![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > hvaddcani | Structured version Visualization version GIF version |
Description: Cancellation law for vector 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 |
---|---|
hvaddcani | โข ((๐ด +โ ๐ต) = (๐ด +โ ๐ถ) โ ๐ต = ๐ถ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7427 | . . 3 โข ((๐ด +โ ๐ต) = (๐ด +โ ๐ถ) โ ((๐ด +โ ๐ต) +โ (-1 ยทโ ๐ด)) = ((๐ด +โ ๐ถ) +โ (-1 ยทโ ๐ด))) | |
2 | hvnegdi.1 | . . . . 5 โข ๐ด โ โ | |
3 | hvnegdi.2 | . . . . 5 โข ๐ต โ โ | |
4 | neg1cn 12357 | . . . . . 6 โข -1 โ โ | |
5 | 4, 2 | hvmulcli 30837 | . . . . 5 โข (-1 ยทโ ๐ด) โ โ |
6 | 2, 3, 5 | hvadd32i 30877 | . . . 4 โข ((๐ด +โ ๐ต) +โ (-1 ยทโ ๐ด)) = ((๐ด +โ (-1 ยทโ ๐ด)) +โ ๐ต) |
7 | 2 | hvnegidi 30853 | . . . . 5 โข (๐ด +โ (-1 ยทโ ๐ด)) = 0โ |
8 | 7 | oveq1i 7430 | . . . 4 โข ((๐ด +โ (-1 ยทโ ๐ด)) +โ ๐ต) = (0โ +โ ๐ต) |
9 | 3 | hvaddlidi 30852 | . . . 4 โข (0โ +โ ๐ต) = ๐ต |
10 | 6, 8, 9 | 3eqtri 2760 | . . 3 โข ((๐ด +โ ๐ต) +โ (-1 ยทโ ๐ด)) = ๐ต |
11 | hvaddcan.3 | . . . . 5 โข ๐ถ โ โ | |
12 | 2, 11, 5 | hvadd32i 30877 | . . . 4 โข ((๐ด +โ ๐ถ) +โ (-1 ยทโ ๐ด)) = ((๐ด +โ (-1 ยทโ ๐ด)) +โ ๐ถ) |
13 | 7 | oveq1i 7430 | . . . 4 โข ((๐ด +โ (-1 ยทโ ๐ด)) +โ ๐ถ) = (0โ +โ ๐ถ) |
14 | 11 | hvaddlidi 30852 | . . . 4 โข (0โ +โ ๐ถ) = ๐ถ |
15 | 12, 13, 14 | 3eqtri 2760 | . . 3 โข ((๐ด +โ ๐ถ) +โ (-1 ยทโ ๐ด)) = ๐ถ |
16 | 1, 10, 15 | 3eqtr3g 2791 | . 2 โข ((๐ด +โ ๐ต) = (๐ด +โ ๐ถ) โ ๐ต = ๐ถ) |
17 | oveq2 7428 | . 2 โข (๐ต = ๐ถ โ (๐ด +โ ๐ต) = (๐ด +โ ๐ถ)) | |
18 | 16, 17 | impbii 208 | 1 โข ((๐ด +โ ๐ต) = (๐ด +โ ๐ถ) โ ๐ต = ๐ถ) |
Colors of variables: wff setvar class |
Syntax hints: โ wb 205 = wceq 1534 โ wcel 2099 (class class class)co 7420 1c1 11140 -cneg 11476 โchba 30742 +โ cva 30743 ยทโ csm 30744 0โc0v 30747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-hvcom 30824 ax-hvass 30825 ax-hv0cl 30826 ax-hvaddid 30827 ax-hfvmul 30828 ax-hvmulid 30829 ax-hvdistr2 30832 ax-hvmul0 30833 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-sub 11477 df-neg 11478 df-hvsub 30794 |
This theorem is referenced by: hvsubaddi 30889 hvaddcan 30893 |
Copyright terms: Public domain | W3C validator |