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Mirrors > Home > HSE Home > Th. List > pjhtheu | Structured version Visualization version GIF version |
Description: Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102. See pjhtheu2 28582 for the uniqueness of 𝑦. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjhtheu | ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃!𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjhth 28559 | . . . . 5 ⊢ (𝐻 ∈ Cℋ → (𝐻 +ℋ (⊥‘𝐻)) = ℋ) | |
2 | 1 | eleq2d 2823 | . . . 4 ⊢ (𝐻 ∈ Cℋ → (𝐴 ∈ (𝐻 +ℋ (⊥‘𝐻)) ↔ 𝐴 ∈ ℋ)) |
3 | chsh 28388 | . . . . 5 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | |
4 | shocsh 28450 | . . . . . 6 ⊢ (𝐻 ∈ Sℋ → (⊥‘𝐻) ∈ Sℋ ) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐻 ∈ Cℋ → (⊥‘𝐻) ∈ Sℋ ) |
6 | shsel 28480 | . . . . 5 ⊢ ((𝐻 ∈ Sℋ ∧ (⊥‘𝐻) ∈ Sℋ ) → (𝐴 ∈ (𝐻 +ℋ (⊥‘𝐻)) ↔ ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) | |
7 | 3, 5, 6 | syl2anc 696 | . . . 4 ⊢ (𝐻 ∈ Cℋ → (𝐴 ∈ (𝐻 +ℋ (⊥‘𝐻)) ↔ ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
8 | 2, 7 | bitr3d 270 | . . 3 ⊢ (𝐻 ∈ Cℋ → (𝐴 ∈ ℋ ↔ ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
9 | 8 | biimpa 502 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
10 | ocin 28462 | . . . . 5 ⊢ (𝐻 ∈ Sℋ → (𝐻 ∩ (⊥‘𝐻)) = 0ℋ) | |
11 | 3, 10 | syl 17 | . . . 4 ⊢ (𝐻 ∈ Cℋ → (𝐻 ∩ (⊥‘𝐻)) = 0ℋ) |
12 | pjhthmo 28468 | . . . 4 ⊢ ((𝐻 ∈ Sℋ ∧ (⊥‘𝐻) ∈ Sℋ ∧ (𝐻 ∩ (⊥‘𝐻)) = 0ℋ) → ∃*𝑥(𝑥 ∈ 𝐻 ∧ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) | |
13 | 3, 5, 11, 12 | syl3anc 1477 | . . 3 ⊢ (𝐻 ∈ Cℋ → ∃*𝑥(𝑥 ∈ 𝐻 ∧ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
14 | 13 | adantr 472 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃*𝑥(𝑥 ∈ 𝐻 ∧ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
15 | reu5 3296 | . . 3 ⊢ (∃!𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦) ↔ (∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦) ∧ ∃*𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) | |
16 | df-rmo 3056 | . . . 4 ⊢ (∃*𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦) ↔ ∃*𝑥(𝑥 ∈ 𝐻 ∧ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) | |
17 | 16 | anbi2i 732 | . . 3 ⊢ ((∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦) ∧ ∃*𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) ↔ (∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦) ∧ ∃*𝑥(𝑥 ∈ 𝐻 ∧ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))) |
18 | 15, 17 | bitri 264 | . 2 ⊢ (∃!𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦) ↔ (∃𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦) ∧ ∃*𝑥(𝑥 ∈ 𝐻 ∧ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))) |
19 | 9, 14, 18 | sylanbrc 701 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃!𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1630 ∈ wcel 2137 ∃*wmo 2606 ∃wrex 3049 ∃!wreu 3050 ∃*wrmo 3051 ∩ cin 3712 ‘cfv 6047 (class class class)co 6811 ℋchil 28083 +ℎ cva 28084 Sℋ csh 28092 Cℋ cch 28093 ⊥cort 28094 +ℋ cph 28095 0ℋc0h 28099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-rep 4921 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-inf2 8709 ax-cc 9447 ax-cnex 10182 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203 ax-pre-sup 10204 ax-addf 10205 ax-mulf 10206 ax-hilex 28163 ax-hfvadd 28164 ax-hvcom 28165 ax-hvass 28166 ax-hv0cl 28167 ax-hvaddid 28168 ax-hfvmul 28169 ax-hvmulid 28170 ax-hvmulass 28171 ax-hvdistr1 28172 ax-hvdistr2 28173 ax-hvmul0 28174 ax-hfi 28243 ax-his1 28246 ax-his2 28247 ax-his3 28248 ax-his4 28249 ax-hcompl 28366 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rmo 3056 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-uni 4587 df-int 4626 df-iun 4672 df-iin 4673 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-se 5224 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-isom 6056 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-om 7229 df-1st 7331 df-2nd 7332 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-oadd 7731 df-omul 7732 df-er 7909 df-map 8023 df-pm 8024 df-en 8120 df-dom 8121 df-sdom 8122 df-fin 8123 df-fi 8480 df-sup 8511 df-inf 8512 df-oi 8578 df-card 8953 df-acn 8956 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-div 10875 df-nn 11211 df-2 11269 df-3 11270 df-4 11271 df-n0 11483 df-z 11568 df-uz 11878 df-q 11980 df-rp 12024 df-xneg 12137 df-xadd 12138 df-xmul 12139 df-ico 12372 df-icc 12373 df-fz 12518 df-fl 12785 df-seq 12994 df-exp 13053 df-cj 14036 df-re 14037 df-im 14038 df-sqrt 14172 df-abs 14173 df-clim 14416 df-rlim 14417 df-rest 16283 df-topgen 16304 df-psmet 19938 df-xmet 19939 df-met 19940 df-bl 19941 df-mopn 19942 df-fbas 19943 df-fg 19944 df-top 20899 df-topon 20916 df-bases 20950 df-cld 21023 df-ntr 21024 df-cls 21025 df-nei 21102 df-lm 21233 df-haus 21319 df-fil 21849 df-fm 21941 df-flim 21942 df-flf 21943 df-cfil 23251 df-cau 23252 df-cmet 23253 df-grpo 27654 df-gid 27655 df-ginv 27656 df-gdiv 27657 df-ablo 27706 df-vc 27721 df-nv 27754 df-va 27757 df-ba 27758 df-sm 27759 df-0v 27760 df-vs 27761 df-nmcv 27762 df-ims 27763 df-ssp 27884 df-ph 27975 df-cbn 28026 df-hnorm 28132 df-hba 28133 df-hvsub 28135 df-hlim 28136 df-hcau 28137 df-sh 28371 df-ch 28385 df-oc 28416 df-ch0 28417 df-shs 28474 |
This theorem is referenced by: pjhtheu2 28582 |
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