![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ringmneg1 | Structured version Visualization version GIF version |
Description: Negation of a product in a ring. (mulneg1 11651 analog.) Compared with rngmneg1 20070, the proof is shorter making use of the existence of a ring unity. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
Ref | Expression |
---|---|
ringneglmul.b | โข ๐ต = (Baseโ๐ ) |
ringneglmul.t | โข ยท = (.rโ๐ ) |
ringneglmul.n | โข ๐ = (invgโ๐ ) |
ringneglmul.r | โข (๐ โ ๐ โ Ring) |
ringneglmul.x | โข (๐ โ ๐ โ ๐ต) |
ringneglmul.y | โข (๐ โ ๐ โ ๐ต) |
Ref | Expression |
---|---|
ringmneg1 | โข (๐ โ ((๐โ๐) ยท ๐) = (๐โ(๐ ยท ๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringneglmul.r | . . 3 โข (๐ โ ๐ โ Ring) | |
2 | ringgrp 20141 | . . . . 5 โข (๐ โ Ring โ ๐ โ Grp) | |
3 | 1, 2 | syl 17 | . . . 4 โข (๐ โ ๐ โ Grp) |
4 | ringneglmul.b | . . . . . 6 โข ๐ต = (Baseโ๐ ) | |
5 | eqid 2726 | . . . . . 6 โข (1rโ๐ ) = (1rโ๐ ) | |
6 | 4, 5 | ringidcl 20163 | . . . . 5 โข (๐ โ Ring โ (1rโ๐ ) โ ๐ต) |
7 | 1, 6 | syl 17 | . . . 4 โข (๐ โ (1rโ๐ ) โ ๐ต) |
8 | ringneglmul.n | . . . . 5 โข ๐ = (invgโ๐ ) | |
9 | 4, 8 | grpinvcl 18915 | . . . 4 โข ((๐ โ Grp โง (1rโ๐ ) โ ๐ต) โ (๐โ(1rโ๐ )) โ ๐ต) |
10 | 3, 7, 9 | syl2anc 583 | . . 3 โข (๐ โ (๐โ(1rโ๐ )) โ ๐ต) |
11 | ringneglmul.x | . . 3 โข (๐ โ ๐ โ ๐ต) | |
12 | ringneglmul.y | . . 3 โข (๐ โ ๐ โ ๐ต) | |
13 | ringneglmul.t | . . . 4 โข ยท = (.rโ๐ ) | |
14 | 4, 13 | ringass 20156 | . . 3 โข ((๐ โ Ring โง ((๐โ(1rโ๐ )) โ ๐ต โง ๐ โ ๐ต โง ๐ โ ๐ต)) โ (((๐โ(1rโ๐ )) ยท ๐) ยท ๐) = ((๐โ(1rโ๐ )) ยท (๐ ยท ๐))) |
15 | 1, 10, 11, 12, 14 | syl13anc 1369 | . 2 โข (๐ โ (((๐โ(1rโ๐ )) ยท ๐) ยท ๐) = ((๐โ(1rโ๐ )) ยท (๐ ยท ๐))) |
16 | 4, 13, 5, 8, 1, 11 | ringnegl 20199 | . . 3 โข (๐ โ ((๐โ(1rโ๐ )) ยท ๐) = (๐โ๐)) |
17 | 16 | oveq1d 7419 | . 2 โข (๐ โ (((๐โ(1rโ๐ )) ยท ๐) ยท ๐) = ((๐โ๐) ยท ๐)) |
18 | 4, 13 | ringcl 20153 | . . . 4 โข ((๐ โ Ring โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
19 | 1, 11, 12, 18 | syl3anc 1368 | . . 3 โข (๐ โ (๐ ยท ๐) โ ๐ต) |
20 | 4, 13, 5, 8, 1, 19 | ringnegl 20199 | . 2 โข (๐ โ ((๐โ(1rโ๐ )) ยท (๐ ยท ๐)) = (๐โ(๐ ยท ๐))) |
21 | 15, 17, 20 | 3eqtr3d 2774 | 1 โข (๐ โ ((๐โ๐) ยท ๐) = (๐โ(๐ ยท ๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โcfv 6536 (class class class)co 7404 Basecbs 17151 .rcmulr 17205 Grpcgrp 18861 invgcminusg 18862 1rcur 20084 Ringcrg 20136 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 |
This theorem is referenced by: mulgass2 20206 cntzsubr 20506 mdetunilem7 22471 r1padd1 33183 |
Copyright terms: Public domain | W3C validator |