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Mirrors > Home > HSE Home > Th. List > lnopeqi | Structured version Visualization version GIF version |
Description: Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopeq.1 | ⊢ 𝑇 ∈ LinOp |
lnopeq.2 | ⊢ 𝑈 ∈ LinOp |
Ref | Expression |
---|---|
lnopeqi | ⊢ (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnopeq.1 | . . . . . . . 8 ⊢ 𝑇 ∈ LinOp | |
2 | 1 | lnopfi 30004 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ |
3 | 2 | ffvelrni 6881 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
4 | hicl 29115 | . . . . . 6 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑥) ∈ ℂ) | |
5 | 3, 4 | mpancom 688 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑇‘𝑥) ·ih 𝑥) ∈ ℂ) |
6 | lnopeq.2 | . . . . . . . 8 ⊢ 𝑈 ∈ LinOp | |
7 | 6 | lnopfi 30004 | . . . . . . 7 ⊢ 𝑈: ℋ⟶ ℋ |
8 | 7 | ffvelrni 6881 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝑈‘𝑥) ∈ ℋ) |
9 | hicl 29115 | . . . . . 6 ⊢ (((𝑈‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈‘𝑥) ·ih 𝑥) ∈ ℂ) | |
10 | 8, 9 | mpancom 688 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑈‘𝑥) ·ih 𝑥) ∈ ℂ) |
11 | 5, 10 | subeq0ad 11164 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((((𝑇‘𝑥) ·ih 𝑥) − ((𝑈‘𝑥) ·ih 𝑥)) = 0 ↔ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥))) |
12 | hodval 29777 | . . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 −op 𝑈)‘𝑥) = ((𝑇‘𝑥) −ℎ (𝑈‘𝑥))) | |
13 | 2, 7, 12 | mp3an12 1453 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑇 −op 𝑈)‘𝑥) = ((𝑇‘𝑥) −ℎ (𝑈‘𝑥))) |
14 | 13 | oveq1d 7206 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (((𝑇 −op 𝑈)‘𝑥) ·ih 𝑥) = (((𝑇‘𝑥) −ℎ (𝑈‘𝑥)) ·ih 𝑥)) |
15 | id 22 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → 𝑥 ∈ ℋ) | |
16 | his2sub 29127 | . . . . . . 7 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑈‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑇‘𝑥) −ℎ (𝑈‘𝑥)) ·ih 𝑥) = (((𝑇‘𝑥) ·ih 𝑥) − ((𝑈‘𝑥) ·ih 𝑥))) | |
17 | 3, 8, 15, 16 | syl3anc 1373 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (((𝑇‘𝑥) −ℎ (𝑈‘𝑥)) ·ih 𝑥) = (((𝑇‘𝑥) ·ih 𝑥) − ((𝑈‘𝑥) ·ih 𝑥))) |
18 | 14, 17 | eqtr2d 2772 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑇‘𝑥) ·ih 𝑥) − ((𝑈‘𝑥) ·ih 𝑥)) = (((𝑇 −op 𝑈)‘𝑥) ·ih 𝑥)) |
19 | 18 | eqeq1d 2738 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((((𝑇‘𝑥) ·ih 𝑥) − ((𝑈‘𝑥) ·ih 𝑥)) = 0 ↔ (((𝑇 −op 𝑈)‘𝑥) ·ih 𝑥) = 0)) |
20 | 11, 19 | bitr3d 284 | . . 3 ⊢ (𝑥 ∈ ℋ → (((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ (((𝑇 −op 𝑈)‘𝑥) ·ih 𝑥) = 0)) |
21 | 20 | ralbiia 3077 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ (((𝑇 −op 𝑈)‘𝑥) ·ih 𝑥) = 0) |
22 | 1, 6 | lnophdi 30037 | . . 3 ⊢ (𝑇 −op 𝑈) ∈ LinOp |
23 | 22 | lnopeq0i 30042 | . 2 ⊢ (∀𝑥 ∈ ℋ (((𝑇 −op 𝑈)‘𝑥) ·ih 𝑥) = 0 ↔ (𝑇 −op 𝑈) = 0hop ) |
24 | 2, 7 | hosubeq0i 29861 | . 2 ⊢ ((𝑇 −op 𝑈) = 0hop ↔ 𝑇 = 𝑈) |
25 | 21, 23, 24 | 3bitri 300 | 1 ⊢ (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 − cmin 11027 ℋchba 28954 ·ih csp 28957 −ℎ cmv 28960 −op chod 28975 0hop ch0o 28978 LinOpclo 28982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cc 10014 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 ax-hilex 29034 ax-hfvadd 29035 ax-hvcom 29036 ax-hvass 29037 ax-hv0cl 29038 ax-hvaddid 29039 ax-hfvmul 29040 ax-hvmulid 29041 ax-hvmulass 29042 ax-hvdistr1 29043 ax-hvdistr2 29044 ax-hvmul0 29045 ax-hfi 29114 ax-his1 29117 ax-his2 29118 ax-his3 29119 ax-his4 29120 ax-hcompl 29237 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-oadd 8184 df-omul 8185 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-acn 9523 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-rlim 15015 df-sum 15215 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-mulg 18443 df-cntz 18665 df-cmn 19126 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-cn 22078 df-cnp 22079 df-lm 22080 df-haus 22166 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-xms 23172 df-ms 23173 df-tms 23174 df-cfil 24106 df-cau 24107 df-cmet 24108 df-grpo 28528 df-gid 28529 df-ginv 28530 df-gdiv 28531 df-ablo 28580 df-vc 28594 df-nv 28627 df-va 28630 df-ba 28631 df-sm 28632 df-0v 28633 df-vs 28634 df-nmcv 28635 df-ims 28636 df-dip 28736 df-ssp 28757 df-ph 28848 df-cbn 28898 df-hnorm 29003 df-hba 29004 df-hvsub 29006 df-hlim 29007 df-hcau 29008 df-sh 29242 df-ch 29256 df-oc 29287 df-ch0 29288 df-shs 29343 df-pjh 29430 df-hosum 29765 df-homul 29766 df-hodif 29767 df-h0op 29783 df-lnop 29876 |
This theorem is referenced by: lnopeq 30044 |
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