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Mirrors > Home > MPE Home > Th. List > ringmneg2 | Structured version Visualization version GIF version |
Description: Negation of a product in a ring. (mulneg2 11689 analog.) Compared with rngmneg2 20115, 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 |
---|---|
ringmneg2 | โข (๐ โ (๐ ยท (๐โ๐)) = (๐โ(๐ ยท ๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringneglmul.r | . . 3 โข (๐ โ ๐ โ Ring) | |
2 | ringneglmul.x | . . 3 โข (๐ โ ๐ โ ๐ต) | |
3 | ringneglmul.y | . . 3 โข (๐ โ ๐ โ ๐ต) | |
4 | ringgrp 20185 | . . . . 5 โข (๐ โ Ring โ ๐ โ Grp) | |
5 | 1, 4 | syl 17 | . . . 4 โข (๐ โ ๐ โ Grp) |
6 | ringneglmul.b | . . . . . 6 โข ๐ต = (Baseโ๐ ) | |
7 | eqid 2728 | . . . . . 6 โข (1rโ๐ ) = (1rโ๐ ) | |
8 | 6, 7 | ringidcl 20209 | . . . . 5 โข (๐ โ Ring โ (1rโ๐ ) โ ๐ต) |
9 | 1, 8 | syl 17 | . . . 4 โข (๐ โ (1rโ๐ ) โ ๐ต) |
10 | ringneglmul.n | . . . . 5 โข ๐ = (invgโ๐ ) | |
11 | 6, 10 | grpinvcl 18951 | . . . 4 โข ((๐ โ Grp โง (1rโ๐ ) โ ๐ต) โ (๐โ(1rโ๐ )) โ ๐ต) |
12 | 5, 9, 11 | syl2anc 582 | . . 3 โข (๐ โ (๐โ(1rโ๐ )) โ ๐ต) |
13 | ringneglmul.t | . . . 4 โข ยท = (.rโ๐ ) | |
14 | 6, 13 | ringass 20200 | . . 3 โข ((๐ โ Ring โง (๐ โ ๐ต โง ๐ โ ๐ต โง (๐โ(1rโ๐ )) โ ๐ต)) โ ((๐ ยท ๐) ยท (๐โ(1rโ๐ ))) = (๐ ยท (๐ ยท (๐โ(1rโ๐ ))))) |
15 | 1, 2, 3, 12, 14 | syl13anc 1369 | . 2 โข (๐ โ ((๐ ยท ๐) ยท (๐โ(1rโ๐ ))) = (๐ ยท (๐ ยท (๐โ(1rโ๐ ))))) |
16 | 6, 13 | ringcl 20197 | . . . 4 โข ((๐ โ Ring โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
17 | 1, 2, 3, 16 | syl3anc 1368 | . . 3 โข (๐ โ (๐ ยท ๐) โ ๐ต) |
18 | 6, 13, 7, 10, 1, 17 | ringnegr 20246 | . 2 โข (๐ โ ((๐ ยท ๐) ยท (๐โ(1rโ๐ ))) = (๐โ(๐ ยท ๐))) |
19 | 6, 13, 7, 10, 1, 3 | ringnegr 20246 | . . 3 โข (๐ โ (๐ ยท (๐โ(1rโ๐ ))) = (๐โ๐)) |
20 | 19 | oveq2d 7442 | . 2 โข (๐ โ (๐ ยท (๐ ยท (๐โ(1rโ๐ )))) = (๐ ยท (๐โ๐))) |
21 | 15, 18, 20 | 3eqtr3rd 2777 | 1 โข (๐ โ (๐ ยท (๐โ๐)) = (๐โ(๐ ยท ๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โcfv 6553 (class class class)co 7426 Basecbs 17187 .rcmulr 17241 Grpcgrp 18897 invgcminusg 18898 1rcur 20128 Ringcrg 20180 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 |
This theorem is referenced by: cntzsubr 20552 abvneg 20721 erler 33004 lflnegcl 38579 |
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