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Mirrors > Home > HSE Home > Th. List > rnbra | Structured version Visualization version GIF version |
Description: The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rnbra | β’ ran bra = (LinFn β© ContFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnfncnbd 31805 | . . . 4 β’ (π‘ β LinFn β (π‘ β ContFn β (normfnβπ‘) β β)) | |
2 | 1 | pm5.32i 574 | . . 3 β’ ((π‘ β LinFn β§ π‘ β ContFn) β (π‘ β LinFn β§ (normfnβπ‘) β β)) |
3 | elin 3957 | . . 3 β’ (π‘ β (LinFn β© ContFn) β (π‘ β LinFn β§ π‘ β ContFn)) | |
4 | ax-hilex 30747 | . . . . . . 7 β’ β β V | |
5 | 4 | mptex 7217 | . . . . . 6 β’ (π¦ β β β¦ (π¦ Β·ih π₯)) β V |
6 | df-bra 31598 | . . . . . 6 β’ bra = (π₯ β β β¦ (π¦ β β β¦ (π¦ Β·ih π₯))) | |
7 | 5, 6 | fnmpti 6684 | . . . . 5 β’ bra Fn β |
8 | fvelrnb 6943 | . . . . 5 β’ (bra Fn β β (π‘ β ran bra β βπ₯ β β (braβπ₯) = π‘)) | |
9 | 7, 8 | ax-mp 5 | . . . 4 β’ (π‘ β ran bra β βπ₯ β β (braβπ₯) = π‘) |
10 | bralnfn 31696 | . . . . . . . 8 β’ (π₯ β β β (braβπ₯) β LinFn) | |
11 | brabn 31854 | . . . . . . . 8 β’ (π₯ β β β (normfnβ(braβπ₯)) β β) | |
12 | 10, 11 | jca 511 | . . . . . . 7 β’ (π₯ β β β ((braβπ₯) β LinFn β§ (normfnβ(braβπ₯)) β β)) |
13 | eleq1 2813 | . . . . . . . 8 β’ ((braβπ₯) = π‘ β ((braβπ₯) β LinFn β π‘ β LinFn)) | |
14 | fveq2 6882 | . . . . . . . . 9 β’ ((braβπ₯) = π‘ β (normfnβ(braβπ₯)) = (normfnβπ‘)) | |
15 | 14 | eleq1d 2810 | . . . . . . . 8 β’ ((braβπ₯) = π‘ β ((normfnβ(braβπ₯)) β β β (normfnβπ‘) β β)) |
16 | 13, 15 | anbi12d 630 | . . . . . . 7 β’ ((braβπ₯) = π‘ β (((braβπ₯) β LinFn β§ (normfnβ(braβπ₯)) β β) β (π‘ β LinFn β§ (normfnβπ‘) β β))) |
17 | 12, 16 | syl5ibcom 244 | . . . . . 6 β’ (π₯ β β β ((braβπ₯) = π‘ β (π‘ β LinFn β§ (normfnβπ‘) β β))) |
18 | 17 | rexlimiv 3140 | . . . . 5 β’ (βπ₯ β β (braβπ₯) = π‘ β (π‘ β LinFn β§ (normfnβπ‘) β β)) |
19 | riesz1 31813 | . . . . . . 7 β’ (π‘ β LinFn β ((normfnβπ‘) β β β βπ₯ β β βπ¦ β β (π‘βπ¦) = (π¦ Β·ih π₯))) | |
20 | 19 | biimpa 476 | . . . . . 6 β’ ((π‘ β LinFn β§ (normfnβπ‘) β β) β βπ₯ β β βπ¦ β β (π‘βπ¦) = (π¦ Β·ih π₯)) |
21 | braval 31692 | . . . . . . . . . . 11 β’ ((π₯ β β β§ π¦ β β) β ((braβπ₯)βπ¦) = (π¦ Β·ih π₯)) | |
22 | eqtr3 2750 | . . . . . . . . . . . 12 β’ ((((braβπ₯)βπ¦) = (π¦ Β·ih π₯) β§ (π‘βπ¦) = (π¦ Β·ih π₯)) β ((braβπ₯)βπ¦) = (π‘βπ¦)) | |
23 | 22 | ex 412 | . . . . . . . . . . 11 β’ (((braβπ₯)βπ¦) = (π¦ Β·ih π₯) β ((π‘βπ¦) = (π¦ Β·ih π₯) β ((braβπ₯)βπ¦) = (π‘βπ¦))) |
24 | 21, 23 | syl 17 | . . . . . . . . . 10 β’ ((π₯ β β β§ π¦ β β) β ((π‘βπ¦) = (π¦ Β·ih π₯) β ((braβπ₯)βπ¦) = (π‘βπ¦))) |
25 | 24 | ralimdva 3159 | . . . . . . . . 9 β’ (π₯ β β β (βπ¦ β β (π‘βπ¦) = (π¦ Β·ih π₯) β βπ¦ β β ((braβπ₯)βπ¦) = (π‘βπ¦))) |
26 | 25 | adantl 481 | . . . . . . . 8 β’ (((π‘ β LinFn β§ (normfnβπ‘) β β) β§ π₯ β β) β (βπ¦ β β (π‘βπ¦) = (π¦ Β·ih π₯) β βπ¦ β β ((braβπ₯)βπ¦) = (π‘βπ¦))) |
27 | brafn 31695 | . . . . . . . . 9 β’ (π₯ β β β (braβπ₯): ββΆβ) | |
28 | lnfnf 31632 | . . . . . . . . . 10 β’ (π‘ β LinFn β π‘: ββΆβ) | |
29 | 28 | adantr 480 | . . . . . . . . 9 β’ ((π‘ β LinFn β§ (normfnβπ‘) β β) β π‘: ββΆβ) |
