![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
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 7433 | . 2 โข (๐ฅ = ๐ด โ (๐ฅ +โ (-1 ยทโ ๐ฆ)) = (๐ด +โ (-1 ยทโ ๐ฆ))) | |
2 | oveq2 7434 | . . 3 โข (๐ฆ = ๐ต โ (-1 ยทโ ๐ฆ) = (-1 ยทโ ๐ต)) | |
3 | 2 | oveq2d 7442 | . 2 โข (๐ฆ = ๐ต โ (๐ด +โ (-1 ยทโ ๐ฆ)) = (๐ด +โ (-1 ยทโ ๐ต))) |
4 | df-hvsub 30809 | . 2 โข โโ = (๐ฅ โ โ, ๐ฆ โ โ โฆ (๐ฅ +โ (-1 ยทโ ๐ฆ))) | |
5 | ovex 7459 | . 2 โข (๐ด +โ (-1 ยทโ ๐ต)) โ V | |
6 | 1, 3, 4, 5 | ovmpo 7588 | 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 394 = wceq 1533 โ wcel 2098 (class class class)co 7426 1c1 11149 -cneg 11485 โchba 30757 +โ cva 30758 ยทโ csm 30759 โโ cmv 30763 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-hvsub 30809 |
This theorem is referenced by: hvsubcl 30855 hvsubvali 30858 hvsubid 30864 hvnegid 30865 hv2neg 30866 hvaddsubval 30871 hvsub4 30875 hvaddsub12 30876 hvpncan 30877 hvaddsubass 30879 hvsubass 30882 hvsubdistr1 30887 hvsubdistr2 30888 hvsubcan 30912 hvsub0 30914 his2sub 30930 hhph 31016 shsubcl 31058 shsel3 31153 honegsubi 31634 lnopsubi 31812 lnfnsubi 31884 superpos 32192 cdj1i 32271 |
Copyright terms: Public domain | W3C validator |