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Mirrors > Home > HSE Home > Th. List > pjhth | 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 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjhth | ⊢ (𝐻 ∈ Cℋ → (𝐻 +ℋ (⊥‘𝐻)) = ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chsh 30742 | . . 3 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | |
2 | shocsh 30802 | . . 3 ⊢ (𝐻 ∈ Sℋ → (⊥‘𝐻) ∈ Sℋ ) | |
3 | shsss 30831 | . . 3 ⊢ ((𝐻 ∈ Sℋ ∧ (⊥‘𝐻) ∈ Sℋ ) → (𝐻 +ℋ (⊥‘𝐻)) ⊆ ℋ) | |
4 | 1, 2, 3 | syl2anc2 583 | . 2 ⊢ (𝐻 ∈ Cℋ → (𝐻 +ℋ (⊥‘𝐻)) ⊆ ℋ) |
5 | fveq2 6892 | . . . . . . . 8 ⊢ (𝐻 = if(𝐻 ∈ Cℋ , 𝐻, ℋ) → (⊥‘𝐻) = (⊥‘if(𝐻 ∈ Cℋ , 𝐻, ℋ))) | |
6 | 5 | rexeqdv 3324 | . . . . . . 7 ⊢ (𝐻 = if(𝐻 ∈ Cℋ , 𝐻, ℋ) → (∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧) ↔ ∃𝑧 ∈ (⊥‘if(𝐻 ∈ Cℋ , 𝐻, ℋ))𝑥 = (𝑦 +ℎ 𝑧))) |
7 | 6 | rexeqbi1dv 3332 | . . . . . 6 ⊢ (𝐻 = if(𝐻 ∈ Cℋ , 𝐻, ℋ) → (∃𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧) ↔ ∃𝑦 ∈ if (𝐻 ∈ Cℋ , 𝐻, ℋ)∃𝑧 ∈ (⊥‘if(𝐻 ∈ Cℋ , 𝐻, ℋ))𝑥 = (𝑦 +ℎ 𝑧))) |
8 | 7 | imbi2d 339 | . . . . 5 ⊢ (𝐻 = if(𝐻 ∈ Cℋ , 𝐻, ℋ) → ((𝑥 ∈ ℋ → ∃𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)) ↔ (𝑥 ∈ ℋ → ∃𝑦 ∈ if (𝐻 ∈ Cℋ , 𝐻, ℋ)∃𝑧 ∈ (⊥‘if(𝐻 ∈ Cℋ , 𝐻, ℋ))𝑥 = (𝑦 +ℎ 𝑧)))) |
9 | ifchhv 30762 | . . . . . 6 ⊢ if(𝐻 ∈ Cℋ , 𝐻, ℋ) ∈ Cℋ | |
10 | id 22 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → 𝑥 ∈ ℋ) | |
11 | 9, 10 | pjhthlem2 30910 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ∃𝑦 ∈ if (𝐻 ∈ Cℋ , 𝐻, ℋ)∃𝑧 ∈ (⊥‘if(𝐻 ∈ Cℋ , 𝐻, ℋ))𝑥 = (𝑦 +ℎ 𝑧)) |
12 | 8, 11 | dedth 4587 | . . . 4 ⊢ (𝐻 ∈ Cℋ → (𝑥 ∈ ℋ → ∃𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧))) |
13 | shsel 30832 | . . . . 5 ⊢ ((𝐻 ∈ Sℋ ∧ (⊥‘𝐻) ∈ Sℋ ) → (𝑥 ∈ (𝐻 +ℋ (⊥‘𝐻)) ↔ ∃𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧))) | |
14 | 1, 2, 13 | syl2anc2 583 | . . . 4 ⊢ (𝐻 ∈ Cℋ → (𝑥 ∈ (𝐻 +ℋ (⊥‘𝐻)) ↔ ∃𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧))) |
15 | 12, 14 | sylibrd 258 | . . 3 ⊢ (𝐻 ∈ Cℋ → (𝑥 ∈ ℋ → 𝑥 ∈ (𝐻 +ℋ (⊥‘𝐻)))) |
16 | 15 | ssrdv 3989 | . 2 ⊢ (𝐻 ∈ Cℋ → ℋ ⊆ (𝐻 +ℋ (⊥‘𝐻))) |
17 | 4, 16 | eqssd 4000 | 1 ⊢ (𝐻 ∈ Cℋ → (𝐻 +ℋ (⊥‘𝐻)) = ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 ∃wrex 3068 ⊆ wss 3949 ifcif 4529 ‘cfv 6544 (class class class)co 7413 ℋchba 30437 +ℎ cva 30438 Sℋ csh 30446 Cℋ cch 30447 ⊥cort 30448 +ℋ cph 30449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-inf2 9640 ax-cc 10434 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194 ax-hilex 30517 ax-hfvadd 30518 ax-hvcom 30519 ax-hvass 30520 ax-hv0cl 30521 ax-hvaddid 30522 ax-hfvmul 30523 ax-hvmulid 30524 ax-hvmulass 30525 ax-hvdistr1 30526 ax-hvdistr2 30527 ax-hvmul0 30528 ax-hfi 30597 ax-his1 30600 ax-his2 30601 ax-his3 30602 ax-his4 30603 ax-hcompl 30720 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-omul 8475 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-acn 9941 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-n0 12479 df-z 12565 df-uz 12829 df-q 12939 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ico 13336 df-icc 13337 df-fz 13491 df-fl 13763 df-seq 13973 df-exp 14034 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-rest 17374 df-topgen 17395 df-psmet 21138 df-xmet 21139 df-met 21140 df-bl 21141 df-mopn 21142 df-fbas 21143 df-fg 21144 df-top 22618 df-topon 22635 df-bases 22671 df-cld 22745 df-ntr 22746 df-cls 22747 df-nei 22824 df-lm 22955 df-haus 23041 df-fil 23572 df-fm 23664 df-flim 23665 df-flf 23666 df-cfil 25005 df-cau 25006 df-cmet 25007 df-grpo 30011 df-gid 30012 df-ginv 30013 df-gdiv 30014 df-ablo 30063 df-vc 30077 df-nv 30110 df-va 30113 df-ba 30114 df-sm 30115 df-0v 30116 df-vs 30117 df-nmcv 30118 df-ims 30119 df-ssp 30240 df-ph 30331 df-cbn 30381 df-hnorm 30486 df-hba 30487 df-hvsub 30489 df-hlim 30490 df-hcau 30491 df-sh 30725 df-ch 30739 df-oc 30770 df-ch0 30771 df-shs 30826 |
This theorem is referenced by: pjhtheu 30912 pjeq 30917 axpjpj 30938 |
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