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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoaddneg1 | Structured version Visualization version GIF version |
Description: Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
ringnegcl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ringnegcl.2 | ⊢ 𝑋 = ran 𝐺 |
ringnegcl.3 | ⊢ 𝑁 = (inv‘𝐺) |
ringaddneg.4 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
rngoaddneg1 | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringnegcl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 34627 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ringnegcl.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | ringaddneg.4 | . . 3 ⊢ 𝑍 = (GId‘𝐺) | |
5 | ringnegcl.3 | . . 3 ⊢ 𝑁 = (inv‘𝐺) | |
6 | 3, 4, 5 | grporinv 28081 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑍) |
7 | 2, 6 | sylan 572 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ran crn 5408 ‘cfv 6188 (class class class)co 6976 1st c1st 7499 GrpOpcgr 28043 GIdcgi 28044 invcgn 28045 RingOpscrngo 34611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-1st 7501 df-2nd 7502 df-grpo 28047 df-gid 28048 df-ginv 28049 df-ablo 28099 df-rngo 34612 |
This theorem is referenced by: rngonegmn1l 34658 |
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