rnbra - Hilbert Space Explorer

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Theorem rnbra 31930

Description: The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
rnbra	⊢ ran bra = (LinFn ∩ ContFn)

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

Step	Hyp	Ref	Expression
1		lnfncnbd 31880	. . . 4 ⊢ (𝑡 ∈ LinFn → (𝑡 ∈ ContFn ↔ (norm_fn‘𝑡) ∈ ℝ))
2	1	pm5.32i 574	. . 3 ⊢ ((𝑡 ∈ LinFn ∧ 𝑡 ∈ ContFn) ↔ (𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ))
3		elin 3963	. . 3 ⊢ (𝑡 ∈ (LinFn ∩ ContFn) ↔ (𝑡 ∈ LinFn ∧ 𝑡 ∈ ContFn))
4		ax-hilex 30822	. . . . . . 7 ⊢ ℋ ∈ V
5	4	mptex 7235	. . . . . 6 ⊢ (𝑦 ∈ ℋ ↦ (𝑦 ·_ih 𝑥)) ∈ V
6		df-bra 31673	. . . . . 6 ⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·_ih 𝑥)))
7	5, 6	fnmpti 6698	. . . . 5 ⊢ bra Fn ℋ
8		fvelrnb 6959	. . . . 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 31771	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (bra‘𝑥) ∈ LinFn)
11		brabn 31929	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (norm_fn‘(bra‘𝑥)) ∈ ℝ)
12	10, 11	jca 511	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((bra‘𝑥) ∈ LinFn ∧ (norm_fn‘(bra‘𝑥)) ∈ ℝ))
13		eleq1 2817	. . . . . . . 8 ⊢ ((bra‘𝑥) = 𝑡 → ((bra‘𝑥) ∈ LinFn ↔ 𝑡 ∈ LinFn))
14		fveq2 6897	. . . . . . . . 9 ⊢ ((bra‘𝑥) = 𝑡 → (norm_fn‘(bra‘𝑥)) = (norm_fn‘𝑡))
15	14	eleq1d 2814	. . . . . . . 8 ⊢ ((bra‘𝑥) = 𝑡 → ((norm_fn‘(bra‘𝑥)) ∈ ℝ ↔ (norm_fn‘𝑡) ∈ ℝ))
16	13, 15	anbi12d 631	. . . . . . 7 ⊢ ((bra‘𝑥) = 𝑡 → (((bra‘𝑥) ∈ LinFn ∧ (norm_fn‘(bra‘𝑥)) ∈ ℝ) ↔ (𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ)))
17	12, 16	syl5ibcom 244	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((bra‘𝑥) = 𝑡 → (𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ)))
18	17	rexlimiv 3145	. . . . 5 ⊢ (∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡 → (𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ))
19		riesz1 31888	. . . . . . 7 ⊢ (𝑡 ∈ LinFn → ((norm_fn‘𝑡) ∈ ℝ ↔ ∃𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·_ih 𝑥)))
20	19	biimpa 476	. . . . . 6 ⊢ ((𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ) → ∃𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·_ih 𝑥))
21		braval 31767	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝑥)‘𝑦) = (𝑦 ·_ih 𝑥))
22		eqtr3 2754	. . . . . . . . . . . 12 ⊢ ((((bra‘𝑥)‘𝑦) = (𝑦 ·_ih 𝑥) ∧ (𝑡‘𝑦) = (𝑦 ·_ih 𝑥)) → ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦))
23	22	ex 412	. . . . . . . . . . 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 3164	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·_ih 𝑥) → ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦)))
26	25	adantl 481	. . . . . . . 8 ⊢ (((𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ) ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·_ih 𝑥) → ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦)))
27		brafn 31770	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (bra‘𝑥): ℋ⟶ℂ)
28		lnfnf 31707	. . . . . . . . . 10 ⊢ (𝑡 ∈ LinFn → 𝑡: ℋ⟶ℂ)
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ) → 𝑡: ℋ⟶ℂ)
30		ffn 6722	. . . . . . . . . 10 ⊢ ((bra‘𝑥): ℋ⟶ℂ → (bra‘𝑥) Fn ℋ)
31		ffn 6722	. . . . . . . . . 10 ⊢ (𝑡: ℋ⟶ℂ → 𝑡 Fn ℋ)
32		eqfnfv 7040	. . . . . . . . . 10 ⊢ (((bra‘𝑥) Fn ℋ ∧ 𝑡 Fn ℋ) → ((bra‘𝑥) = 𝑡 ↔ ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦)))
33	30, 31, 32	syl2an 595	. . . . . . . . 9 ⊢ (((bra‘𝑥): ℋ⟶ℂ ∧ 𝑡: ℋ⟶ℂ) → ((bra‘𝑥) = 𝑡 ↔ ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦)))
34	27, 29, 33	syl2anr 596	. . . . . . . 8 ⊢ (((𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ) ∧ 𝑥 ∈ ℋ) → ((bra‘𝑥) = 𝑡 ↔ ∀𝑦 ∈ ℋ ((bra‘𝑥)‘𝑦) = (𝑡‘𝑦)))
35	26, 34	sylibrd 259	. . . . . . 7 ⊢ (((𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ) ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·_ih 𝑥) → (bra‘𝑥) = 𝑡))
36	35	reximdva 3165	. . . . . 6 ⊢ ((𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ) → (∃𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑡‘𝑦) = (𝑦 ·_ih 𝑥) → ∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡))
37	20, 36	mpd 15	. . . . 5 ⊢ ((𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ) → ∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡)
38	18, 37	impbii 208	. . . 4 ⊢ (∃𝑥 ∈ ℋ (bra‘𝑥) = 𝑡 ↔ (𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ))
39	9, 38	bitri 275	. . 3 ⊢ (𝑡 ∈ ran bra ↔ (𝑡 ∈ LinFn ∧ (norm_fn‘𝑡) ∈ ℝ))
40	2, 3, 39	3bitr4ri 304	. 2 ⊢ (𝑡 ∈ ran bra ↔ 𝑡 ∈ (LinFn ∩ ContFn))
41	40	eqriv 2725	1 ⊢ ran bra = (LinFn ∩ ContFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 ∩ cin 3946 ↦ cmpt 5231 ran crn 5679 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 ℝcr 11138 ℋchba 30742 ·_ih csp 30745 norm_fncnmf 30774 ContFnccnfn 30776 LinFnclf 30777 bracbr 30779

