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Mirrors > Home > HSE Home > Th. List > adjbd1o | Structured version Visualization version GIF version |
Description: The mapping of adjoints of bounded linear operators is one-to-one onto. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
adjbd1o | ⊢ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adj1o 30878 | . . . 4 ⊢ adjℎ:dom adjℎ–1-1-onto→dom adjℎ | |
2 | f1of1 6788 | . . . 4 ⊢ (adjℎ:dom adjℎ–1-1-onto→dom adjℎ → adjℎ:dom adjℎ–1-1→dom adjℎ) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ adjℎ:dom adjℎ–1-1→dom adjℎ |
4 | bdopssadj 31065 | . . 3 ⊢ BndLinOp ⊆ dom adjℎ | |
5 | f1ores 6803 | . . 3 ⊢ ((adjℎ:dom adjℎ–1-1→dom adjℎ ∧ BndLinOp ⊆ dom adjℎ) → (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adjℎ “ BndLinOp)) | |
6 | 3, 4, 5 | mp2an 691 | . 2 ⊢ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adjℎ “ BndLinOp) |
7 | vex 3452 | . . . . . 6 ⊢ 𝑦 ∈ V | |
8 | 7 | elima 6023 | . . . . 5 ⊢ (𝑦 ∈ (adjℎ “ BndLinOp) ↔ ∃𝑥 ∈ BndLinOp 𝑥adjℎ𝑦) |
9 | f1ofn 6790 | . . . . . . . 8 ⊢ (adjℎ:dom adjℎ–1-1-onto→dom adjℎ → adjℎ Fn dom adjℎ) | |
10 | 1, 9 | ax-mp 5 | . . . . . . 7 ⊢ adjℎ Fn dom adjℎ |
11 | bdopadj 31066 | . . . . . . 7 ⊢ (𝑥 ∈ BndLinOp → 𝑥 ∈ dom adjℎ) | |
12 | fnbrfvb 6900 | . . . . . . 7 ⊢ ((adjℎ Fn dom adjℎ ∧ 𝑥 ∈ dom adjℎ) → ((adjℎ‘𝑥) = 𝑦 ↔ 𝑥adjℎ𝑦)) | |
13 | 10, 11, 12 | sylancr 588 | . . . . . 6 ⊢ (𝑥 ∈ BndLinOp → ((adjℎ‘𝑥) = 𝑦 ↔ 𝑥adjℎ𝑦)) |
14 | 13 | rexbiia 3096 | . . . . 5 ⊢ (∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ BndLinOp 𝑥adjℎ𝑦) |
15 | adjbdlnb 31068 | . . . . . . . . 9 ⊢ (𝑥 ∈ BndLinOp ↔ (adjℎ‘𝑥) ∈ BndLinOp) | |
16 | eleq1 2826 | . . . . . . . . 9 ⊢ ((adjℎ‘𝑥) = 𝑦 → ((adjℎ‘𝑥) ∈ BndLinOp ↔ 𝑦 ∈ BndLinOp)) | |
17 | 15, 16 | bitrid 283 | . . . . . . . 8 ⊢ ((adjℎ‘𝑥) = 𝑦 → (𝑥 ∈ BndLinOp ↔ 𝑦 ∈ BndLinOp)) |
18 | 17 | biimpcd 249 | . . . . . . 7 ⊢ (𝑥 ∈ BndLinOp → ((adjℎ‘𝑥) = 𝑦 → 𝑦 ∈ BndLinOp)) |
19 | 18 | rexlimiv 3146 | . . . . . 6 ⊢ (∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦 → 𝑦 ∈ BndLinOp) |
20 | adjbdln 31067 | . . . . . . 7 ⊢ (𝑦 ∈ BndLinOp → (adjℎ‘𝑦) ∈ BndLinOp) | |
21 | bdopadj 31066 | . . . . . . . 8 ⊢ (𝑦 ∈ BndLinOp → 𝑦 ∈ dom adjℎ) | |
22 | adjadj 30920 | . . . . . . . 8 ⊢ (𝑦 ∈ dom adjℎ → (adjℎ‘(adjℎ‘𝑦)) = 𝑦) | |
23 | 21, 22 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ BndLinOp → (adjℎ‘(adjℎ‘𝑦)) = 𝑦) |
24 | fveqeq2 6856 | . . . . . . . 8 ⊢ (𝑥 = (adjℎ‘𝑦) → ((adjℎ‘𝑥) = 𝑦 ↔ (adjℎ‘(adjℎ‘𝑦)) = 𝑦)) | |
25 | 24 | rspcev 3584 | . . . . . . 7 ⊢ (((adjℎ‘𝑦) ∈ BndLinOp ∧ (adjℎ‘(adjℎ‘𝑦)) = 𝑦) → ∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦) |
