hhcnf - Hilbert Space Explorer

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Theorem hhcnf 31937

Description: The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hhcn.1	⊢ 𝐷 = (norm_ℎ ∘ −_ℎ )
hhcn.2	⊢ 𝐽 = (MetOpen‘𝐷)
hhcn.4	⊢ 𝐾 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
hhcnf	⊢ ContFn = (𝐽 Cn 𝐾)

Proof of Theorem hhcnf Dummy variables 𝑥 𝑤 𝑦 𝑧 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3444	. 2 ⊢ {𝑡 ∈ (ℂ ↑_m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)} = {𝑡 ∣ (𝑡 ∈ (ℂ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦))}
2		df-cnfn 31879	. 2 ⊢ ContFn = {𝑡 ∈ (ℂ ↑_m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)}
3		hhcn.1	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (norm_ℎ ∘ −_ℎ )
4	3	hilmetdval 31228	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑥𝐷𝑤) = (norm_ℎ‘(𝑥 −_ℎ 𝑤)))
5		normsub 31175	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (norm_ℎ‘(𝑥 −_ℎ 𝑤)) = (norm_ℎ‘(𝑤 −_ℎ 𝑥)))
6	4, 5	eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑥𝐷𝑤) = (norm_ℎ‘(𝑤 −_ℎ 𝑥)))
7	6	adantll 713	. . . . . . . . . . 11 ⊢ (((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥𝐷𝑤) = (norm_ℎ‘(𝑤 −_ℎ 𝑥)))
8	7	breq1d 5176	. . . . . . . . . 10 ⊢ (((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥𝐷𝑤) < 𝑧 ↔ (norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧))
9		ffvelcdm 7115	. . . . . . . . . . . . . 14 ⊢ ((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) → (𝑡‘𝑥) ∈ ℂ)
10		ffvelcdm 7115	. . . . . . . . . . . . . 14 ⊢ ((𝑡: ℋ⟶ℂ ∧ 𝑤 ∈ ℋ) → (𝑡‘𝑤) ∈ ℂ)
11	9, 10	anim12dan 618	. . . . . . . . . . . . 13 ⊢ ((𝑡: ℋ⟶ℂ ∧ (𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ)) → ((𝑡‘𝑥) ∈ ℂ ∧ (𝑡‘𝑤) ∈ ℂ))
12		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
13	12	cnmetdval 24812	. . . . . . . . . . . . . 14 ⊢ (((𝑡‘𝑥) ∈ ℂ ∧ (𝑡‘𝑤) ∈ ℂ) → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) = (abs‘((𝑡‘𝑥) − (𝑡‘𝑤))))
14		abssub 15375	. . . . . . . . . . . . . 14 ⊢ (((𝑡‘𝑥) ∈ ℂ ∧ (𝑡‘𝑤) ∈ ℂ) → (abs‘((𝑡‘𝑥) − (𝑡‘𝑤))) = (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))))
15	13, 14	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ (((𝑡‘𝑥) ∈ ℂ ∧ (𝑡‘𝑤) ∈ ℂ) → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) = (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))))
16	11, 15	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑡: ℋ⟶ℂ ∧ (𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ)) → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) = (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))))
17	16	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) = (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))))
18	17	breq1d 5176	. . . . . . . . . 10 ⊢ (((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) < 𝑦 ↔ (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦))
19	8, 18	imbi12d 344	. . . . . . . . 9 ⊢ (((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (((𝑥𝐷𝑤) < 𝑧 → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) < 𝑦) ↔ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)))
20	19	ralbidva 3182	. . . . . . . 8 ⊢ ((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) → (∀𝑤 ∈ ℋ ((𝑥𝐷𝑤) < 𝑧 → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) < 𝑦) ↔ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)))
21	20	rexbidv 3185	. . . . . . 7 ⊢ ((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) → (∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((𝑥𝐷𝑤) < 𝑧 → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) < 𝑦) ↔ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)))
