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Mirrors > Home > MPE Home > Th. List > rngm2neg | Structured version Visualization version GIF version |
Description: Double negation of a product in a non-unital ring (mul2neg 11683 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 20246. (Revised by AV, 17-Feb-2025.) |
Ref | Expression |
---|---|
rngneglmul.b | โข ๐ต = (Baseโ๐ ) |
rngneglmul.t | โข ยท = (.rโ๐ ) |
rngneglmul.n | โข ๐ = (invgโ๐ ) |
rngneglmul.r | โข (๐ โ ๐ โ Rng) |
rngneglmul.x | โข (๐ โ ๐ โ ๐ต) |
rngneglmul.y | โข (๐ โ ๐ โ ๐ต) |
Ref | Expression |
---|---|
rngm2neg | โข (๐ โ ((๐โ๐) ยท (๐โ๐)) = (๐ ยท ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngneglmul.b | . . 3 โข ๐ต = (Baseโ๐ ) | |
2 | rngneglmul.t | . . 3 โข ยท = (.rโ๐ ) | |
3 | rngneglmul.n | . . 3 โข ๐ = (invgโ๐ ) | |
4 | rngneglmul.r | . . 3 โข (๐ โ ๐ โ Rng) | |
5 | rngneglmul.x | . . 3 โข (๐ โ ๐ โ ๐ต) | |
6 | rnggrp 20102 | . . . . 5 โข (๐ โ Rng โ ๐ โ Grp) | |
7 | 4, 6 | syl 17 | . . . 4 โข (๐ โ ๐ โ Grp) |
8 | rngneglmul.y | . . . 4 โข (๐ โ ๐ โ ๐ต) | |
9 | 1, 3, 7, 8 | grpinvcld 18949 | . . 3 โข (๐ โ (๐โ๐) โ ๐ต) |
10 | 1, 2, 3, 4, 5, 9 | rngmneg1 20111 | . 2 โข (๐ โ ((๐โ๐) ยท (๐โ๐)) = (๐โ(๐ ยท (๐โ๐)))) |
11 | 1, 2, 3, 4, 5, 8 | rngmneg2 20112 | . . 3 โข (๐ โ (๐ ยท (๐โ๐)) = (๐โ(๐ ยท ๐))) |
12 | 11 | fveq2d 6896 | . 2 โข (๐ โ (๐โ(๐ ยท (๐โ๐))) = (๐โ(๐โ(๐ ยท ๐)))) |
13 | 1, 2 | rngcl 20108 | . . . 4 โข ((๐ โ Rng โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
14 | 4, 5, 8, 13 | syl3anc 1368 | . . 3 โข (๐ โ (๐ ยท ๐) โ ๐ต) |
15 | 1, 3 | grpinvinv 18966 | . . 3 โข ((๐ โ Grp โง (๐ ยท ๐) โ ๐ต) โ (๐โ(๐โ(๐ ยท ๐))) = (๐ ยท ๐)) |
16 | 7, 14, 15 | syl2anc 582 | . 2 โข (๐ โ (๐โ(๐โ(๐ ยท ๐))) = (๐ ยท ๐)) |
17 | 10, 12, 16 | 3eqtrd 2769 | 1 โข (๐ โ ((๐โ๐) ยท (๐โ๐)) = (๐ ยท ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โcfv 6543 (class class class)co 7416 Basecbs 17179 .rcmulr 17233 Grpcgrp 18894 invgcminusg 18895 Rngcrng 20096 |
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-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 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-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-abl 19742 df-mgp 20079 df-rng 20097 |
This theorem is referenced by: ringm2neg 20246 |
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