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cc 10459 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 ax-hilex 30822 ax-hfvadd 30823 ax-hvcom 30824 ax-hvass 30825 ax-hv0cl 30826 ax-hvaddid 30827 ax-hfvmul 30828 ax-hvmulid 30829 ax-hvmulass 30830 ax-hvdistr1 30831 ax-hvdistr2 30832 ax-hvmul0 30833 ax-hfi 30902 ax-his1 30905 ax-his2 30906 ax-his3 30907 ax-his4 30908 ax-hcompl 31025

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-acn 9966 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-rlim 15466 df-sum 15666 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-cn 23144 df-cnp 23145 df-lm 23146 df-t1 23231 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cfil 25196 df-cau 25197 df-cmet 25198 df-grpo 30316 df-gid 30317 df-ginv 30318 df-gdiv 30319 df-ablo 30368 df-vc 30382 df-nv 30415 df-va 30418 df-ba 30419 df-sm 30420 df-0v 30421 df-vs 30422 df-nmcv 30423 df-ims 30424 df-dip 30524 df-ssp 30545 df-ph 30636 df-cbn 30686 df-hnorm 30791 df-hba 30792 df-hvsub 30794 df-hlim 30795 df-hcau 30796 df-sh 31030 df-ch 31044 df-oc 31075 df-ch0 31076 df-nmfn 31668 df-nlfn 31669 df-cnfn 31670 df-lnfn 31671 df-bra 31673

This theorem is referenced by: bra11 31931 cnvbraval 31933

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