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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 37439 | . 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 30379 | . 2 β’ ((πΊ β GrpOp β§ π΄ β π) β (π΄πΊ(πβπ΄)) = π) |
| 7 | 2, 6 | sylan 578 | 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 394 = wceq 1533 β wcel 2098 ran crn 5673 βcfv 6542 (class class class)co 7415 1st c1st 7987 GrpOpcgr 30341 GIdcgi 30342 invcgn 30343 RingOpscrngo 37423 |
| 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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7737 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-1st 7989 df-2nd 7990 df-grpo 30345 df-gid 30346 df-ginv 30347 df-ablo 30397 df-rngo 37424 |
| This theorem is referenced by: rngonegmn1l 37470 |
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