![]() |
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 7411 | . . 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 12327 | . . . . . 6 โข -1 โ โ | |
5 | 4, 2 | hvmulcli 30772 | . . . . 5 โข (-1 ยทโ ๐ด) โ โ |
6 | 2, 3, 5 | hvadd32i 30812 | . . . 4 โข ((๐ด +โ ๐ต) +โ (-1 ยทโ ๐ด)) = ((๐ด +โ (-1 ยทโ ๐ด)) +โ ๐ต) |
7 | 2 | hvnegidi 30788 | . . . . 5 โข (๐ด +โ (-1 ยทโ ๐ด)) = 0โ |
8 | 7 | oveq1i 7414 | . . . 4 โข ((๐ด +โ (-1 ยทโ ๐ด)) +โ ๐ต) = (0โ +โ ๐ต) |
9 | 3 | hvaddlidi 30787 | . . . 4 โข (0โ +โ ๐ต) = ๐ต |
10 | 6, 8, 9 | 3eqtri 2758 | . . 3 โข ((๐ด +โ ๐ต) +โ (-1 ยทโ ๐ด)) = ๐ต |
11 | hvaddcan.3 | . . . . 5 โข ๐ถ โ โ | |
12 | 2, 11, 5 | hvadd32i 30812 | . . . 4 โข ((๐ด +โ ๐ถ) +โ (-1 ยทโ ๐ด)) = ((๐ด +โ (-1 ยทโ ๐ด)) +โ ๐ถ) |
13 | 7 | oveq1i 7414 | . . . 4 โข ((๐ด +โ (-1 ยทโ ๐ด)) +โ ๐ถ) = (0โ +โ ๐ถ) |
14 | 11 | hvaddlidi 30787 | . . . 4 โข (0โ +โ ๐ถ) = ๐ถ |
15 | 12, 13, 14 | 3eqtri 2758 | . . 3 โข ((๐ด +โ ๐ถ) +โ (-1 ยทโ ๐ด)) = ๐ถ |
16 | 1, 10, 15 | 3eqtr3g 2789 | . 2 โข ((๐ด +โ ๐ต) = (๐ด +โ ๐ถ) โ ๐ต = ๐ถ) |
17 | oveq2 7412 | . 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 1533 โ wcel 2098 (class class class)co 7404 1c1 11110 -cneg 11446 โchba 30677 +โ cva 30678 ยทโ csm 30679 0โc0v 30682 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-hvcom 30759 ax-hvass 30760 ax-hv0cl 30761 ax-hvaddid 30762 ax-hfvmul 30763 ax-hvmulid 30764 ax-hvdistr2 30767 ax-hvmul0 30768 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-sub 11447 df-neg 11448 df-hvsub 30729 |
This theorem is referenced by: hvsubaddi 30824 hvaddcan 30828 |
Copyright terms: Public domain | W3C validator |