hhcnf - Hilbert Space Explorer

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Theorem hhcnf 31976

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 3390	. 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 31918	. 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 31267	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑥𝐷𝑤) = (norm_ℎ‘(𝑥 −_ℎ 𝑤)))
5		normsub 31214	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (norm_ℎ‘(𝑥 −_ℎ 𝑤)) = (norm_ℎ‘(𝑤 −_ℎ 𝑥)))
6	4, 5	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑥𝐷𝑤) = (norm_ℎ‘(𝑤 −_ℎ 𝑥)))
7	6	adantll 715	. . . . . . . . . . 11 ⊢ (((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥𝐷𝑤) = (norm_ℎ‘(𝑤 −_ℎ 𝑥)))
8	7	breq1d 5095	. . . . . . . . . 10 ⊢ (((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥𝐷𝑤) < 𝑧 ↔ (norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧))
9		ffvelcdm 7033	. . . . . . . . . . . . . 14 ⊢ ((𝑡: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) → (𝑡‘𝑥) ∈ ℂ)
10		ffvelcdm 7033	. . . . . . . . . . . . . 14 ⊢ ((𝑡: ℋ⟶ℂ ∧ 𝑤 ∈ ℋ) → (𝑡‘𝑤) ∈ ℂ)
11	9, 10	anim12dan 620	. . . . . . . . . . . . 13 ⊢ ((𝑡: ℋ⟶ℂ ∧ (𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ)) → ((𝑡‘𝑥) ∈ ℂ ∧ (𝑡‘𝑤) ∈ ℂ))
12		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
13	12	cnmetdval 24735	. . . . . . . . . . . . . 14 ⊢ (((𝑡‘𝑥) ∈ ℂ ∧ (𝑡‘𝑤) ∈ ℂ) → ((𝑡‘𝑥)(abs ∘ − )(𝑡‘𝑤)) = (abs‘((𝑡‘𝑥) − (𝑡‘𝑤))))
14		abssub 15289	. . . . . . . . . . . . . 14 ⊢ (((𝑡‘𝑥) ∈ ℂ ∧ (𝑡‘𝑤) ∈ ℂ) → (abs‘((𝑡‘𝑥) − (𝑡‘𝑤))) = (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))))
15	13, 14	eqtrd 2771	. . . . . . . . . . . . 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 5095	. . . . . . . . . 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 3158	. . . . . . . 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 3161	. . . . . . 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 3160	. . . . . 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 3158	. . . . 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 31266	. . . . 5 ⊢ 𝐷 ∈ (∞Met‘ ℋ)
26		cnxmet 24737	. . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
27		hhcn.2	. . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷)
28		hhcn.4	. . . . . . 7 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
29	28	cnfldtopn 24746	. . . . . 6 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
30	27, 29	metcn 24508	. . . . 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 693	. . . 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 11119	. . . . . 6 ⊢ ℂ ∈ V
33		ax-hilex 31070	. . . . . 6 ⊢ ℋ ∈ V
34	32, 33	elmap 8819	. . . . 5 ⊢ (𝑡 ∈ (ℂ ↑_m ℋ) ↔ 𝑡: ℋ⟶ℂ)
35	34	anbi1i 625	. . . 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 2871	. 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 2769	1 ⊢ ContFn = (𝐽 Cn 𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {cab 2714 ∀wral 3051 ∃wrex 3061 {crab 3389 class class class wbr 5085 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 ℂcc 11036 < clt 11179 − cmin 11377 ℝ⁺crp 12942 abscabs 15196 TopOpenctopn 17384 ∞Metcxmet 21337 MetOpencmopn 21342 ℂ_fldccnfld 21352 Cn ccn 23189 ℋchba 30990 norm_ℎcno 30994 −_ℎ cmv 30996 ContFnccnfn 31024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31070 ax-hfvadd 31071 ax-hvcom 31072 ax-hvass 31073 ax-hv0cl 31074 ax-hvaddid 31075 ax-hfvmul 31076 ax-hvmulid 31077 ax-hvmulass 31078 ax-hvdistr1 31079 ax-hvdistr2 31080 ax-hvmul0 31081 ax-hfi 31150 ax-his1 31153 ax-his2 31154 ax-his3 31155 ax-his4 31156

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-fz 13462 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-topn 17386 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-bases 22911 df-cn 23192 df-cnp 23193 df-grpo 30564 df-gid 30565 df-ginv 30566 df-gdiv 30567 df-ablo 30616 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-vs 30670 df-nmcv 30671 df-ims 30672 df-hnorm 31039 df-hvsub 31042 df-cnfn 31918

This theorem is referenced by: nlelchi 32132

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