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Mirrors > Home > HSE Home > Th. List > hmopidmpji | Structured version Visualization version GIF version |
Description: An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that 𝐻 is a closed subspace, which is not trivial as hmopidmchi 29607 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hmopidmch.1 | ⊢ 𝑇 ∈ HrmOp |
hmopidmch.2 | ⊢ (𝑇 ∘ 𝑇) = 𝑇 |
Ref | Expression |
---|---|
hmopidmpji | ⊢ 𝑇 = (projℎ‘ran 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmopidmch.1 | . . . . . 6 ⊢ 𝑇 ∈ HrmOp | |
2 | hmoplin 29398 | . . . . . 6 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 𝑇 ∈ LinOp |
4 | 3 | lnopfi 29425 | . . . 4 ⊢ 𝑇: ℋ⟶ ℋ |
5 | ffn 6374 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 𝑇 Fn ℋ |
7 | hmopidmch.2 | . . . . 5 ⊢ (𝑇 ∘ 𝑇) = 𝑇 | |
8 | 1, 7 | hmopidmchi 29607 | . . . 4 ⊢ ran 𝑇 ∈ Cℋ |
9 | 8 | pjfni 29157 | . . 3 ⊢ (projℎ‘ran 𝑇) Fn ℋ |
10 | eqfnfv 6658 | . . 3 ⊢ ((𝑇 Fn ℋ ∧ (projℎ‘ran 𝑇) Fn ℋ) → (𝑇 = (projℎ‘ran 𝑇) ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ((projℎ‘ran 𝑇)‘𝑥))) | |
11 | 6, 9, 10 | mp2an 688 | . 2 ⊢ (𝑇 = (projℎ‘ran 𝑇) ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = ((projℎ‘ran 𝑇)‘𝑥)) |
12 | fnfvelrn 6704 | . . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ran 𝑇) | |
13 | 6, 12 | mpan 686 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ran 𝑇) |
14 | id 22 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → 𝑥 ∈ ℋ) | |
15 | 4 | ffvelrni 6706 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
16 | hvsubcl 28473 | . . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → (𝑥 −ℎ (𝑇‘𝑥)) ∈ ℋ) | |
17 | 14, 15, 16 | syl2anc 584 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑥 −ℎ (𝑇‘𝑥)) ∈ ℋ) |
18 | simpl 483 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℋ) | |
19 | 15 | adantr 481 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ) |
20 | 4 | ffvelrni 6706 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ) |
21 | 20 | adantl 482 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ) |
22 | his2sub 28548 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑥 −ℎ (𝑇‘𝑥)) ·ih (𝑇‘𝑦)) = ((𝑥 ·ih (𝑇‘𝑦)) − ((𝑇‘𝑥) ·ih (𝑇‘𝑦)))) | |
23 | 18, 19, 21, 22 | syl3anc 1362 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 −ℎ (𝑇‘𝑥)) ·ih (𝑇‘𝑦)) = ((𝑥 ·ih (𝑇‘𝑦)) − ((𝑇‘𝑥) ·ih (𝑇‘𝑦)))) |
24 | hmop 29378 | . . . . . . . . . . . 12 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑇‘𝑦))) = ((𝑇‘𝑥) ·ih (𝑇‘𝑦))) | |
25 | 1, 24 | mp3an1 1438 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑇‘𝑦))) = ((𝑇‘𝑥) ·ih (𝑇‘𝑦))) |
26 | 20, 25 | sylan2 592 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑇‘𝑦))) = ((𝑇‘𝑥) ·ih (𝑇‘𝑦))) |
27 | 7 | fveq1i 6531 | . . . . . . . . . . . . 13 ⊢ ((𝑇 ∘ 𝑇)‘𝑦) = (𝑇‘𝑦) |
28 | 4, 4 | hocoi 29220 | . . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → ((𝑇 ∘ 𝑇)‘𝑦) = (𝑇‘(𝑇‘𝑦))) |
