![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > pjci | Structured version Visualization version GIF version |
Description: Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjclem1.1 | ⊢ 𝐺 ∈ Cℋ |
pjclem1.2 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjci | ⊢ (𝐺 𝐶ℋ 𝐻 ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjclem1.1 | . . 3 ⊢ 𝐺 ∈ Cℋ | |
2 | pjclem1.2 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
3 | 1, 2 | pjclem2 29285 | . 2 ⊢ (𝐺 𝐶ℋ 𝐻 → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺))) |
4 | 1, 2 | pjclem4 29288 | . . . . . 6 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘(𝐺 ∩ 𝐻))) |
5 | 1, 2 | pjclem3 29286 | . . . . . . 7 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻))) = ((projℎ‘(⊥‘𝐻)) ∘ (projℎ‘𝐺))) |
6 | 2 | choccli 28396 | . . . . . . . 8 ⊢ (⊥‘𝐻) ∈ Cℋ |
7 | 1, 6 | pjclem4 29288 | . . . . . . 7 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻))) = ((projℎ‘(⊥‘𝐻)) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻))) = (projℎ‘(𝐺 ∩ (⊥‘𝐻)))) |
8 | 5, 7 | syl 17 | . . . . . 6 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻))) = (projℎ‘(𝐺 ∩ (⊥‘𝐻)))) |
9 | 4, 8 | oveq12d 6783 | . . . . 5 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) +op ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻)))) = ((projℎ‘(𝐺 ∩ 𝐻)) +op (projℎ‘(𝐺 ∩ (⊥‘𝐻))))) |
10 | df-iop 28838 | . . . . . . . 8 ⊢ Iop = (projℎ‘ ℋ) | |
11 | 10 | coeq2i 5390 | . . . . . . 7 ⊢ ((projℎ‘𝐺) ∘ Iop ) = ((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) |
12 | 1 | pjfi 28793 | . . . . . . . 8 ⊢ (projℎ‘𝐺): ℋ⟶ ℋ |
13 | 12 | hoid1i 28878 | . . . . . . 7 ⊢ ((projℎ‘𝐺) ∘ Iop ) = (projℎ‘𝐺) |
14 | 11, 13 | eqtr3i 2748 | . . . . . 6 ⊢ ((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) = (projℎ‘𝐺) |
15 | 2 | pjtoi 29268 | . . . . . . . 8 ⊢ ((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻))) = (projℎ‘ ℋ) |
16 | 15 | coeq2i 5390 | . . . . . . 7 ⊢ ((projℎ‘𝐺) ∘ ((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻)))) = ((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) |
17 | 2 | pjfi 28793 | . . . . . . . 8 ⊢ (projℎ‘𝐻): ℋ⟶ ℋ |
18 | 6 | pjfi 28793 | . . . . . . . 8 ⊢ (projℎ‘(⊥‘𝐻)): ℋ⟶ ℋ |
19 | 1, 17, 18 | pjsdii 29244 | . . . . . . 7 ⊢ ((projℎ‘𝐺) ∘ ((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻)))) = (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) +op ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻)))) |
20 | 16, 19 | eqtr3i 2748 | . . . . . 6 ⊢ ((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) = (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) +op ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻)))) |