22	21	ralbidv 3184	. . . . . 6 ⊢ ((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((𝑥𝐷𝑤) < 𝑧 → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) < 𝑦) ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)))
23	22	ralbidva 3182	. . . . 5 ⊢ (𝑡: ℋ⟶ℂ → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((𝑥𝐷𝑤) < 𝑧 → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) < 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)))
24	23	pm5.32i 574	. . . 4 ⊢ ((𝑡: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((𝑥𝐷𝑤) < 𝑧 → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) < 𝑦)) ↔ (𝑡: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)))
25	3	hilxmet 31227	. . . . 5 ⊢ 𝐷 ∈ (∞Met‘ ℋ)
26		cnxmet 24814	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
27		hhcn.2	. . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷)
28		hhcn.4	. . . . . . 7 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
29	28	cnfldtopn 24823	. . . . . 6 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
30	27, 29	metcn 24577	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘ ℋ) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ)) → (𝑡 ∈ (𝐽 Cn 𝐾) ↔ (𝑡: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((𝑥𝐷𝑤) < 𝑧 → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) < 𝑦))))
31	25, 26, 30	mp2an 691	. . . 4 ⊢ (𝑡 ∈ (𝐽 Cn 𝐾) ↔ (𝑡: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((𝑥𝐷𝑤) < 𝑧 → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) < 𝑦)))
32		cnex 11265	. . . . . 6 ⊢ ℂ ∈ V
33		ax-hilex 31031	. . . . . 6 ⊢ ℋ ∈ V
34	32, 33	elmap 8929	. . . . 5 ⊢ (𝑡 ∈ (ℂ ↑_m ℋ) ↔ 𝑡: ℋ⟶ℂ)
35	34	anbi1i 623	. . . 4 ⊢ ((𝑡 ∈ (ℂ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)) ↔ (𝑡: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)))
36	24, 31, 35	3bitr4i 303	. . 3 ⊢ (𝑡 ∈ (𝐽 Cn 𝐾) ↔ (𝑡 ∈ (ℂ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)))
37	36	eqabi 2880	. 2 ⊢ (𝐽 Cn 𝐾) = {𝑡 ∣ (𝑡 ∈ (ℂ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦))}
38	1, 2, 37	3eqtr4i 2778	1 ⊢ ContFn = (𝐽 Cn 𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {cab 2717 ∀wral 3067 ∃wrex 3076 {crab 3443 class class class wbr 5166 ∘ ccom 5704 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 ℂcc 11182 < clt 11324 − cmin 11520 ℝ⁺crp 13057 abscabs 15283 TopOpenctopn 17481 ∞Metcxmet 21372 MetOpencmopn 21377 ℂ_fldccnfld 21387 Cn ccn 23253 ℋchba 30951 norm_ℎcno 30955 −_ℎ cmv 30957 ContFnccnfn 30985

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264 ax-hilex 31031 ax-hfvadd 31032 ax-hvcom 31033 ax-hvass 31034 ax-hv0cl 31035 ax-hvaddid 31036 ax-hfvmul 31037 ax-hvmulid 31038 ax-hvmulass 31039 ax-hvdistr1 31040 ax-hvdistr2 31041 ax-hvmul0 31042 ax-hfi 31111 ax-his1 31114 ax-his2 31115 ax-his3 31116 ax-his4 31117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-fz 13568 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-rest 17482 df-topn 17483 df-topgen 17503 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-bases 22974 df-cn 23256 df-cnp 23257 df-grpo 30525 df-gid 30526 df-ginv 30527 df-gdiv 30528 df-ablo 30577 df-vc 30591 df-nv 30624 df-va 30627 df-ba 30628 df-sm 30629 df-0v 30630 df-vs 30631 df-nmcv 30632 df-ims 30633 df-hnorm 31000 df-hvsub 31003 df-cnfn 31879

This theorem is referenced by: nlelchi 32093

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