Theorem nmopun 31950

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 31856	. . . . 5 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		lnopf 31795	. . . . 5 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
3	1, 2	syl 17	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4		nmopval 31792	. . . 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 31800	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
8		ressxr 11225	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
9	7, 8	sstrdi 3962	. . . . . 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 11240	. . . 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 3454	. . . . . . 7 ⊢ 𝑧 ∈ V
15		eqeq1 2734	. . . . . . . . 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 3158	. . . . . . 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 3649	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))))
19		unopnorm 31853	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
20	19	eqeq2d 2741	. . . . . . . . . 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 5113	. . . . . . . . . 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 3135	. . . . . . 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 3126	. . . 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 31483	. . . . . . . . . . 11 ⊢ ( ℋ ≠ 0ℋ ↔ ∃𝑦 ∈ ℋ 𝑦 ≠ 0ℎ)
31		norm1hex 31187	. . . . . . . . . . 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 11813	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 1
35		breq1 5113	. . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 3147	. . . . . . . . 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 11177	. . . . . . . . 9 ⊢ 1 ∈ V
49		eqeq1 2734	. . . . . . . . . . 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 3158	. . . . . . . . 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 3649	. . . . . . . 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 5114	. . . . . . 7 ⊢ (𝑤 = 1 → (𝑧 < 𝑤 ↔ 𝑧 < 1))
56	55	rspcev 3591	. . . . . 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 3126	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 < 𝑤))
60		supxr2 13281	. . 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 836	. 2 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) = 1)
62	6, 61	eqtrd 2765	1 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = 1)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917   class class class wbr 5110  ⟶wf 6510  ‘cfv 6514  supcsup 9398  ℝcr 11074  1c1 11076  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216   ℋchba 30855  norm_ℎcno 30859  0_ℎc0v 30860  0_ℋc0h 30871  norm_opcnop 30881  LinOpclo 30883  UniOpcuo 30885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-hilex 30935  ax-hfvadd 30936  ax-hvcom 30937  ax-hvass 30938  ax-hv0cl 30939  ax-hvaddid 30940  ax-hfvmul 30941  ax-hvmulid 30942  ax-hvmulass 30943  ax-hvdistr1 30944  ax-hvdistr2 30945  ax-hvmul0 30946  ax-hfi 31015  ax-his1 31018  ax-his2 31019  ax-his3 31020  ax-his4 31021

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-sup 9400  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-seq 13974  df-exp 14034  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-grpo 30429  df-gid 30430  df-ablo 30481  df-vc 30495  df-nv 30528  df-va 30531  df-ba 30532  df-sm 30533  df-0v 30534  df-nmcv 30536  df-hnorm 30904  df-hba 30905  df-hvsub 30907  df-hlim 30908  df-sh 31143  df-ch 31157  df-ch0 31189  df-nmop 31775  df-lnop 31777  df-unop 31779

This theorem is referenced by:  unopbd  31951  unierri  32040