Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rngonegmn1l | Structured version Visualization version Unicode version |
Description: Negation in a ring is the same as left multiplication by . (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
ringneg.1 | |
ringneg.2 | |
ringneg.3 | |
ringneg.4 | |
ringneg.5 | GId |
Ref | Expression |
---|---|
rngonegmn1l |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringneg.3 | . . . . . . 7 | |
2 | ringneg.1 | . . . . . . . 8 | |
3 | 2 | rneqi 5352 | . . . . . . 7 |
4 | 1, 3 | eqtri 2644 | . . . . . 6 |
5 | ringneg.2 | . . . . . 6 | |
6 | ringneg.5 | . . . . . 6 GId | |
7 | 4, 5, 6 | rngo1cl 33738 | . . . . 5 |
8 | ringneg.4 | . . . . . . 7 | |
9 | 2, 1, 8 | rngonegcl 33726 | . . . . . 6 |
10 | 7, 9 | mpdan 702 | . . . . 5 |
11 | 7, 10 | jca 554 | . . . 4 |
12 | 2, 5, 1 | rngodir 33704 | . . . . . . 7 |
13 | 12 | 3exp2 1285 | . . . . . 6 |
14 | 13 | imp42 620 | . . . . 5 |
15 | 14 | an32s 846 | . . . 4 |
16 | 11, 15 | mpidan 704 | . . 3 |
17 | eqid 2622 | . . . . . . . 8 GId GId | |
18 | 2, 1, 8, 17 | rngoaddneg1 33727 | . . . . . . 7 GId |
19 | 7, 18 | mpdan 702 | . . . . . 6 GId |
20 | 19 | adantr 481 | . . . . 5 GId |
21 | 20 | oveq1d 6665 | . . . 4 GId |
22 | 17, 1, 2, 5 | rngolz 33721 | . . . 4 GId GId |
23 | 21, 22 | eqtrd 2656 | . . 3 GId |
24 | 5, 4, 6 | rngolidm 33736 | . . . 4 |
25 | 24 | oveq1d 6665 | . . 3 |
26 | 16, 23, 25 | 3eqtr3rd 2665 | . 2 GId |
27 | 2, 5, 1 | rngocl 33700 | . . . . . 6 |
28 | 27 | 3expa 1265 | . . . . 5 |
29 | 28 | an32s 846 | . . . 4 |
30 | 10, 29 | mpidan 704 | . . 3 |
31 | 2 | rngogrpo 33709 | . . . 4 |
32 | 1, 17, 8 | grpoinvid1 27382 | . . . 4 GId |
33 | 31, 32 | syl3an1 1359 | . . 3 GId |
34 | 30, 33 | mpd3an3 1425 | . 2 GId |
35 | 26, 34 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 crn 5115 cfv 5888 (class class class)co 6650 c1st 7166 c2nd 7167 cgr 27343 GIdcgi 27344 cgn 27345 crngo 33693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-1st 7168 df-2nd 7169 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-ass 33642 df-exid 33644 df-mgmOLD 33648 df-sgrOLD 33660 df-mndo 33666 df-rngo 33694 |
This theorem is referenced by: rngoneglmul 33742 idlnegcl 33821 |
Copyright terms: Public domain | W3C validator |