26 | 20, 23, 25 | syl2anc 585 | . . . . . 6 ⊢ (𝑦 ∈ BndLinOp → ∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦) |
27 | 19, 26 | impbii 208 | . . . . 5 ⊢ (∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦 ↔ 𝑦 ∈ BndLinOp) |
28 | 8, 14, 27 | 3bitr2i 299 | . . . 4 ⊢ (𝑦 ∈ (adjℎ “ BndLinOp) ↔ 𝑦 ∈ BndLinOp) |
29 | 28 | eqriv 2734 | . . 3 ⊢ (adjℎ “ BndLinOp) = BndLinOp |
30 | f1oeq3 6779 | . . 3 ⊢ ((adjℎ “ BndLinOp) = BndLinOp → ((adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adjℎ “ BndLinOp) ↔ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp)) | |
31 | 29, 30 | ax-mp 5 | . 2 ⊢ ((adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adjℎ “ BndLinOp) ↔ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp) |
32 | 6, 31 | mpbi 229 | 1 ⊢ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∃wrex 3074 ⊆ wss 3915 class class class wbr 5110 dom cdm 5638 ↾ cres 5640 “ cima 5641 Fn wfn 6496 –1-1→wf1 6498 –1-1-onto→wf1o 6500 ‘cfv 6501 BndLinOpcbo 29932 adjℎcado 29939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cc 10378 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 ax-hilex 29983 ax-hfvadd 29984 ax-hvcom 29985 ax-hvass 29986 ax-hv0cl 29987 ax-hvaddid 29988 ax-hfvmul 29989 ax-hvmulid 29990 ax-hvmulass 29991 ax-hvdistr1 29992 ax-hvdistr2 29993 ax-hvmul0 29994 ax-hfi 30063 ax-his1 30066 ax-his2 30067 ax-his3 30068 ax-his4 30069 ax-hcompl 30186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-omul 8422 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-acn 9885 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-rlim 15378 df-sum 15578 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-mulg 18880 df-cntz 19104 df-cmn 19571 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-cn 22594 df-cnp 22595 df-lm 22596 df-t1 22681 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cfil 24635 df-cau 24636 df-cmet 24637 df-grpo 29477 df-gid 29478 df-ginv 29479 df-gdiv 29480 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-vs 29583 df-nmcv 29584 df-ims 29585 df-dip 29685 df-ssp 29706 df-ph 29797 df-cbn 29847 df-hnorm 29952 df-hba 29953 df-hvsub 29955 df-hlim 29956 df-hcau 29957 df-sh 30191 df-ch 30205 df-oc 30236 df-ch0 30237 df-shs 30292 df-pjh 30379 df-h0op 30732 df-nmop 30823 df-cnop 30824 df-lnop 30825 df-bdop 30826 df-unop 30827 df-hmop 30828 df-nmfn 30829 df-nlfn 30830 df-cnfn 30831 df-lnfn 30832 df-adjh 30833 |
This theorem is referenced by: (None) |
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