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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 35190 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ringlz.2 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
4 | ringlz.1 | . . . . . . 7 ⊢ 𝑍 = (GId‘𝐺) | |
5 | 3, 4 | grpoidcl 28293 | . . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋) |
6 | 3, 4 | grpolid 28295 | . . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍) |
7 | 2, 5, 6 | syl2anc2 587 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍) |
8 | 7 | adantr 483 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝑍) = 𝑍) |
9 | 8 | oveq2d 7174 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = (𝐴𝐻𝑍)) |
10 | simpr 487 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
11 | 1, 3, 4 | rngo0cl 35199 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋) |
12 | 11 | adantr 483 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
13 | 10, 12, 12 | 3jca 1124 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋)) |
14 | ringlz.4 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
15 | 1, 14, 3 | rngodi 35184 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋)) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍))) |
16 | 13, 15 | syldan 593 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍))) |
17 | 2 | adantr 483 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp) |
18 | 1, 14, 3 | rngocl 35181 | . . . . 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 28295 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝑍𝐺(𝐴𝐻𝑍)) = (𝐴𝐻𝑍)) |
21 | 20 | eqcomd 2829 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍))) |
22 | 17, 19, 21 | syl2anc 586 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = (𝑍𝐺(𝐴𝐻𝑍))) |
23 | 9, 16, 22 | 3eqtr3d 2866 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍))) |
24 | 3 | grporcan 28297 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐻𝑍) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ (𝐴𝐻𝑍) ∈ 𝑋)) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍)) |
25 | 17, 19, 12, 19, 24 | syl13anc 1368 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍)) |
26 | 23, 25 | mpbid 234 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ran crn 5558 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 GrpOpcgr 28268 GIdcgi 28269 RingOpscrngo 35174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-riota 7116 df-ov 7161 df-1st 7691 df-2nd 7692 df-grpo 28272 df-gid 28273 df-ablo 28324 df-rngo 35175 |
This theorem is referenced by: rngoueqz 35220 rngonegmn1r 35222 zerdivemp1x 35227 0idl 35305 keridl 35312 |
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