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Mirrors > Home > HSE Home > Th. List > hvsubaddi | Structured version Visualization version GIF version |
Description: Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvnegdi.1 | โข ๐ด โ โ |
hvnegdi.2 | โข ๐ต โ โ |
hvaddcan.3 | โข ๐ถ โ โ |
Ref | Expression |
---|---|
hvsubaddi | โข ((๐ด โโ ๐ต) = ๐ถ โ (๐ต +โ ๐ถ) = ๐ด) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvnegdi.1 | . . . 4 โข ๐ด โ โ | |
2 | hvnegdi.2 | . . . 4 โข ๐ต โ โ | |
3 | 1, 2 | hvsubvali 29427 | . . 3 โข (๐ด โโ ๐ต) = (๐ด +โ (-1 ยทโ ๐ต)) |
4 | 3 | eqeq1i 2741 | . 2 โข ((๐ด โโ ๐ต) = ๐ถ โ (๐ด +โ (-1 ยทโ ๐ต)) = ๐ถ) |
5 | neg1cn 12133 | . . . . . . 7 โข -1 โ โ | |
6 | 5, 2 | hvmulcli 29421 | . . . . . 6 โข (-1 ยทโ ๐ต) โ โ |
7 | 2, 1, 6 | hvadd12i 29464 | . . . . 5 โข (๐ต +โ (๐ด +โ (-1 ยทโ ๐ต))) = (๐ด +โ (๐ต +โ (-1 ยทโ ๐ต))) |
8 | 2 | hvnegidi 29437 | . . . . . 6 โข (๐ต +โ (-1 ยทโ ๐ต)) = 0โ |
9 | 8 | oveq2i 7318 | . . . . 5 โข (๐ด +โ (๐ต +โ (-1 ยทโ ๐ต))) = (๐ด +โ 0โ) |
10 | ax-hvaddid 29411 | . . . . . 6 โข (๐ด โ โ โ (๐ด +โ 0โ) = ๐ด) | |
11 | 1, 10 | ax-mp 5 | . . . . 5 โข (๐ด +โ 0โ) = ๐ด |
12 | 7, 9, 11 | 3eqtri 2768 | . . . 4 โข (๐ต +โ (๐ด +โ (-1 ยทโ ๐ต))) = ๐ด |
13 | 12 | eqeq1i 2741 | . . 3 โข ((๐ต +โ (๐ด +โ (-1 ยทโ ๐ต))) = (๐ต +โ ๐ถ) โ ๐ด = (๐ต +โ ๐ถ)) |
14 | 1, 6 | hvaddcli 29425 | . . . 4 โข (๐ด +โ (-1 ยทโ ๐ต)) โ โ |
15 | hvaddcan.3 | . . . 4 โข ๐ถ โ โ | |
16 | 2, 14, 15 | hvaddcani 29472 | . . 3 โข ((๐ต +โ (๐ด +โ (-1 ยทโ ๐ต))) = (๐ต +โ ๐ถ) โ (๐ด +โ (-1 ยทโ ๐ต)) = ๐ถ) |
17 | eqcom 2743 | . . 3 โข (๐ด = (๐ต +โ ๐ถ) โ (๐ต +โ ๐ถ) = ๐ด) | |
18 | 13, 16, 17 | 3bitr3i 301 | . 2 โข ((๐ด +โ (-1 ยทโ ๐ต)) = ๐ถ โ (๐ต +โ ๐ถ) = ๐ด) |
19 | 4, 18 | bitri 275 | 1 โข ((๐ด โโ ๐ต) = ๐ถ โ (๐ต +โ ๐ถ) = ๐ด) |
Colors of variables: wff setvar class |
Syntax hints: โ wb 205 = wceq 1539 โ wcel 2104 (class class class)co 7307 1c1 10918 -cneg 11252 โchba 29326 +โ cva 29327 ยทโ csm 29328 0โc0v 29331 โโ cmv 29332 |
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 7620 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-hfvadd 29407 ax-hvcom 29408 ax-hvass 29409 ax-hv0cl 29410 ax-hvaddid 29411 ax-hfvmul 29412 ax-hvmulid 29413 ax-hvdistr2 29416 ax-hvmul0 29417 |
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 5500 df-po 5514 df-so 5515 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-pnf 11057 df-mnf 11058 df-ltxr 11060 df-sub 11253 df-neg 11254 df-hvsub 29378 |
This theorem is referenced by: hvsubadd 29484 omlsilem 29809 |
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