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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 28725 | . . 3 ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ NrmCVec) |
3 | resss 5762 | . . . 4 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ | |
4 | resss 5762 | . . . 4 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ | |
5 | resss 5762 | . . . 4 ⊢ (normℎ ↾ 𝐻) ⊆ normℎ | |
6 | 3, 4, 5 | 3pm3.2i 1332 | . . 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 28625 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
10 | 8 | hhva 28626 | . . . 4 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
11 | 1 | hhssva 28717 | . . . 4 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) |
12 | 8 | hhsm 28629 | . . . 4 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
13 | 1 | hhsssm 28718 | . . . 4 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·𝑠OLD ‘𝑊) |
14 | 8 | hhnm 28631 | . . . 4 ⊢ normℎ = (normCV‘𝑈) |
15 | 1 | hhssnm 28719 | . . . 4 ⊢ (normℎ ↾ 𝐻) = (normCV‘𝑊) |
16 | eqid 2794 | . . . 4 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
17 | 10, 11, 12, 13, 14, 15, 16 | isssp 28184 | . . 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 235 | 1 ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ (SubSp‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2080 ⊆ wss 3861 〈cop 4480 × cxp 5444 ↾ cres 5448 ‘cfv 6228 ℂcc 10384 NrmCVeccnv 28044 SubSpcss 28181 +ℎ cva 28380 ·ℎ csm 28381 normℎcno 28383 Sℋ csh 28388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 ax-addf 10465 ax-mulf 10466 ax-hilex 28459 ax-hfvadd 28460 ax-hvcom 28461 ax-hvass 28462 ax-hv0cl 28463 ax-hvaddid 28464 ax-hfvmul 28465 ax-hvmulid 28466 ax-hvmulass 28467 ax-hvdistr1 28468 ax-hvdistr2 28469 ax-hvmul0 28470 ax-hfi 28539 ax-his1 28542 ax-his2 28543 ax-his3 28544 ax-his4 28545 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-er 8142 df-map 8261 df-pm 8262 df-en 8361 df-dom 8362 df-sdom 8363 df-sup 8755 df-inf 8756 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-n0 11748 df-z 11832 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-icc 12595 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-topgen 16546 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-top 21186 df-topon 21203 df-bases 21238 df-lm 21521 df-haus 21607 df-grpo 27953 df-gid 27954 df-ginv 27955 df-gdiv 27956 df-ablo 28005 df-vc 28019 df-nv 28052 df-va 28055 df-ba 28056 df-sm 28057 df-0v 28058 df-vs 28059 df-nmcv 28060 df-ims 28061 df-ssp 28182 df-hnorm 28428 df-hba 28429 df-hvsub 28431 df-hlim 28432 df-sh 28667 df-ch 28681 df-ch0 28713 |
This theorem is referenced by: hhsssh 28729 hhssba 28731 hhssvs 28732 pjhthlem2 28852 |
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