nmcopexi - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > nmcopexi		Structured version Visualization version GIF version

Theorem nmcopexi 31784

Description: The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 5-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmcopex.1	⊢ 𝑇 ∈ LinOp
nmcopex.2	⊢ 𝑇 ∈ ContOp

**Assertion**
Ref	Expression
nmcopexi	⊢ (norm_op‘𝑇) ∈ ℝ

Proof of Theorem nmcopexi Dummy variables 𝑥 𝑚 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmcopex.2	. . . 4 ⊢ 𝑇 ∈ ContOp
2		ax-hv0cl 30760	. . . 4 ⊢ 0_ℎ ∈ ℋ
3		1rp 12981	. . . 4 ⊢ 1 ∈ ℝ⁺
4		cnopc 31670	. . . 4 ⊢ ((𝑇 ∈ ContOp ∧ 0_ℎ ∈ ℋ ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ ℋ ((norm_ℎ‘(𝑧 −_ℎ 0_ℎ)) < 𝑦 → (norm_ℎ‘((𝑇‘𝑧) −_ℎ (𝑇‘0_ℎ))) < 1))
5	1, 2, 3, 4	mp3an 1457	. . 3 ⊢ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ ℋ ((norm_ℎ‘(𝑧 −_ℎ 0_ℎ)) < 𝑦 → (norm_ℎ‘((𝑇‘𝑧) −_ℎ (𝑇‘0_ℎ))) < 1)
6		hvsub0 30833	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑧 −_ℎ 0_ℎ) = 𝑧)
7	6	fveq2d 6888	. . . . . . 7 ⊢ (𝑧 ∈ ℋ → (norm_ℎ‘(𝑧 −_ℎ 0_ℎ)) = (norm_ℎ‘𝑧))
8	7	breq1d 5151	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((norm_ℎ‘(𝑧 −_ℎ 0_ℎ)) < 𝑦 ↔ (norm_ℎ‘𝑧) < 𝑦))
9		nmcopex.1	. . . . . . . . . . 11 ⊢ 𝑇 ∈ LinOp
10	9	lnop0i 31727	. . . . . . . . . 10 ⊢ (𝑇‘0_ℎ) = 0_ℎ
11	10	oveq2i 7415	. . . . . . . . 9 ⊢ ((𝑇‘𝑧) −_ℎ (𝑇‘0_ℎ)) = ((𝑇‘𝑧) −_ℎ 0_ℎ)
12	9	lnopfi 31726	. . . . . . . . . . 11 ⊢ 𝑇: ℋ⟶ ℋ
13	12	ffvelcdmi 7078	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
14		hvsub0 30833	. . . . . . . . . 10 ⊢ ((𝑇‘𝑧) ∈ ℋ → ((𝑇‘𝑧) −_ℎ 0_ℎ) = (𝑇‘𝑧))
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) −_ℎ 0_ℎ) = (𝑇‘𝑧))
16	11, 15	eqtrid 2778	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) −_ℎ (𝑇‘0_ℎ)) = (𝑇‘𝑧))
17	16	fveq2d 6888	. . . . . . 7 ⊢ (𝑧 ∈ ℋ → (norm_ℎ‘((𝑇‘𝑧) −_ℎ (𝑇‘0_ℎ))) = (norm_ℎ‘(𝑇‘𝑧)))
18	17	breq1d 5151	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((norm_ℎ‘((𝑇‘𝑧) −_ℎ (𝑇‘0_ℎ))) < 1 ↔ (norm_ℎ‘(𝑇‘𝑧)) < 1))
19	8, 18	imbi12d 344	. . . . 5 ⊢ (𝑧 ∈ ℋ → (((norm_ℎ‘(𝑧 −_ℎ 0_ℎ)) < 𝑦 → (norm_ℎ‘((𝑇‘𝑧) −_ℎ (𝑇‘0_ℎ))) < 1) ↔ ((norm_ℎ‘𝑧) < 𝑦 → (norm_ℎ‘(𝑇‘𝑧)) < 1)))
20	19	ralbiia 3085	. . . 4 ⊢ (∀𝑧 ∈ ℋ ((norm_ℎ‘(𝑧 −_ℎ 0_ℎ)) < 𝑦 → (norm_ℎ‘((𝑇‘𝑧) −_ℎ (𝑇‘0_ℎ))) < 1) ↔ ∀𝑧 ∈ ℋ ((norm_ℎ‘𝑧) < 𝑦 → (norm_ℎ‘(𝑇‘𝑧)) < 1))
21	20	rexbii 3088	. . 3 ⊢ (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ ℋ ((norm_ℎ‘(𝑧 −_ℎ 0_ℎ)) < 𝑦 → (norm_ℎ‘((𝑇‘𝑧) −_ℎ (𝑇‘0_ℎ))) < 1) ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ ℋ ((norm_ℎ‘𝑧) < 𝑦 → (norm_ℎ‘(𝑇‘𝑧)) < 1))
22	5, 21	mpbi 229	. 2 ⊢ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ ℋ ((norm_ℎ‘𝑧) < 𝑦 → (norm_ℎ‘(𝑇‘𝑧)) < 1)
23		nmopval 31613	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (norm_op‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm_ℎ‘𝑥) ≤ 1 ∧ 𝑚 = (norm_ℎ‘(𝑇‘𝑥)))}, ℝ^*, < ))
24	12, 23	ax-mp 5	. 2 ⊢ (norm_op‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm_ℎ‘𝑥) ≤ 1 ∧ 𝑚 = (norm_ℎ‘(𝑇‘𝑥)))}, ℝ^*, < )
25	12	ffvelcdmi 7078	. . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
