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Mirrors > Home > MPE Home > Th. List > clmvz | Structured version Visualization version GIF version |
Description: Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (Revised by AV, 7-Oct-2021.) |
Ref | Expression |
---|---|
clmvz.v | β’ π = (Baseβπ) |
clmvz.m | β’ β = (-gβπ) |
clmvz.s | β’ Β· = ( Β·π βπ) |
clmvz.0 | β’ 0 = (0gβπ) |
Ref | Expression |
---|---|
clmvz | β’ ((π β βMod β§ π΄ β π) β ( 0 β π΄) = (-1 Β· π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . 3 β’ ((π β βMod β§ π΄ β π) β π β βMod) | |
2 | clmgrp 24939 | . . . . 5 β’ (π β βMod β π β Grp) | |
3 | clmvz.v | . . . . . 6 β’ π = (Baseβπ) | |
4 | clmvz.0 | . . . . . 6 β’ 0 = (0gβπ) | |
5 | 3, 4 | grpidcl 18891 | . . . . 5 β’ (π β Grp β 0 β π) |
6 | 2, 5 | syl 17 | . . . 4 β’ (π β βMod β 0 β π) |
7 | 6 | adantr 480 | . . 3 β’ ((π β βMod β§ π΄ β π) β 0 β π) |
8 | simpr 484 | . . 3 β’ ((π β βMod β§ π΄ β π) β π΄ β π) | |
9 | eqid 2724 | . . . 4 β’ (+gβπ) = (+gβπ) | |
10 | clmvz.m | . . . 4 β’ β = (-gβπ) | |
11 | eqid 2724 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
12 | clmvz.s | . . . 4 β’ Β· = ( Β·π βπ) | |
13 | 3, 9, 10, 11, 12 | clmvsubval2 24981 | . . 3 β’ ((π β βMod β§ 0 β π β§ π΄ β π) β ( 0 β π΄) = ((-1 Β· π΄)(+gβπ) 0 )) |
14 | 1, 7, 8, 13 | syl3anc 1368 | . 2 β’ ((π β βMod β§ π΄ β π) β ( 0 β π΄) = ((-1 Β· π΄)(+gβπ) 0 )) |
15 | eqid 2724 | . . . . . 6 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
16 | 11, 15 | clmneg1 24953 | . . . . 5 β’ (π β βMod β -1 β (Baseβ(Scalarβπ))) |
17 | 16 | adantr 480 | . . . 4 β’ ((π β βMod β§ π΄ β π) β -1 β (Baseβ(Scalarβπ))) |
18 | 3, 11, 12, 15 | clmvscl 24959 | . . . 4 β’ ((π β βMod β§ -1 β (Baseβ(Scalarβπ)) β§ π΄ β π) β (-1 Β· π΄) β π) |
19 | 1, 17, 8, 18 | syl3anc 1368 | . . 3 β’ ((π β βMod β§ π΄ β π) β (-1 Β· π΄) β π) |
20 | 3, 9, 4 | grprid 18894 | . . 3 β’ ((π β Grp β§ (-1 Β· π΄) β π) β ((-1 Β· π΄)(+gβπ) 0 ) = (-1 Β· π΄)) |
21 | 2, 19, 20 | syl2an2r 682 | . 2 β’ ((π β βMod β§ π΄ β π) β ((-1 Β· π΄)(+gβπ) 0 ) = (-1 Β· π΄)) |
22 | 14, 21 | eqtrd 2764 | 1 β’ ((π β βMod β§ π΄ β π) β ( 0 β π΄) = (-1 Β· π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 1c1 11108 -cneg 11444 Basecbs 17149 +gcplusg 17202 Scalarcsca 17205 Β·π cvsca 17206 0gc0g 17390 Grpcgrp 18859 -gcsg 18861 βModcclm 24933 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-seq 13968 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-ur 20083 df-ring 20136 df-cring 20137 df-subrg 20467 df-lmod 20704 df-cnfld 21235 df-clm 24934 |
This theorem is referenced by: (None) |
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