21 | 14, 20 | eqtr3i 2748 | . . . . 5 ⊢ (projℎ‘𝐺) = (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) +op ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻)))) |
22 | inss2 3942 | . . . . . . . 8 ⊢ (𝐺 ∩ 𝐻) ⊆ 𝐻 | |
23 | 1 | choccli 28396 | . . . . . . . . 9 ⊢ (⊥‘𝐺) ∈ Cℋ |
24 | 2, 23 | chub2i 28559 | . . . . . . . 8 ⊢ 𝐻 ⊆ ((⊥‘𝐺) ∨ℋ 𝐻) |
25 | 22, 24 | sstri 3718 | . . . . . . 7 ⊢ (𝐺 ∩ 𝐻) ⊆ ((⊥‘𝐺) ∨ℋ 𝐻) |
26 | 1, 2 | chdmm3i 28568 | . . . . . . 7 ⊢ (⊥‘(𝐺 ∩ (⊥‘𝐻))) = ((⊥‘𝐺) ∨ℋ 𝐻) |
27 | 25, 26 | sseqtr4i 3744 | . . . . . 6 ⊢ (𝐺 ∩ 𝐻) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻))) |
28 | 1, 2 | chincli 28549 | . . . . . . 7 ⊢ (𝐺 ∩ 𝐻) ∈ Cℋ |
29 | 1, 6 | chincli 28549 | . . . . . . 7 ⊢ (𝐺 ∩ (⊥‘𝐻)) ∈ Cℋ |
30 | 28, 29 | pjscji 29259 | . . . . . 6 ⊢ ((𝐺 ∩ 𝐻) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻))) → (projℎ‘((𝐺 ∩ 𝐻) ∨ℋ (𝐺 ∩ (⊥‘𝐻)))) = ((projℎ‘(𝐺 ∩ 𝐻)) +op (projℎ‘(𝐺 ∩ (⊥‘𝐻))))) |
31 | 27, 30 | ax-mp 5 | . . . . 5 ⊢ (projℎ‘((𝐺 ∩ 𝐻) ∨ℋ (𝐺 ∩ (⊥‘𝐻)))) = ((projℎ‘(𝐺 ∩ 𝐻)) +op (projℎ‘(𝐺 ∩ (⊥‘𝐻)))) |
32 | 9, 21, 31 | 3eqtr4g 2783 | . . . 4 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → (projℎ‘𝐺) = (projℎ‘((𝐺 ∩ 𝐻) ∨ℋ (𝐺 ∩ (⊥‘𝐻))))) |
33 | 28, 29 | chjcli 28546 | . . . . 5 ⊢ ((𝐺 ∩ 𝐻) ∨ℋ (𝐺 ∩ (⊥‘𝐻))) ∈ Cℋ |
34 | 1, 33 | pj11i 28800 | . . . 4 ⊢ ((projℎ‘𝐺) = (projℎ‘((𝐺 ∩ 𝐻) ∨ℋ (𝐺 ∩ (⊥‘𝐻)))) ↔ 𝐺 = ((𝐺 ∩ 𝐻) ∨ℋ (𝐺 ∩ (⊥‘𝐻)))) |
35 | 32, 34 | sylib 208 | . . 3 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → 𝐺 = ((𝐺 ∩ 𝐻) ∨ℋ (𝐺 ∩ (⊥‘𝐻)))) |
36 | 1, 2 | cmbri 28679 | . . 3 ⊢ (𝐺 𝐶ℋ 𝐻 ↔ 𝐺 = ((𝐺 ∩ 𝐻) ∨ℋ (𝐺 ∩ (⊥‘𝐻)))) |
37 | 35, 36 | sylibr 224 | . 2 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → 𝐺 𝐶ℋ 𝐻) |
38 | 3, 37 | impbii 199 | 1 ⊢ (𝐺 𝐶ℋ 𝐻 ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1596 ∈ wcel 2103 ∩ cin 3679 ⊆ wss 3680 class class class wbr 4760 ∘ ccom 5222 ‘cfv 6001 (class class class)co 6765 ℋchil 28006 Cℋ cch 28016 ⊥cort 28017 ∨ℋ chj 28020 𝐶ℋ ccm 28023 projℎcpjh 28024 +op chos 28025 Iop chio 28031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-inf2 8651 ax-cc 9370 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 ax-pre-sup 10127 ax-addf 10128 ax-mulf 10129 ax-hilex 28086 ax-hfvadd 28087 ax-hvcom 28088 ax-hvass 28089 ax-hv0cl 28090 ax-hvaddid 28091 ax-hfvmul 28092 ax-hvmulid 28093 ax-hvmulass 28094 ax-hvdistr1 28095 ax-hvdistr2 28096 ax-hvmul0 28097 ax-hfi 28166 ax-his1 28169 ax-his2 28170 ax-his3 28171 ax-his4 28172 ax-hcompl 28289 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-fal 1602 