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Mirrors > Home > MPE Home > Th. List > ringm2neg | Structured version Visualization version GIF version |
Description: Double negation of a product in a ring. (mul2neg 11602 analog.) (Contributed by Mario Carneiro, 4-Dec-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 |
---|---|
ringm2neg | โข (๐ โ ((๐โ๐) ยท (๐โ๐)) = (๐ ยท ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringneglmul.b | . . 3 โข ๐ต = (Baseโ๐ ) | |
2 | ringneglmul.t | . . 3 โข ยท = (.rโ๐ ) | |
3 | ringneglmul.n | . . 3 โข ๐ = (invgโ๐ ) | |
4 | ringneglmul.r | . . 3 โข (๐ โ ๐ โ Ring) | |
5 | ringneglmul.x | . . 3 โข (๐ โ ๐ โ ๐ต) | |
6 | ringgrp 19977 | . . . . 5 โข (๐ โ Ring โ ๐ โ Grp) | |
7 | 4, 6 | syl 17 | . . . 4 โข (๐ โ ๐ โ Grp) |
8 | ringneglmul.y | . . . 4 โข (๐ โ ๐ โ ๐ต) | |
9 | 1, 3 | grpinvcl 18806 | . . . 4 โข ((๐ โ Grp โง ๐ โ ๐ต) โ (๐โ๐) โ ๐ต) |
10 | 7, 8, 9 | syl2anc 585 | . . 3 โข (๐ โ (๐โ๐) โ ๐ต) |
11 | 1, 2, 3, 4, 5, 10 | ringmneg1 20028 | . 2 โข (๐ โ ((๐โ๐) ยท (๐โ๐)) = (๐โ(๐ ยท (๐โ๐)))) |
12 | 1, 2, 3, 4, 5, 8 | ringmneg2 20029 | . . 3 โข (๐ โ (๐ ยท (๐โ๐)) = (๐โ(๐ ยท ๐))) |
13 | 12 | fveq2d 6850 | . 2 โข (๐ โ (๐โ(๐ ยท (๐โ๐))) = (๐โ(๐โ(๐ ยท ๐)))) |
14 | 1, 2 | ringcl 19989 | . . . 4 โข ((๐ โ Ring โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
15 | 4, 5, 8, 14 | syl3anc 1372 | . . 3 โข (๐ โ (๐ ยท ๐) โ ๐ต) |
16 | 1, 3 | grpinvinv 18822 | . . 3 โข ((๐ โ Grp โง (๐ ยท ๐) โ ๐ต) โ (๐โ(๐โ(๐ ยท ๐))) = (๐ ยท ๐)) |
17 | 7, 15, 16 | syl2anc 585 | . 2 โข (๐ โ (๐โ(๐โ(๐ ยท ๐))) = (๐ ยท ๐)) |
18 | 11, 13, 17 | 3eqtrd 2777 | 1 โข (๐ โ ((๐โ๐) ยท (๐โ๐)) = (๐ ยท ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โcfv 6500 (class class class)co 7361 Basecbs 17091 .rcmulr 17142 Grpcgrp 18756 invgcminusg 18757 Ringcrg 19972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-mgp 19905 df-ur 19922 df-ring 19974 |
This theorem is referenced by: orngsqr 32153 |
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