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Mirrors > Home > HSE Home > Th. List > hhsst | Structured version Visualization version GIF version |
Description: A member of Sℋ is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhsst.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
hhsst.2 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
Ref | Expression |
---|---|
hhsst | ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ (SubSp‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhsst.2 | . . . 4 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
2 | 1 | hhssnvt 30036 | . . 3 ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ NrmCVec) |
3 | resss 5960 | . . . 4 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ | |
4 | resss 5960 | . . . 4 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ | |
5 | resss 5960 | . . . 4 ⊢ (normℎ ↾ 𝐻) ⊆ normℎ | |
6 | 3, 4, 5 | 3pm3.2i 1339 | . . 3 ⊢ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ) |
7 | 2, 6 | jctir 521 | . 2 ⊢ (𝐻 ∈ Sℋ → (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ))) |
8 | hhsst.1 | . . . 4 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
9 | 8 | hhnv 29936 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
10 | 8 | hhva 29937 | . . . 4 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
11 | 1 | hhssva 30028 | . . . 4 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) |
12 | 8 | hhsm 29940 | . . . 4 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
13 | 1 | hhsssm 30029 | . . . 4 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·𝑠OLD ‘𝑊) |
14 | 8 | hhnm 29942 | . . . 4 ⊢ normℎ = (normCV‘𝑈) |
15 | 1 | hhssnm 30030 | . . . 4 ⊢ (normℎ ↾ 𝐻) = (normCV‘𝑊) |
16 | eqid 2737 | . . . 4 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
17 | 10, 11, 12, 13, 14, 15, 16 | isssp 29495 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ (SubSp‘𝑈) ↔ (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ)))) |
18 | 9, 17 | ax-mp 5 | . 2 ⊢ (𝑊 ∈ (SubSp‘𝑈) ↔ (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ))) |
19 | 7, 18 | sylibr 233 | 1 ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ (SubSp‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 〈cop 4590 × cxp 5629 ↾ cres 5633 ‘cfv 6493 ℂcc 11007 NrmCVeccnv 29355 SubSpcss 29492 +ℎ cva 29691 ·ℎ csm 29692 normℎcno 29694 Sℋ csh 29699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 ax-hilex 29770 ax-hfvadd 29771 ax-hvcom 29772 ax-hvass 29773 ax-hv0cl 29774 ax-hvaddid 29775 ax-hfvmul 29776 ax-hvmulid 29777 ax-hvmulass 29778 ax-hvdistr1 29779 ax-hvdistr2 29780 ax-hvmul0 29781 ax-hfi 29850 ax-his1 29853 ax-his2 29854 ax-his3 29855 ax-his4 29856 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-map 8725 df-pm 8726 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-n0 12372 df-z 12458 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-icc 13225 df-seq 13861 df-exp 13922 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-topgen 17285 df-psmet 20741 df-xmet 20742 df-met 20743 df-bl 20744 df-mopn 20745 df-top 22195 df-topon 22212 df-bases 22248 df-lm 22532 df-haus 22618 df-grpo 29264 df-gid 29265 df-ginv 29266 df-gdiv 29267 df-ablo 29316 df-vc 29330 df-nv 29363 df-va 29366 df-ba 29367 df-sm 29368 df-0v 29369 df-vs 29370 df-nmcv 29371 df-ims 29372 df-ssp 29493 df-hnorm 29739 df-hba 29740 df-hvsub 29742 df-hlim 29743 df-sh 29978 df-ch 29992 df-ch0 30024 |
This theorem is referenced by: hhsssh 30040 hhssba 30042 hhssvs 30043 pjhthlem2 30163 |
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