Theorem nmopun 32043

Description: Norm of a unitary Hilbert space operator. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
nmopun	⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = 1)

Proof of Theorem nmopun

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unoplin 31949	. . . . 5 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		lnopf 31888	. . . . 5 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
3	1, 2	syl 17	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4		nmopval 31885	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
5	3, 4	syl 17	. . 3 ⊢ (𝑇 ∈ UniOp → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
6	5	adantl 481	. 2 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
7		nmopsetretHIL 31893	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
8		ressxr 11303	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
9	7, 8	sstrdi 4008	. . . . . 6 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
10	3, 9	syl 17	. . . . 5 ⊢ (𝑇 ∈ UniOp → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
11	10	adantl 481	. . . 4 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
12		1xr 11318	. . . 4 ⊢ 1 ∈ ℝ*
13	11, 12	jctir 520	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ* ∧ 1 ∈ ℝ*))
14		vex 3482	. . . . . . 7 ⊢ 𝑧 ∈ V
15		eqeq1 2739	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ 𝑧 = (normℎ‘(𝑇‘𝑦))))
16	15	anbi2d 630	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦)))))
17	16	rexbidv 3177	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦)))))
18	14, 17	elab 3681	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))))
19		unopnorm 31946	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
20	19	eqeq2d 2746	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑧 = (normℎ‘(𝑇‘𝑦)) ↔ 𝑧 = (normℎ‘𝑦)))
21	20	anbi2d 630	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘𝑦))))
22		breq1 5151	. . . . . . . . . 10 ⊢ (𝑧 = (normℎ‘𝑦) → (𝑧 ≤ 1 ↔ (normℎ‘𝑦) ≤ 1))
23	22	biimparc 479	. . . . . . . . 9 ⊢ (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘𝑦)) → 𝑧 ≤ 1)
24	21, 23	biimtrdi 253	. . . . . . . 8 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) → 𝑧 ≤ 1))
25	24	rexlimdva 3153	. . . . . . 7 ⊢ (𝑇 ∈ UniOp → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) → 𝑧 ≤ 1))
26	25	imp 406	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦)))) → 𝑧 ≤ 1)
27	18, 26	sylan2b 594	. . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}) → 𝑧 ≤ 1)
28	27	ralrimiva 3144	. . . 4 ⊢ (𝑇 ∈ UniOp → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 ≤ 1)
29	28	adantl 481	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 ≤ 1)
30		hne0 31576	. . . . . . . . . . 11 ⊢ ( ℋ ≠ 0ℋ ↔ ∃𝑦 ∈ ℋ 𝑦 ≠ 0ℎ)
31		norm1hex 31280	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℋ 𝑦 ≠ 0ℎ ↔ ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
32	30, 31	sylbb 219	. . . . . . . . . 10 ⊢ ( ℋ ≠ 0ℋ → ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
33	32	adantr 480	. . . . . . . . 9 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
34		1le1 11889	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 1
35		breq1 5151	. . . . . . . . . . . . . 14 ⊢ ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ↔ 1 ≤ 1))
36	34, 35	mpbiri 258	. . . . . . . . . . . . 13 ⊢ ((normℎ‘𝑦) = 1 → (normℎ‘𝑦) ≤ 1)
37	36	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → (normℎ‘𝑦) ≤ 1))
38	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
39		eqeq2 2747	. . . . . . . . . . . . . . . 16 ⊢ ((normℎ‘𝑦) = 1 → ((normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦) ↔ (normℎ‘(𝑇‘𝑦)) = 1))
40	39	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → ((normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦) ↔ (normℎ‘(𝑇‘𝑦)) = 1))
41	38, 40	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → (normℎ‘(𝑇‘𝑦)) = 1)
42	41	eqcomd 2741	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → 1 = (normℎ‘(𝑇‘𝑦)))
43	42	ex 412	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → 1 = (normℎ‘(𝑇‘𝑦))))
44	37, 43	jcad 512	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
45	44	adantll 714	. . . . . . . . . 10 ⊢ ((( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
46	45	reximdva 3166	. . . . . . . . 9 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1 → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
47	33, 46	mpd 15	. . . . . . . 8 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦))))
48		1ex 11255	. . . . . . . . 9 ⊢ 1 ∈ V
49		eqeq1 2739	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ 1 = (normℎ‘(𝑇‘𝑦))))
50	49	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
51	50	rexbidv 3177	. . . . . . . . 9 ⊢ (𝑥 = 1 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
52	48, 51	elab 3681	. . . . . . . 8 ⊢ (1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦))))
53	47, 52	sylibr 234	. . . . . . 7 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
54	53	adantr 480	. . . . . 6 ⊢ ((( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
55		breq2 5152	. . . . . . 7 ⊢ (𝑤 = 1 → (𝑧 < 𝑤 ↔ 𝑧 < 1))
56	55	rspcev 3622	. . . . . 6 ⊢ ((1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ∧ 𝑧 < 1) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 < 𝑤)
57	54, 56	sylan 580	. . . . 5 ⊢ (((( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < 1) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 < 𝑤)
58	57	ex 412	. . . 4 ⊢ ((( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) → (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 < 𝑤))
59	58	ralrimiva 3144	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 < 𝑤))
60		supxr2 13353	. . 3 ⊢ ((({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ* ∧ 1 ∈ ℝ*) ∧ (∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 ≤ 1 ∧ ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 < 𝑤))) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) = 1)
61	13, 29, 59, 60	syl12anc 837	. 2 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) = 1)
62	6, 61	eqtrd 2775	1 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = 1)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963   class class class wbr 5148  ⟶wf 6559  ‘cfv 6563  supcsup 9478  ℝcr 11152  1c1 11154  ℝ^*cxr 11292   < clt 11293   ≤ cle 11294   ℋchba 30948  norm_ℎcno 30952  0_ℎc0v 30953  0_ℋc0h 30964  norm_opcnop 30974  LinOpclo 30976  UniOpcuo 30978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-hilex 31028  ax-hfvadd 31029  ax-hvcom 31030  ax-hvass 31031  ax-hv0cl 31032  ax-hvaddid 31033  ax-hfvmul 31034  ax-hvmulid 31035  ax-hvmulass 31036  ax-hvdistr1 31037  ax-hvdistr2 31038  ax-hvmul0 31039  ax-hfi 31108  ax-his1 31111  ax-his2 31112  ax-his3 31113  ax-his4 31114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-grpo 30522  df-gid 30523  df-ablo 30574  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629  df-hnorm 30997  df-hba 30998  df-hvsub 31000  df-hlim 31001  df-sh 31236  df-ch 31250  df-ch0 31282  df-nmop 31868  df-lnop 31870  df-unop 31872

This theorem is referenced by:  unopbd  32044  unierri  32133