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Mirrors > Home > HSE Home > Th. List > hv2neg | Structured version Visualization version GIF version |
Description: Two ways to express the negative of a vector. (Contributed by NM, 23-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hv2neg | โข (๐ด โ โ โ (0โ โโ ๐ด) = (-1 ยทโ ๐ด)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 29406 | . . 3 โข 0โ โ โ | |
2 | hvsubval 29419 | . . 3 โข ((0โ โ โ โง ๐ด โ โ) โ (0โ โโ ๐ด) = (0โ +โ (-1 ยทโ ๐ด))) | |
3 | 1, 2 | mpan 688 | . 2 โข (๐ด โ โ โ (0โ โโ ๐ด) = (0โ +โ (-1 ยทโ ๐ด))) |
4 | neg1cn 12129 | . . . 4 โข -1 โ โ | |
5 | hvmulcl 29416 | . . . 4 โข ((-1 โ โ โง ๐ด โ โ) โ (-1 ยทโ ๐ด) โ โ) | |
6 | 4, 5 | mpan 688 | . . 3 โข (๐ด โ โ โ (-1 ยทโ ๐ด) โ โ) |
7 | hvaddid2 29426 | . . 3 โข ((-1 ยทโ ๐ด) โ โ โ (0โ +โ (-1 ยทโ ๐ด)) = (-1 ยทโ ๐ด)) | |
8 | 6, 7 | syl 17 | . 2 โข (๐ด โ โ โ (0โ +โ (-1 ยทโ ๐ด)) = (-1 ยทโ ๐ด)) |
9 | 3, 8 | eqtrd 2776 | 1 โข (๐ด โ โ โ (0โ โโ ๐ด) = (-1 ยทโ ๐ด)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1539 โ wcel 2104 (class class class)co 7303 โcc 10911 1c1 10914 -cneg 11248 โchba 29322 +โ cva 29323 ยทโ csm 29324 0โc0v 29327 โโ cmv 29328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-hvcom 29404 ax-hv0cl 29406 ax-hvaddid 29407 ax-hfvmul 29408 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5496 df-po 5510 df-so 5511 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-pnf 11053 df-mnf 11054 df-ltxr 11056 df-sub 11249 df-neg 11250 df-hvsub 29374 |
This theorem is referenced by: hv2negi 29434 normneg 29547 |
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