29 | 27, 28 | syl5reqr 2844 | . . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℋ → (𝑇‘(𝑇‘𝑦)) = (𝑇‘𝑦)) |
30 | 29 | adantl 482 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑇‘𝑦)) = (𝑇‘𝑦)) |
31 | 30 | oveq2d 7023 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑇‘𝑦))) = (𝑥 ·ih (𝑇‘𝑦))) |
32 | 26, 31 | eqtr3d 2831 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦))) |
33 | 32 | oveq2d 7023 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑇‘𝑦)) − ((𝑇‘𝑥) ·ih (𝑇‘𝑦))) = ((𝑥 ·ih (𝑇‘𝑦)) − (𝑥 ·ih (𝑇‘𝑦)))) |
34 | hicl 28536 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇‘𝑦)) ∈ ℂ) | |
35 | 20, 34 | sylan2 592 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih (𝑇‘𝑦)) ∈ ℂ) |
36 | 35 | subidd 10822 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑇‘𝑦)) − (𝑥 ·ih (𝑇‘𝑦))) = 0) |
37 | 23, 33, 36 | 3eqtrd 2833 | . . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 −ℎ (𝑇‘𝑥)) ·ih (𝑇‘𝑦)) = 0) |
38 | 37 | ralrimiva 3147 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ∀𝑦 ∈ ℋ ((𝑥 −ℎ (𝑇‘𝑥)) ·ih (𝑇‘𝑦)) = 0) |
39 | oveq2 7015 | . . . . . . . . 9 ⊢ (𝑧 = (𝑇‘𝑦) → ((𝑥 −ℎ (𝑇‘𝑥)) ·ih 𝑧) = ((𝑥 −ℎ (𝑇‘𝑥)) ·ih (𝑇‘𝑦))) | |
40 | 39 | eqeq1d 2795 | . . . . . . . 8 ⊢ (𝑧 = (𝑇‘𝑦) → (((𝑥 −ℎ (𝑇‘𝑥)) ·ih 𝑧) = 0 ↔ ((𝑥 −ℎ (𝑇‘𝑥)) ·ih (𝑇‘𝑦)) = 0)) |
41 | 40 | ralrn 6710 | . . . . . . 7 ⊢ (𝑇 Fn ℋ → (∀𝑧 ∈ ran 𝑇((𝑥 −ℎ (𝑇‘𝑥)) ·ih 𝑧) = 0 ↔ ∀𝑦 ∈ ℋ ((𝑥 −ℎ (𝑇‘𝑥)) ·ih (𝑇‘𝑦)) = 0)) |
42 | 6, 41 | ax-mp 5 | . . . . . 6 ⊢ (∀𝑧 ∈ ran 𝑇((𝑥 −ℎ (𝑇‘𝑥)) ·ih 𝑧) = 0 ↔ ∀𝑦 ∈ ℋ ((𝑥 −ℎ (𝑇‘𝑥)) ·ih (𝑇‘𝑦)) = 0) |
43 | 38, 42 | sylibr 235 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ∀𝑧 ∈ ran 𝑇((𝑥 −ℎ (𝑇‘𝑥)) ·ih 𝑧) = 0) |
44 | 8 | chssii 28687 | . . . . . 6 ⊢ ran 𝑇 ⊆ ℋ |
45 | ocel 28737 | . . . . . 6 ⊢ (ran 𝑇 ⊆ ℋ → ((𝑥 −ℎ (𝑇‘𝑥)) ∈ (⊥‘ran 𝑇) ↔ ((𝑥 −ℎ (𝑇‘𝑥)) ∈ ℋ ∧ ∀𝑧 ∈ ran 𝑇((𝑥 −ℎ (𝑇‘𝑥)) ·ih 𝑧) = 0))) | |
46 | 44, 45 | ax-mp 5 | . . . . 5 ⊢ ((𝑥 −ℎ (𝑇‘𝑥)) ∈ (⊥‘ran 𝑇) ↔ ((𝑥 −ℎ (𝑇‘𝑥)) ∈ ℋ ∧ ∀𝑧 ∈ ran 𝑇((𝑥 −ℎ (𝑇‘𝑥)) ·ih 𝑧) = 0)) |
47 | 17, 43, 46 | sylanbrc 583 | . . . 4 ⊢ (𝑥 ∈ ℋ → (𝑥 −ℎ (𝑇‘𝑥)) ∈ (⊥‘ran 𝑇)) |
48 | 8 | pjcompi 29128 | . . . 4 ⊢ (((𝑇‘𝑥) ∈ ran 𝑇 ∧ (𝑥 −ℎ (𝑇‘𝑥)) ∈ (⊥‘ran 𝑇)) → ((projℎ‘ran 𝑇)‘((𝑇‘𝑥) +ℎ (𝑥 −ℎ (𝑇‘𝑥)))) = (𝑇‘𝑥)) |
49 | 13, 47, 48 | syl2anc 584 | . . 3 ⊢ (𝑥 ∈ ℋ → ((projℎ‘ran 𝑇)‘((𝑇‘𝑥) +ℎ (𝑥 −ℎ (𝑇‘𝑥)))) = (𝑇‘𝑥)) |
50 | hvpncan3 28498 | . . . . 5 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) +ℎ (𝑥 −ℎ (𝑇‘𝑥))) = 𝑥) | |
51 | 15, 14, 50 | syl2anc 584 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑇‘𝑥) +ℎ (𝑥 −ℎ (𝑇‘𝑥))) = 𝑥) |
52 | 51 | fveq2d 6534 | . . 3 ⊢ (𝑥 ∈ ℋ → ((projℎ‘ran 𝑇)‘((𝑇‘𝑥) +ℎ (𝑥 −ℎ (𝑇‘𝑥)))) = ((projℎ‘ran 𝑇)‘𝑥)) |
53 | 49, 52 | eqtr3d 2831 | . 2 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) = ((projℎ‘ran 𝑇)‘𝑥)) |
