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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 29329 | . . . 4 ⊢ adjℎ:dom adjℎ–1-1-onto→dom adjℎ | |
2 | f1of1 6392 | . . . 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 29516 | . . 3 ⊢ BndLinOp ⊆ dom adjℎ | |
5 | f1ores 6407 | . . 3 ⊢ ((adjℎ:dom adjℎ–1-1→dom adjℎ ∧ BndLinOp ⊆ dom adjℎ) → (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adjℎ “ BndLinOp)) | |
6 | 3, 4, 5 | mp2an 682 | . 2 ⊢ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adjℎ “ BndLinOp) |
7 | vex 3401 | . . . . . 6 ⊢ 𝑦 ∈ V | |
8 | 7 | elima 5727 | . . . . 5 ⊢ (𝑦 ∈ (adjℎ “ BndLinOp) ↔ ∃𝑥 ∈ BndLinOp 𝑥adjℎ𝑦) |
9 | f1ofn 6394 | . . . . . . . 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 29517 | . . . . . . 7 ⊢ (𝑥 ∈ BndLinOp → 𝑥 ∈ dom adjℎ) | |
12 | fnbrfvb 6497 | . . . . . . 7 ⊢ ((adjℎ Fn dom adjℎ ∧ 𝑥 ∈ dom adjℎ) → ((adjℎ‘𝑥) = 𝑦 ↔ 𝑥adjℎ𝑦)) | |
13 | 10, 11, 12 | sylancr 581 | . . . . . 6 ⊢ (𝑥 ∈ BndLinOp → ((adjℎ‘𝑥) = 𝑦 ↔ 𝑥adjℎ𝑦)) |
14 | 13 | rexbiia 3223 | . . . . 5 ⊢ (∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ BndLinOp 𝑥adjℎ𝑦) |
15 | adjbdlnb 29519 | . . . . . . . . 9 ⊢ (𝑥 ∈ BndLinOp ↔ (adjℎ‘𝑥) ∈ BndLinOp) | |
16 | eleq1 2847 | . . . . . . . . 9 ⊢ ((adjℎ‘𝑥) = 𝑦 → ((adjℎ‘𝑥) ∈ BndLinOp ↔ 𝑦 ∈ BndLinOp)) | |
17 | 15, 16 | syl5bb 275 | . . . . . . . 8 ⊢ ((adjℎ‘𝑥) = 𝑦 → (𝑥 ∈ BndLinOp ↔ 𝑦 ∈ BndLinOp)) |
18 | 17 | biimpcd 241 | . . . . . . 7 ⊢ (𝑥 ∈ BndLinOp → ((adjℎ‘𝑥) = 𝑦 → 𝑦 ∈ BndLinOp)) |
19 | 18 | rexlimiv 3209 | . . . . . 6 ⊢ (∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦 → 𝑦 ∈ BndLinOp) |
20 | adjbdln 29518 | . . . . . . 7 ⊢ (𝑦 ∈ BndLinOp → (adjℎ‘𝑦) ∈ BndLinOp) | |
21 | bdopadj 29517 | . . . . . . . 8 ⊢ (𝑦 ∈ BndLinOp → 𝑦 ∈ dom adjℎ) | |
22 | adjadj 29371 | . . . . . . . 8 ⊢ (𝑦 ∈ dom adjℎ → (adjℎ‘(adjℎ‘𝑦)) = 𝑦) | |
23 | 21, 22 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ BndLinOp → (adjℎ‘(adjℎ‘𝑦)) = 𝑦) |
24 | fveqeq2 6457 | . . . . . . . 8 ⊢ (𝑥 = (adjℎ‘𝑦) → ((adjℎ‘𝑥) = 𝑦 ↔ (adjℎ‘(adjℎ‘𝑦)) = 𝑦)) | |
25 | 24 | rspcev 3511 | . . . . . . 7 ⊢ (((adjℎ‘𝑦) ∈ BndLinOp ∧ (adjℎ‘(adjℎ‘𝑦)) = 𝑦) → ∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦) |
26 | 20, 23, 25 | syl2anc 579 | . . . . . 6 ⊢ (𝑦 ∈ BndLinOp → ∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦) |
27 | 19, 26 | impbii 201 | . . . . 5 ⊢ (∃𝑥 ∈ BndLinOp (adjℎ‘𝑥) = 𝑦 ↔ 𝑦 ∈ BndLinOp) |
28 | 8, 14, 27 | 3bitr2i 291 | . . . 4 ⊢ (𝑦 ∈ (adjℎ “ BndLinOp) ↔ 𝑦 ∈ BndLinOp) |
29 | 28 | eqriv 2775 | . . 3 ⊢ (adjℎ “ BndLinOp) = BndLinOp |
30 | f1oeq3 6384 | . . 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 222 | 1 ⊢ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ⊆ wss 3792 class class class wbr 4888 dom cdm 5357 ↾ cres 5359 “ cima 5360 Fn wfn 6132 –1-1→wf1 6134 –1-1-onto→wf1o 6136 ‘cfv 6137 BndLinOpcbo 28381 adjℎcado 28388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cc 9594 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 ax-hilex 28432 ax-hfvadd 28433 ax-hvcom 28434 ax-hvass 28435 ax-hv0cl 28436 ax-hvaddid 28437 ax-hfvmul 28438 ax-hvmulid 28439 ax-hvmulass 28440 ax-hvdistr1 28441 ax-hvdistr2 28442 ax-hvmul0 28443 ax-hfi 28512 ax-his1 28515 ax-his2 28516 ax-his3 28517 ax-his4 28518 ax-hcompl 28635 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-omul 7850 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-acn 9103 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ioo 12495 df-ico 12497 df-icc 12498 df-fz 12648 df-fzo 12789 df-fl 12916 df-seq 13124 df-exp 13183 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-clim 14631 df-rlim 14632 df-sum 14829 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-sca 16358 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-hom 16366 df-cco 16367 df-rest 16473 df-topn 16474 df-0g 16492 df-gsum 16493 df-topgen 16494 df-pt 16495 df-prds 16498 df-xrs 16552 df-qtop 16557 df-imas 16558 df-xps 16560 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-mulg 17932 df-cntz 18137 df-cmn 18585 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-fbas 20143 df-fg 20144 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-cn 21443 df-cnp 21444 df-lm 21445 df-t1 21530 df-haus 21531 df-tx 21778 df-hmeo 21971 df-fil 22062 df-fm 22154 df-flim 22155 df-flf 22156 df-xms 22537 df-ms 22538 df-tms 22539 df-cfil 23465 df-cau 23466 df-cmet 23467 df-grpo 27924 df-gid 27925 df-ginv 27926 df-gdiv 27927 df-ablo 27976 df-vc 27990 df-nv 28023 df-va 28026 df-ba 28027 df-sm 28028 df-0v 28029 df-vs 28030 df-nmcv 28031 df-ims 28032 df-dip 28132 df-ssp 28153 df-ph 28244 df-cbn 28295 df-hnorm 28401 df-hba 28402 df-hvsub 28404 df-hlim 28405 df-hcau 28406 df-sh 28640 df-ch 28654 df-oc 28685 df-ch0 28686 df-shs 28743 df-pjh 28830 df-h0op 29183 df-nmop 29274 df-cnop 29275 df-lnop 29276 df-bdop 29277 df-unop 29278 df-hmop 29279 df-nmfn 29280 df-nlfn 29281 df-cnfn 29282 df-lnfn 29283 df-adjh 29284 |
This theorem is referenced by: (None) |
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