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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 | β’ πΊ = (1st βπ ) |
| ringnegcl.2 | β’ π = ran πΊ |
| ringnegcl.3 | β’ π = (invβπΊ) |
| ringaddneg.4 | β’ π = (GIdβπΊ) |
| Ref | Expression |
|---|---|
| rngoaddneg1 | β’ ((π β RingOps β§ π΄ β π) β (π΄πΊ(πβπ΄)) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ringnegcl.1 | . . 3 β’ πΊ = (1st βπ ) | |
| 2 | 1 | rngogrpo 36778 | . 2 β’ (π β RingOps β πΊ β GrpOp) |
| 3 | ringnegcl.2 | . . 3 β’ π = ran πΊ | |
| 4 | ringaddneg.4 | . . 3 β’ π = (GIdβπΊ) | |
| 5 | ringnegcl.3 | . . 3 β’ π = (invβπΊ) | |
| 6 | 3, 4, 5 | grporinv 29780 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π) β (π΄πΊ(πβπ΄)) = π) |
| 7 | 2, 6 | sylan 581 | 1 β’ ((π β RingOps β§ π΄ β π) β (π΄πΊ(πβπ΄)) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 ran crn 5678 βcfv 6544 (class class class)co 7409 1st c1st 7973 GrpOpcgr 29742 GIdcgi 29743 invcgn 29744 RingOpscrngo 36762 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-1st 7975 df-2nd 7976 df-grpo 29746 df-gid 29747 df-ginv 29748 df-ablo 29798 df-rngo 36763 |
| This theorem is referenced by: rngonegmn1l 36809 |
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