30 | ffn 6708 | . . . . . . . . . 10 β’ ((braβπ₯): ββΆβ β (braβπ₯) Fn β) | |
31 | ffn 6708 | . . . . . . . . . 10 β’ (π‘: ββΆβ β π‘ Fn β) | |
32 | eqfnfv 7023 | . . . . . . . . . 10 β’ (((braβπ₯) Fn β β§ π‘ Fn β) β ((braβπ₯) = π‘ β βπ¦ β β ((braβπ₯)βπ¦) = (π‘βπ¦))) | |
33 | 30, 31, 32 | syl2an 595 | . . . . . . . . 9 β’ (((braβπ₯): ββΆβ β§ π‘: ββΆβ) β ((braβπ₯) = π‘ β βπ¦ β β ((braβπ₯)βπ¦) = (π‘βπ¦))) |
34 | 27, 29, 33 | syl2anr 596 | . . . . . . . 8 β’ (((π‘ β LinFn β§ (normfnβπ‘) β β) β§ π₯ β β) β ((braβπ₯) = π‘ β βπ¦ β β ((braβπ₯)βπ¦) = (π‘βπ¦))) |
35 | 26, 34 | sylibrd 259 | . . . . . . 7 β’ (((π‘ β LinFn β§ (normfnβπ‘) β β) β§ π₯ β β) β (βπ¦ β β (π‘βπ¦) = (π¦ Β·ih π₯) β (braβπ₯) = π‘)) |
36 | 35 | reximdva 3160 | . . . . . 6 β’ ((π‘ β LinFn β§ (normfnβπ‘) β β) β (βπ₯ β β βπ¦ β β (π‘βπ¦) = (π¦ Β·ih π₯) β βπ₯ β β (braβπ₯) = π‘)) |
37 | 20, 36 | mpd 15 | . . . . 5 β’ ((π‘ β LinFn β§ (normfnβπ‘) β β) β βπ₯ β β (braβπ₯) = π‘) |
38 | 18, 37 | impbii 208 | . . . 4 β’ (βπ₯ β β (braβπ₯) = π‘ β (π‘ β LinFn β§ (normfnβπ‘) β β)) |
39 | 9, 38 | bitri 275 | . . 3 β’ (π‘ β ran bra β (π‘ β LinFn β§ (normfnβπ‘) β β)) |
40 | 2, 3, 39 | 3bitr4ri 304 | . 2 β’ (π‘ β ran bra β π‘ β (LinFn β© ContFn)) |
41 | 40 | eqriv 2721 | 1 β’ ran bra = (LinFn β© ContFn) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3053 βwrex 3062 β© cin 3940 β¦ cmpt 5222 ran crn 5668 Fn wfn 6529 βΆwf 6530 βcfv 6534 (class class class)co 7402 βcc 11105 βcr 11106 βchba 30667 Β·ih csp 30670 normfncnmf 30699 ContFnccnfn 30701 LinFnclf 30702 bracbr 30704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cc 10427 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 ax-hilex 30747 ax-hfvadd 30748 ax-hvcom 30749 ax-hvass 30750 ax-hv0cl 30751 ax-hvaddid 30752 ax-hfvmul 30753 ax-hvmulid 30754 ax-hvmulass 30755 ax-hvdistr1 30756 ax-hvdistr2 30757 ax-hvmul0 30758 ax-hfi 30827 ax-his1 30830 ax-his2 30831 ax-his3 30832 ax-his4 30833 ax-hcompl 30950 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-acn 9934 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ioo 13329 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-fl 13758 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-mulg 18992 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-cnfld 21235 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cld 22867 df-ntr 22868 df-cls 22869 df-nei 22946 df-cn 23075 df-cnp 23076 df-lm 23077 df-t1 23162 df-haus 23163 df-tx 23410 df-hmeo 23603 df-fil 23694 df-fm 23786 df-flim 23787 df-flf 23788 df-xms 24170 df-ms 24171 df-tms 24172 df-cfil 25127 df-cau 25128 df-cmet 25129 df-grpo 30241 df-gid 30242 df-ginv 30243 df-gdiv 30244 df-ablo 30293 df-vc 30307 df-nv 30340 df-va 30343 df-ba 30344 df-sm 30345 df-0v 30346 df-vs 30347 df-nmcv 30348 df-ims 30349 df-dip 30449 df-ssp 30470 df-ph 30561 df-cbn 30611 df-hnorm 30716 df-hba 30717 df-hvsub 30719 df-hlim 30720 df-hcau 30721 df-sh 30955 df-ch 30969 df-oc 31000 df-ch0 31001 df-nmfn 31593 df-nlfn 31594 df-cnfn 31595 df-lnfn 31596 df-bra 31598 |
This theorem is referenced by: bra11 31856 cnvbraval 31858 |
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