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-int 4584 df-iun 4630 df-iin 4631 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-se 5178 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-isom 6010 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-of 7014 df-om 7183 df-1st 7285 df-2nd 7286 df-supp 7416 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-1o 7680 df-2o 7681 df-oadd 7684 df-omul 7685 df-er 7862 df-map 7976 df-pm 7977 df-ixp 8026 df-en 8073 df-dom 8074 df-sdom 8075 df-fin 8076 df-fsupp 8392 df-fi 8433 df-sup 8464 df-inf 8465 df-oi 8531 df-card 8878 df-acn 8881 df-cda 9103 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-nn 11134 df-2 11192 df-3 11193 df-4 11194 df-5 11195 df-6 11196 df-7 11197 df-8 11198 df-9 11199 df-n0 11406 df-z 11491 df-dec 11607 df-uz 11801 df-q 11903 df-rp 11947 df-xneg 12060 df-xadd 12061 df-xmul 12062 df-ioo 12293 df-ico 12295 df-icc 12296 df-fz 12441 df-fzo 12581 df-fl 12708 df-seq 12917 df-exp 12976 df-hash 13233 df-cj 13959 df-re 13960 df-im 13961 df-sqrt 14095 df-abs 14096 df-clim 14339 df-rlim 14340 df-sum 14537 df-struct 15982 df-ndx 15983 df-slot 15984 df-base 15986 df-sets 15987 df-ress 15988 df-plusg 16077 df-mulr 16078 df-starv 16079 df-sca 16080 df-vsca 16081 df-ip 16082 df-tset 16083 df-ple 16084 df-ds 16087 df-unif 16088 df-hom 16089 df-cco 16090 df-rest 16206 df-topn 16207 df-0g 16225 df-gsum 16226 df-topgen 16227 df-pt 16228 df-prds 16231 df-xrs 16285 df-qtop 16290 df-imas 16291 df-xps 16293 df-mre 16369 df-mrc 16370 df-acs 16372 df-mgm 17364 df-sgrp 17406 df-mnd 17417 df-submnd 17458 df-mulg 17663 df-cntz 17871 df-cmn 18316 df-psmet 19861 df-xmet 19862 df-met 19863 df-bl 19864 df-mopn 19865 df-fbas 19866 df-fg 19867 df-cnfld 19870 df-top 20822 df-topon 20839 df-topsp 20860 df-bases 20873 df-cld 20946 df-ntr 20947 df-cls 20948 df-nei 21025 df-cn 21154 df-cnp 21155 df-lm 21156 df-haus 21242 df-tx 21488 df-hmeo 21681 df-fil 21772 df-fm 21864 df-flim 21865 df-flf 21866 df-xms 22247 df-ms 22248 df-tms 22249 df-cfil 23174 df-cau 23175 df-cmet 23176 df-grpo 27577 df-gid 27578 df-ginv 27579 df-gdiv 27580 df-ablo 27629 df-vc 27644 df-nv 27677 df-va 27680 df-ba 27681 df-sm 27682 df-0v 27683 df-vs 27684 df-nmcv 27685 df-ims 27686 df-dip 27786 df-ssp 27807 df-ph 27898 df-cbn 27949 df-hnorm 28055 df-hba 28056 df-hvsub 28058 df-hlim 28059 df-hcau 28060 df-sh 28294 df-ch 28308 df-oc 28339 df-ch0 28340 df-shs 28397 df-chj 28399 df-pjh 28484 df-cm 28672 df-hosum 28819 df-hodif 28821 df-h0op 28837 df-iop 28838 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |