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Mirrors > Home > HSE Home > Th. List > hhsssh2 | Structured version Visualization version GIF version |
Description: The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhsssh2.1 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
Ref | Expression |
---|---|
hhsssh2 | ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | hhsssh2.1 | . . 3 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
3 | 1, 2 | hhsssh 31171 | . 2 ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∧ 𝐻 ⊆ ℋ)) |
4 | resss 6007 | . . . . 5 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ | |
5 | resss 6007 | . . . . 5 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ | |
6 | resss 6007 | . . . . 5 ⊢ (normℎ ↾ 𝐻) ⊆ normℎ | |
7 | 4, 5, 6 | 3pm3.2i 1336 | . . . 4 ⊢ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ) |
8 | 1 | hhnv 31067 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
9 | 1 | hhva 31068 | . . . . . 6 ⊢ +ℎ = ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
10 | 2 | hhssva 31159 | . . . . . 6 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) |
11 | 1 | hhsm 31071 | . . . . . 6 ⊢ ·ℎ = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
12 | 2 | hhsssm 31160 | . . . . . 6 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·𝑠OLD ‘𝑊) |
13 | 1 | hhnm 31073 | . . . . . 6 ⊢ normℎ = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
14 | 2 | hhssnm 31161 | . . . . . 6 ⊢ (normℎ ↾ 𝐻) = (normCV‘𝑊) |
15 | eqid 2725 | . . . . . 6 ⊢ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
16 | 9, 10, 11, 12, 13, 14, 15 | isssp 30626 | . . . . 5 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec → (𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ↔ (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ)))) |
17 | 8, 16 | ax-mp 5 | . . . 4 ⊢ (𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ↔ (𝑊 ∈ NrmCVec ∧ (( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ∧ ( ·ℎ ↾ (ℂ × 𝐻)) ⊆ ·ℎ ∧ (normℎ ↾ 𝐻) ⊆ normℎ))) |
18 | 7, 17 | mpbiran2 708 | . . 3 ⊢ (𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ↔ 𝑊 ∈ NrmCVec) |
19 | 18 | anbi1i 622 | . 2 ⊢ ((𝑊 ∈ (SubSp‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∧ 𝐻 ⊆ ℋ) ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ)) |
20 | 3, 19 | bitri 274 | 1 ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 〈cop 4636 × cxp 5676 ↾ cres 5680 ‘cfv 6549 ℂcc 11143 NrmCVeccnv 30486 SubSpcss 30623 ℋchba 30821 +ℎ cva 30822 ·ℎ csm 30823 normℎcno 30825 Sℋ csh 30830 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 ax-mulf 11225 ax-hilex 30901 ax-hfvadd 30902 ax-hvcom 30903 ax-hvass 30904 ax-hv0cl 30905 ax-hvaddid 30906 ax-hfvmul 30907 ax-hvmulid 30908 ax-hvmulass 30909 ax-hvdistr1 30910 ax-hvdistr2 30911 ax-hvmul0 30912 ax-hfi 30981 ax-his1 30984 ax-his2 30985 ax-his3 30986 ax-his4 30987 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9472 df-inf 9473 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-icc 13371 df-seq 14008 df-exp 14068 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-topgen 17444 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-top 22857 df-topon 22874 df-bases 22910 df-lm 23194 df-haus 23280 df-grpo 30395 df-gid 30396 df-ginv 30397 df-gdiv 30398 df-ablo 30447 df-vc 30461 df-nv 30494 df-va 30497 df-ba 30498 df-sm 30499 df-0v 30500 df-vs 30501 df-nmcv 30502 df-ims 30503 df-ssp 30624 df-hnorm 30870 df-hba 30871 df-hvsub 30873 df-hlim 30874 df-sh 31109 df-ch 31123 df-ch0 31155 |
This theorem is referenced by: (None) |
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