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Mirrors > Home > HSE Home > Th. List > hhssvs | Structured version Visualization version GIF version |
Description: The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhsssh2.1 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
hhssba.2 | ⊢ 𝐻 ∈ Sℋ |
Ref | Expression |
---|---|
hhssvs | ⊢ ( −ℎ ↾ (𝐻 × 𝐻)) = ( −𝑣 ‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | 1 | hhnv 30396 | . . 3 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
3 | hhssba.2 | . . . 4 ⊢ 𝐻 ∈ Sℋ | |
4 | hhsssh2.1 | . . . . 5 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
5 | 1, 4 | hhsst 30497 | . . . 4 ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
6 | 3, 5 | ax-mp 5 | . . 3 ⊢ 𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
7 | 4, 3 | hhssba 30502 | . . . 4 ⊢ 𝐻 = (BaseSet‘𝑊) |
8 | 1 | hhvs 30401 | . . . 4 ⊢ −ℎ = ( −𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
9 | eqid 2733 | . . . 4 ⊢ ( −𝑣 ‘𝑊) = ( −𝑣 ‘𝑊) | |
10 | eqid 2733 | . . . 4 ⊢ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
11 | 7, 8, 9, 10 | sspm 29965 | . . 3 ⊢ ((〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) → ( −𝑣 ‘𝑊) = ( −ℎ ↾ (𝐻 × 𝐻))) |
12 | 2, 6, 11 | mp2an 691 | . 2 ⊢ ( −𝑣 ‘𝑊) = ( −ℎ ↾ (𝐻 × 𝐻)) |
13 | 12 | eqcomi 2742 | 1 ⊢ ( −ℎ ↾ (𝐻 × 𝐻)) = ( −𝑣 ‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 〈cop 4633 × cxp 5673 ↾ cres 5677 ‘cfv 6540 ℂcc 11104 NrmCVeccnv 29815 −𝑣 cnsb 29820 SubSpcss 29952 +ℎ cva 30151 ·ℎ csm 30152 normℎcno 30154 −ℎ cmv 30156 Sℋ csh 30159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 ax-hilex 30230 ax-hfvadd 30231 ax-hvcom 30232 ax-hvass 30233 ax-hv0cl 30234 ax-hvaddid 30235 ax-hfvmul 30236 ax-hvmulid 30237 ax-hvmulass 30238 ax-hvdistr1 30239 ax-hvdistr2 30240 ax-hvmul0 30241 ax-hfi 30310 ax-his1 30313 ax-his2 30314 ax-his3 30315 ax-his4 30316 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-topgen 17385 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-top 22378 df-topon 22395 df-bases 22431 df-lm 22715 df-haus 22801 df-grpo 29724 df-gid 29725 df-ginv 29726 df-gdiv 29727 df-ablo 29776 df-vc 29790 df-nv 29823 df-va 29826 df-ba 29827 df-sm 29828 df-0v 29829 df-vs 29830 df-nmcv 29831 df-ims 29832 df-ssp 29953 df-hnorm 30199 df-hba 30200 df-hvsub 30202 df-hlim 30203 df-sh 30438 df-ch 30452 df-ch0 30484 |
This theorem is referenced by: hhssvsf 30504 hhssims 30505 hhssmetdval 30508 |
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