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Mirrors > Home > HSE Home > Th. List > hvsubval | Structured version Visualization version GIF version |
Description: Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsubval | โข ((๐ด โ โ โง ๐ต โ โ) โ (๐ด โโ ๐ต) = (๐ด +โ (-1 ยทโ ๐ต))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7412 | . 2 โข (๐ฅ = ๐ด โ (๐ฅ +โ (-1 ยทโ ๐ฆ)) = (๐ด +โ (-1 ยทโ ๐ฆ))) | |
2 | oveq2 7413 | . . 3 โข (๐ฆ = ๐ต โ (-1 ยทโ ๐ฆ) = (-1 ยทโ ๐ต)) | |
3 | 2 | oveq2d 7421 | . 2 โข (๐ฆ = ๐ต โ (๐ด +โ (-1 ยทโ ๐ฆ)) = (๐ด +โ (-1 ยทโ ๐ต))) |
4 | df-hvsub 30733 | . 2 โข โโ = (๐ฅ โ โ, ๐ฆ โ โ โฆ (๐ฅ +โ (-1 ยทโ ๐ฆ))) | |
5 | ovex 7438 | . 2 โข (๐ด +โ (-1 ยทโ ๐ต)) โ V | |
6 | 1, 3, 4, 5 | ovmpo 7564 | 1 โข ((๐ด โ โ โง ๐ต โ โ) โ (๐ด โโ ๐ต) = (๐ด +โ (-1 ยทโ ๐ต))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1533 โ wcel 2098 (class class class)co 7405 1c1 11113 -cneg 11449 โchba 30681 +โ cva 30682 ยทโ csm 30683 โโ cmv 30687 |
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-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-hvsub 30733 |
This theorem is referenced by: hvsubcl 30779 hvsubvali 30782 hvsubid 30788 hvnegid 30789 hv2neg 30790 hvaddsubval 30795 hvsub4 30799 hvaddsub12 30800 hvpncan 30801 hvaddsubass 30803 hvsubass 30806 hvsubdistr1 30811 hvsubdistr2 30812 hvsubcan 30836 hvsub0 30838 his2sub 30854 hhph 30940 shsubcl 30982 shsel3 31077 honegsubi 31558 lnopsubi 31736 lnfnsubi 31808 superpos 32116 cdj1i 32195 |
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