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Mirrors > Home > HSE Home > Th. List > pjtoi | Structured version Visualization version GIF version |
Description: Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjidmco.1 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjtoi | ⊢ ((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻))) = (projℎ‘ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjidmco.1 | . . . . 5 ⊢ 𝐻 ∈ Cℋ | |
2 | axpjpj 29357 | . . . . 5 ⊢ ((𝐻 ∈ Cℋ ∧ 𝑥 ∈ ℋ) → 𝑥 = (((projℎ‘𝐻)‘𝑥) +ℎ ((projℎ‘(⊥‘𝐻))‘𝑥))) | |
3 | 1, 2 | mpan 690 | . . . 4 ⊢ (𝑥 ∈ ℋ → 𝑥 = (((projℎ‘𝐻)‘𝑥) +ℎ ((projℎ‘(⊥‘𝐻))‘𝑥))) |
4 | pjch1 29607 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((projℎ‘ ℋ)‘𝑥) = 𝑥) | |
5 | 1 | pjfi 29641 | . . . . 5 ⊢ (projℎ‘𝐻): ℋ⟶ ℋ |
6 | 1 | choccli 29244 | . . . . . 6 ⊢ (⊥‘𝐻) ∈ Cℋ |
7 | 6 | pjfi 29641 | . . . . 5 ⊢ (projℎ‘(⊥‘𝐻)): ℋ⟶ ℋ |
8 | hosval 29677 | . . . . 5 ⊢ (((projℎ‘𝐻): ℋ⟶ ℋ ∧ (projℎ‘(⊥‘𝐻)): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻)))‘𝑥) = (((projℎ‘𝐻)‘𝑥) +ℎ ((projℎ‘(⊥‘𝐻))‘𝑥))) | |
9 | 5, 7, 8 | mp3an12 1452 | . . . 4 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻)))‘𝑥) = (((projℎ‘𝐻)‘𝑥) +ℎ ((projℎ‘(⊥‘𝐻))‘𝑥))) |
10 | 3, 4, 9 | 3eqtr4rd 2784 | . . 3 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻)))‘𝑥) = ((projℎ‘ ℋ)‘𝑥)) |
11 | 10 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ ℋ (((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻)))‘𝑥) = ((projℎ‘ ℋ)‘𝑥) |
12 | 5, 7 | hoaddcli 29705 | . . 3 ⊢ ((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻))): ℋ⟶ ℋ |
13 | helch 29180 | . . . 4 ⊢ ℋ ∈ Cℋ | |
14 | 13 | pjfi 29641 | . . 3 ⊢ (projℎ‘ ℋ): ℋ⟶ ℋ |
15 | 12, 14 | hoeqi 29698 | . 2 ⊢ (∀𝑥 ∈ ℋ (((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻)))‘𝑥) = ((projℎ‘ ℋ)‘𝑥) ↔ ((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻))) = (projℎ‘ ℋ)) |
16 | 11, 15 | mpbi 233 | 1 ⊢ ((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻))) = (projℎ‘ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 ∀wral 3053 ⟶wf 6335 ‘cfv 6339 (class class class)co 7172 ℋchba 28856 +ℎ cva 28857 Cℋ cch 28866 ⊥cort 28867 projℎcpjh 28874 +op chos 28875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-inf2 9179 ax-cc 9937 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 ax-addf 10696 ax-mulf 10697 ax-hilex 28936 ax-hfvadd 28937 ax-hvcom 28938 ax-hvass 28939 ax-hv0cl 28940 ax-hvaddid 28941 ax-hfvmul 28942 ax-hvmulid 28943 ax-hvmulass 28944 ax-hvdistr1 28945 ax-hvdistr2 28946 ax-hvmul0 28947 ax-hfi 29016 ax-his1 29019 ax-his2 29020 ax-his3 29021 ax-his4 29022 ax-hcompl 29139 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-om 7602 df-1st 7716 df-2nd 7717 df-supp 7859 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-2o 8134 df-oadd 8137 df-omul 8138 df-er 8322 df-map 8441 df-pm 8442 df-ixp 8510 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-fsupp 8909 df-fi 8950 df-sup 8981 df-inf 8982 df-oi 9049 df-card 9443 df-acn 9446 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-dec 12182 df-uz 12327 df-q 12433 df-rp 12475 df-xneg 12592 df-xadd 12593 df-xmul 12594 df-ioo 12827 df-ico 12829 df-icc 12830 df-fz 12984 df-fzo 13127 df-fl 13255 df-seq 13463 df-exp 13524 df-hash 13785 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 df-clim 14937 df-rlim 14938 df-sum 15138 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-mulr 16684 df-starv 16685 df-sca 16686 df-vsca 16687 df-ip 16688 df-tset 16689 df-ple 16690 df-ds 16692 df-unif 16693 df-hom 16694 df-cco 16695 df-rest 16801 df-topn 16802 df-0g 16820 df-gsum 16821 df-topgen 16822 df-pt 16823 df-prds 16826 df-xrs 16880 df-qtop 16885 df-imas 16886 df-xps 16888 df-mre 16962 df-mrc 16963 df-acs 16965 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-submnd 18075 df-mulg 18345 df-cntz 18567 df-cmn 19028 df-psmet 20211 df-xmet 20212 df-met 20213 df-bl 20214 df-mopn 20215 df-fbas 20216 df-fg 20217 df-cnfld 20220 df-top 21647 df-topon 21664 df-topsp 21686 df-bases 21699 df-cld 21772 df-ntr 21773 df-cls 21774 df-nei 21851 df-cn 21980 df-cnp 21981 df-lm 21982 df-haus 22068 df-tx 22315 df-hmeo 22508 df-fil 22599 df-fm 22691 df-flim 22692 df-flf 22693 df-xms 23075 df-ms 23076 df-tms 23077 df-cfil 24009 df-cau 24010 df-cmet 24011 df-grpo 28430 df-gid 28431 df-ginv 28432 df-gdiv 28433 df-ablo 28482 df-vc 28496 df-nv 28529 df-va 28532 df-ba 28533 df-sm 28534 df-0v 28535 df-vs 28536 df-nmcv 28537 df-ims 28538 df-dip 28638 df-ssp 28659 df-ph 28750 df-cbn 28800 df-hnorm 28905 df-hba 28906 df-hvsub 28908 df-hlim 28909 df-hcau 28910 df-sh 29144 df-ch 29158 df-oc 29189 df-ch0 29190 df-shs 29245 df-pjh 29332 df-hosum 29667 |
This theorem is referenced by: pjoci 30117 pjclem1 30132 pjci 30137 |
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