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Mirrors > Home > HSE Home > Th. List > pjoc1i | Structured version Visualization version GIF version |
Description: Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjop.1 | ⊢ 𝐻 ∈ Cℋ |
pjop.2 | ⊢ 𝐴 ∈ ℋ |
Ref | Expression |
---|---|
pjoc1i | ⊢ (𝐴 ∈ 𝐻 ↔ ((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjop.1 | . . . . . . . 8 ⊢ 𝐻 ∈ Cℋ | |
2 | pjop.2 | . . . . . . . 8 ⊢ 𝐴 ∈ ℋ | |
3 | 1, 2 | pjopi 29198 | . . . . . . 7 ⊢ ((projℎ‘(⊥‘𝐻))‘𝐴) = (𝐴 −ℎ ((projℎ‘𝐻)‘𝐴)) |
4 | 1 | chshii 28996 | . . . . . . . 8 ⊢ 𝐻 ∈ Sℋ |
5 | 1, 2 | pjclii 29190 | . . . . . . . 8 ⊢ ((projℎ‘𝐻)‘𝐴) ∈ 𝐻 |
6 | shsubcl 28989 | . . . . . . . 8 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ ((projℎ‘𝐻)‘𝐴) ∈ 𝐻) → (𝐴 −ℎ ((projℎ‘𝐻)‘𝐴)) ∈ 𝐻) | |
7 | 4, 5, 6 | mp3an13 1446 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐻 → (𝐴 −ℎ ((projℎ‘𝐻)‘𝐴)) ∈ 𝐻) |
8 | 3, 7 | eqeltrid 2915 | . . . . . 6 ⊢ (𝐴 ∈ 𝐻 → ((projℎ‘(⊥‘𝐻))‘𝐴) ∈ 𝐻) |
9 | 1 | choccli 29076 | . . . . . . 7 ⊢ (⊥‘𝐻) ∈ Cℋ |
10 | 9, 2 | pjclii 29190 | . . . . . 6 ⊢ ((projℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻) |
11 | 8, 10 | jctir 523 | . . . . 5 ⊢ (𝐴 ∈ 𝐻 → (((projℎ‘(⊥‘𝐻))‘𝐴) ∈ 𝐻 ∧ ((projℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻))) |
12 | elin 4167 | . . . . 5 ⊢ (((projℎ‘(⊥‘𝐻))‘𝐴) ∈ (𝐻 ∩ (⊥‘𝐻)) ↔ (((projℎ‘(⊥‘𝐻))‘𝐴) ∈ 𝐻 ∧ ((projℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻))) | |
13 | 11, 12 | sylibr 236 | . . . 4 ⊢ (𝐴 ∈ 𝐻 → ((projℎ‘(⊥‘𝐻))‘𝐴) ∈ (𝐻 ∩ (⊥‘𝐻))) |
14 | ocin 29065 | . . . . 5 ⊢ (𝐻 ∈ Sℋ → (𝐻 ∩ (⊥‘𝐻)) = 0ℋ) | |
15 | 4, 14 | ax-mp 5 | . . . 4 ⊢ (𝐻 ∩ (⊥‘𝐻)) = 0ℋ |
16 | 13, 15 | eleqtrdi 2921 | . . 3 ⊢ (𝐴 ∈ 𝐻 → ((projℎ‘(⊥‘𝐻))‘𝐴) ∈ 0ℋ) |
17 | elch0 29023 | . . 3 ⊢ (((projℎ‘(⊥‘𝐻))‘𝐴) ∈ 0ℋ ↔ ((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ) | |
18 | 16, 17 | sylib 220 | . 2 ⊢ (𝐴 ∈ 𝐻 → ((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ) |
19 | 1, 2 | pjpji 29193 | . . . . 5 ⊢ 𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐴)) |
20 | oveq2 7156 | . . . . 5 ⊢ (((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ → (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐴)) = (((projℎ‘𝐻)‘𝐴) +ℎ 0ℎ)) | |
21 | 19, 20 | syl5eq 2866 | . . . 4 ⊢ (((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ → 𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ 0ℎ)) |
22 | 1, 2 | pjhclii 29191 | . . . . 5 ⊢ ((projℎ‘𝐻)‘𝐴) ∈ ℋ |
23 | ax-hvaddid 28773 | . . . . 5 ⊢ (((projℎ‘𝐻)‘𝐴) ∈ ℋ → (((projℎ‘𝐻)‘𝐴) +ℎ 0ℎ) = ((projℎ‘𝐻)‘𝐴)) | |
24 | 22, 23 | ax-mp 5 | . . . 4 ⊢ (((projℎ‘𝐻)‘𝐴) +ℎ 0ℎ) = ((projℎ‘𝐻)‘𝐴) |
25 | 21, 24 | syl6eq 2870 | . . 3 ⊢ (((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ → 𝐴 = ((projℎ‘𝐻)‘𝐴)) |
26 | 25, 5 | syl6eqel 2919 | . 2 ⊢ (((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ → 𝐴 ∈ 𝐻) |
27 | 18, 26 | impbii 211 | 1 ⊢ (𝐴 ∈ 𝐻 ↔ ((projℎ‘(⊥‘𝐻))‘𝐴) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∩ cin 3933 ‘cfv 6348 (class class class)co 7148 ℋchba 28688 +ℎ cva 28689 0ℎc0v 28693 −ℎ cmv 28694 Sℋ csh 28697 Cℋ cch 28698 ⊥cort 28699 0ℋc0h 28704 projℎcpjh 28706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cc 9849 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 ax-hilex 28768 ax-hfvadd 28769 ax-hvcom 28770 ax-hvass 28771 ax-hv0cl 28772 ax-hvaddid 28773 ax-hfvmul 28774 ax-hvmulid 28775 ax-hvmulass 28776 ax-hvdistr1 28777 ax-hvdistr2 28778 ax-hvmul0 28779 ax-hfi 28848 ax-his1 28851 ax-his2 28852 ax-his3 28853 ax-his4 28854 ax-hcompl 28971 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-omul 8099 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-acn 9363 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-fbas 20534 df-fg 20535 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cld 21619 df-ntr 21620 df-cls 21621 df-nei 21698 df-cn 21827 df-cnp 21828 df-lm 21829 df-haus 21915 df-tx 22162 df-hmeo 22355 df-fil 22446 df-fm 22538 df-flim 22539 df-flf 22540 df-xms 22922 df-ms 22923 df-tms 22924 df-cfil 23850 df-cau 23851 df-cmet 23852 df-grpo 28262 df-gid 28263 df-ginv 28264 df-gdiv 28265 df-ablo 28314 df-vc 28328 df-nv 28361 df-va 28364 df-ba 28365 df-sm 28366 df-0v 28367 df-vs 28368 df-nmcv 28369 df-ims 28370 df-dip 28470 df-ssp 28491 df-ph 28582 df-cbn 28632 df-hnorm 28737 df-hba 28738 df-hvsub 28740 df-hlim 28741 df-hcau 28742 df-sh 28976 df-ch 28990 df-oc 29021 df-ch0 29022 df-shs 29077 df-pjh 29164 |
This theorem is referenced by: pjchi 29201 pjoc1 29203 pjoc2i 29207 pjneli 29492 |
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