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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 | ⊢ ran bra = (LinFn ∩ ContFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnfncnbd 29836 | . . . 4 ⊢ (𝑡 ∈ LinFn → (𝑡 ∈ ContFn ↔ (normfn‘𝑡) ∈ ℝ)) | |
2 | 1 | pm5.32i 577 | . . 3 ⊢ ((𝑡 ∈ LinFn ∧ 𝑡 ∈ ContFn) ↔ (𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ)) |
3 | elin 4171 | . . 3 ⊢ (𝑡 ∈ (LinFn ∩ ContFn) ↔ (𝑡 ∈ LinFn ∧ 𝑡 ∈ ContFn)) | |
4 | ax-hilex 28778 | . . . . . . 7 ⊢ ℋ ∈ V | |
5 | 4 | mptex 6988 | . . . . . 6 ⊢ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥)) ∈ V |
6 | df-bra 29629 | . . . . . 6 ⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥))) | |
7 | 5, 6 | fnmpti 6493 | . . . . 5 ⊢ bra Fn ℋ |
8 | fvelrnb 6728 | . . . . 5 ⊢ (bra Fn ℋ → (𝑡 ∈ ran bra ↔ ∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡)) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (𝑡 ∈ ran bra ↔ ∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡) |
10 | bralnfn 29727 | . . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (bra‘𝑥) ∈ LinFn) | |
11 | brabn 29885 | . . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (normfn‘(bra‘𝑥)) ∈ ℝ) | |
12 | 10, 11 | jca 514 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((bra‘𝑥) ∈ LinFn ∧ (normfn‘(bra‘𝑥)) ∈ ℝ)) |
13 | eleq1 2902 | . . . . . . . 8 ⊢ ((bra‘𝑥) = 𝑡 → ((bra‘𝑥) ∈ LinFn ↔ 𝑡 ∈ LinFn)) | |
14 | fveq2 6672 | . . . . . . . . 9 ⊢ ((bra‘𝑥) = 𝑡 → (normfn‘(bra‘𝑥)) = (normfn‘𝑡)) | |
15 | 14 | eleq1d 2899 | . . . . . . . 8 ⊢ ((bra‘𝑥) = 𝑡 → ((normfn‘(bra‘𝑥)) ∈ ℝ ↔ (normfn‘𝑡) ∈ ℝ)) |
16 | 13, 15 | anbi12d 632 | . . . . . . 7 ⊢ ((bra‘𝑥) = 𝑡 → (((bra‘𝑥) ∈ LinFn ∧ (normfn‘(bra‘𝑥)) ∈ ℝ) ↔ (𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ))) |
17 | 12, 16 | syl5ibcom 247 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((bra‘𝑥) = 𝑡 → (𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ))) |
18 | 17 | rexlimiv 3282 | . . . . 5 ⊢ (∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡 → (𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ)) |
19 | riesz1 29844 | . . . . . . 7 ⊢ (𝑡 ∈ LinFn → ((normfn‘𝑡) ∈ ℝ ↔ ∃𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·ih 𝑥))) | |
20 | 19 | biimpa 479 | . . . . . 6 ⊢ ((𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ) → ∃𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·ih 𝑥)) |
21 | braval 29723 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝑥)‘𝑦) = (𝑦 ·ih 𝑥)) | |
22 | eqtr3 2845 | . . . . . . . . . . . 12 ⊢ ((((bra‘𝑥)‘𝑦) = (𝑦 ·ih 𝑥) ∧ (𝑡‘𝑦) = (𝑦 ·ih 𝑥)) → ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦)) | |
23 | 22 | ex 415 | . . . . . . . . . . 11 ⊢ (((bra‘𝑥)‘𝑦) = (𝑦 ·ih 𝑥) → ((𝑡‘𝑦) = (𝑦 ·ih 𝑥) → ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦))) |
24 | 21, 23 | syl 17 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑡‘𝑦) = (𝑦 ·ih 𝑥) → ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦))) |
25 | 24 | ralimdva 3179 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·ih 𝑥) → ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦))) |
26 | 25 | adantl 484 | . . . . . . . 8 ⊢ (((𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ) ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·ih 𝑥) → ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦))) |
27 | brafn 29726 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (bra‘𝑥): ℋ⟶ℂ) | |
28 | lnfnf 29663 | . . . . . . . . . 10 ⊢ (𝑡 ∈ LinFn → 𝑡: ℋ⟶ℂ) | |
29 | 28 | adantr 483 | . . . . . . . . 9 ⊢ ((𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ) → 𝑡: ℋ⟶ℂ) |
30 | ffn 6516 | . . . . . . . . . 10 ⊢ ((bra‘𝑥): ℋ⟶ℂ → (bra‘𝑥) Fn ℋ) | |
31 | ffn 6516 | . . . . . . . . . 10 ⊢ (𝑡: ℋ⟶ℂ → 𝑡 Fn ℋ) | |
32 | eqfnfv 6804 | . . . . . . . . . 10 ⊢ (((bra‘𝑥) Fn ℋ ∧ 𝑡 Fn ℋ) → ((bra‘𝑥) = 𝑡 ↔ ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦))) | |
33 | 30, 31, 32 | syl2an 597 | . . . . . . . . 9 ⊢ (((bra‘𝑥): ℋ⟶ℂ ∧ 𝑡: ℋ⟶ℂ) → ((bra‘𝑥) = 𝑡 ↔ ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦))) |
34 | 27, 29, 33 | syl2anr 598 | . . . . . . . 8 ⊢ (((𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ) ∧ 𝑥 ∈ ℋ) → ((bra‘𝑥) = 𝑡 ↔ ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦))) |
35 | 26, 34 | sylibrd 261 | . . . . . . 7 ⊢ (((𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ) ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·ih 𝑥) → (bra‘𝑥) = 𝑡)) |
36 | 35 | reximdva 3276 | . . . . . 6 ⊢ ((𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ) → (∃𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·ih 𝑥) → ∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡)) |
37 | 20, 36 | mpd 15 | . . . . 5 ⊢ ((𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ) → ∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡) |
38 | 18, 37 | impbii 211 | . . . 4 ⊢ (∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡 ↔ (𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ)) |
39 | 9, 38 | bitri 277 | . . 3 ⊢ (𝑡 ∈ ran bra ↔ (𝑡 ∈ LinFn ∧ (normfn‘𝑡) ∈ ℝ)) |
40 | 2, 3, 39 | 3bitr4ri 306 | . 2 ⊢ (𝑡 ∈ ran bra ↔ 𝑡 ∈ (LinFn ∩ ContFn)) |
41 | 40 | eqriv 2820 | 1 ⊢ ran bra = (LinFn ∩ ContFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 ↦ cmpt 5148 ran crn 5558 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 ℋchba 28698 ·ih csp 28701 normfncnmf 28730 ContFnccnfn 28732 LinFnclf 28733 bracbr 28735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 ax-hcompl 28981 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-cn 21837 df-cnp 21838 df-lm 21839 df-t1 21924 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cfil 23860 df-cau 23861 df-cmet 23862 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-dip 28480 df-ssp 28501 df-ph 28592 df-cbn 28642 df-hnorm 28747 df-hba 28748 df-hvsub 28750 df-hlim 28751 df-hcau 28752 df-sh 28986 df-ch 29000 df-oc 29031 df-ch0 29032 df-nmfn 29624 df-nlfn 29625 df-cnfn 29626 df-lnfn 29627 df-bra 29629 |
This theorem is referenced by: bra11 29887 cnvbraval 29889 |
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