54 | 11, 53 | mprgbir 3118 | 1 ⊢ 𝑇 = (projℎ‘ran 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ∀wral 3103 ⊆ wss 3854 ran crn 5436 ∘ ccom 5439 Fn wfn 6212 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 ℂcc 10370 0cc0 10372 − cmin 10706 ℋchba 28375 +ℎ cva 28376 ·ih csp 28378 −ℎ cmv 28381 ⊥cort 28386 projℎcpjh 28393 LinOpclo 28403 HrmOpcho 28406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-inf2 8939 ax-cc 9692 ax-dc 9703 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 ax-addf 10451 ax-mulf 10452 ax-hilex 28455 ax-hfvadd 28456 ax-hvcom 28457 ax-hvass 28458 ax-hv0cl 28459 ax-hvaddid 28460 ax-hfvmul 28461 ax-hvmulid 28462 ax-hvmulass 28463 ax-hvdistr1 28464 ax-hvdistr2 28465 ax-hvmul0 28466 ax-hfi 28535 ax-his1 28538 ax-his2 28539 ax-his3 28540 ax-his4 28541 ax-hcompl 28658 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-omul 7949 df-er 8130 df-map 8249 df-pm 8250 df-ixp 8301 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-fi 8711 df-sup 8742 df-inf 8743 df-oi 8810 df-card 9203 df-acn 9206 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-q 12187 df-rp 12229 df-xneg 12346 df-xadd 12347 df-xmul 12348 df-ioo 12581 df-ico 12583 df-icc 12584 df-fz 12732 df-fzo 12873 df-fl 13000 df-seq 13208 df-exp 13268 df-hash 13529 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-clim 14667 df-rlim 14668 df-sum 14865 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-starv 16397 df-sca 16398 df-vsca 16399 df-ip 16400 df-tset 16401 df-ple 16402 df-ds 16404 df-unif 16405 df-hom 16406 df-cco 16407 df-rest 16513 df-topn 16514 df-0g 16532 df-gsum 16533 df-topgen 16534 df-pt 16535 df-prds 16538 df-xrs 16592 df-qtop 16597 df-imas 16598 df-xps 16600 df-mre 16674 df-mrc 16675 df-acs 16677 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-submnd 17763 df-mulg 17970 df-cntz 18176 df-cmn 18623 df-psmet 20207 df-xmet 20208 df-met 20209 df-bl 20210 df-mopn 20211 df-fbas 20212 df-fg 20213 df-cnfld 20216 df-top 21174 df-topon 21191 df-topsp 21213 df-bases 21226 df-cld 21299 df-ntr 21300 df-cls 21301 df-nei 21378 df-cn 21507 df-cnp 21508 df-lm 21509 df-t1 21594 df-haus 21595 df-cmp 21667 df-tx 21842 df-hmeo 22035 df-fil 22126 df-fm 22218 df-flim 22219 df-flf 22220 df-fcls 22221 df-xms 22601 df-ms 22602 df-tms 22603 df-cncf 23157 df-cfil 23529 df-cau 23530 df-cmet 23531 df-grpo 27949 df-gid 27950 df-ginv 27951 df-gdiv 27952 df-ablo 28001 df-vc 28015 df-nv 28048 df-va 28051 df-ba 28052 df-sm 28053 df-0v 28054 df-vs 28055 df-nmcv 28056 df-ims 28057 df-dip 28157 df-ssp 28178 df-lno 28200 df-nmoo 28201 df-blo 28202 df-0o 28203 df-ph 28269 df-cbn 28319 df-hlo 28342 df-hnorm 28424 df-hba 28425 df-hvsub 28427 df-hlim 28428 df-hcau 28429 df-sh 28663 df-ch 28677 df-oc 28708 df-ch0 28709 df-shs 28764 df-pjh 28851 df-h0op 29204 df-nmop 29295 df-cnop 29296 df-lnop 29297 df-bdop 29298 df-unop 29299 df-hmop 29300 |
This theorem is referenced by: hmopidmpj 29610 |
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