
26 | 23, 25 | mpbid 231 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ran crn 5670 ‘cfv 6536 (class class class)co 7404 1st c1st 7969 2nd c2nd 7970 GrpOpcgr 30247 GIdcgi 30248 RingOpscrngo 37273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-riota 7360 df-ov 7407 df-1st 7971 df-2nd 7972 df-grpo 30251 df-gid 30252 df-ablo 30303 df-rngo 37274 |
This theorem is referenced by: rngoueqz 37319 rngonegmn1r 37321 zerdivemp1x 37326 0idl 37404 keridl 37411 |
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