26		normcl 30882	. . 3 ⊢ ((𝑇‘𝑥) ∈ ℋ → (norm_ℎ‘(𝑇‘𝑥)) ∈ ℝ)
27	25, 26	syl 17	. 2 ⊢ (𝑥 ∈ ℋ → (norm_ℎ‘(𝑇‘𝑥)) ∈ ℝ)
28	10	fveq2i 6887	. . 3 ⊢ (norm_ℎ‘(𝑇‘0_ℎ)) = (norm_ℎ‘0_ℎ)
29		norm0 30885	. . 3 ⊢ (norm_ℎ‘0_ℎ) = 0
30	28, 29	eqtri 2754	. 2 ⊢ (norm_ℎ‘(𝑇‘0_ℎ)) = 0
31		rpcn 12987	. . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ⁺ → (𝑦 / 2) ∈ ℂ)
32	9	lnopmuli 31729	. . . . 5 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·_ℎ 𝑥)) = ((𝑦 / 2) ·_ℎ (𝑇‘𝑥)))
33	31, 32	sylan 579	. . . 4 ⊢ (((𝑦 / 2) ∈ ℝ⁺ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·_ℎ 𝑥)) = ((𝑦 / 2) ·_ℎ (𝑇‘𝑥)))
34	33	fveq2d 6888	. . 3 ⊢ (((𝑦 / 2) ∈ ℝ⁺ ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘(𝑇‘((𝑦 / 2) ·_ℎ 𝑥))) = (norm_ℎ‘((𝑦 / 2) ·_ℎ (𝑇‘𝑥))))
35		norm-iii 30897	. . . 4 ⊢ (((𝑦 / 2) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → (norm_ℎ‘((𝑦 / 2) ·_ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (norm_ℎ‘(𝑇‘𝑥))))
36	31, 25, 35	syl2an 595	. . 3 ⊢ (((𝑦 / 2) ∈ ℝ⁺ ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘((𝑦 / 2) ·_ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (norm_ℎ‘(𝑇‘𝑥))))
37		rpre 12985	. . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ⁺ → (𝑦 / 2) ∈ ℝ)
38		rpge0 12990	. . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ⁺ → 0 ≤ (𝑦 / 2))
39	37, 38	absidd 15372	. . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ⁺ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
40	39	adantr 480	. . . 4 ⊢ (((𝑦 / 2) ∈ ℝ⁺ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑦 / 2)) = (𝑦 / 2))
41	40	oveq1d 7419	. . 3 ⊢ (((𝑦 / 2) ∈ ℝ⁺ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (norm_ℎ‘(𝑇‘𝑥))) = ((𝑦 / 2) · (norm_ℎ‘(𝑇‘𝑥))))
42	34, 36, 41	3eqtrrd 2771	. 2 ⊢ (((𝑦 / 2) ∈ ℝ⁺ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (norm_ℎ‘(𝑇‘𝑥))) = (norm_ℎ‘(𝑇‘((𝑦 / 2) ·_ℎ 𝑥))))
43	22, 24, 27, 30, 42	nmcexi 31783	1 ⊢ (norm_op‘𝑇) ∈ ℝ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 ∀wral 3055 ∃wrex 3064 class class class wbr 5141 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 supcsup 9434 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 · cmul 11114 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 / cdiv 11872 2c2 12268 ℝ⁺crp 12977 abscabs 15184 ℋchba 30676 ·_ℎ csm 30678 norm_ℎcno 30680 0_ℎc0v 30681 −_ℎ cmv 30682 norm_opcnop 30702 ContOpccop 30703 LinOpclo 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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-hilex 30756 ax-hfvadd 30757 ax-hvass 30759 ax-hv0cl 30760 ax-hvaddid 30761 ax-hfvmul 30762 ax-hvmulid 30763 ax-hvmulass 30764 ax-hvdistr2 30766 ax-hvmul0 30767 ax-hfi 30836 ax-his1 30839 ax-his3 30841 ax-his4 30842

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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-hnorm 30725 df-hvsub 30728 df-nmop 31596 df-cnop 31597 df-lnop 31598

This theorem is referenced by: nmcoplbi 31785 nmcopex 31786 cnlnadjlem2 31825 cnlnadjlem7 31830 cnlnadjlem8 31831

W3C validator