![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
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 37383 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ringlz.2 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
4 | ringlz.1 | . . . . . . 7 ⊢ 𝑍 = (GId‘𝐺) | |
5 | 3, 4 | grpoidcl 30337 | . . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋) |
6 | 3, 4 | grpolid 30339 | . . . . . 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 7436 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻(𝑍𝐺𝑍)) = (𝐴𝐻𝑍)) |
10 | simpr 484 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
11 | 1, 3, 4 | rngo0cl 37392 | . . . . . 6 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋) |
12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → 𝑍 ∈ 𝑋) |
13 | 10, 12, 12 | 3jca 1126 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋)) |
14 | ringlz.4 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
15 | 1, 14, 3 | rngodi 37377 | . . . 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 37374 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋) |
19 | 12, 18 | mpd3an3 1459 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑍) ∈ 𝑋) |
20 | 3, 4 | grpolid 30339 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝑍) ∈ 𝑋) → (𝑍𝐺(𝐴𝐻𝑍)) = (𝐴𝐻𝑍)) |
21 | 20 | eqcomd 2734 | . . . 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 2776 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍))) |
24 | 3 | grporcan 30341 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐻𝑍) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ (𝐴𝐻𝑍) ∈ 𝑋)) → (((𝐴𝐻𝑍)𝐺(𝐴𝐻𝑍)) = (𝑍𝐺(𝐴𝐻𝑍)) ↔ (𝐴𝐻𝑍) = 𝑍)) |
25 | 17, 19, 12, 19, 24 | syl13anc 1370 | . 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 1085 = wceq 1534 ∈ wcel 2099 ran crn 5679 ‘cfv 6548 (class class class)co 7420 1st c1st 7991 2nd c2nd 7992 GrpOpcgr 30312 GIdcgi 30313 RingOpscrngo 37367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fo 6554 df-fv 6556 df-riota 7376 df-ov 7423 df-1st 7993 df-2nd 7994 df-grpo 30316 df-gid 30317 df-ablo 30368 df-rngo 37368 |
This theorem is referenced by: rngoueqz 37413 rngonegmn1r 37415 zerdivemp1x 37420 0idl 37498 keridl 37505 |
Copyright terms: Public domain | W3C validator |