Theorem nmopun 29588

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 29494	. . . . 5 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		lnopf 29433	. . . . 5 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
3	1, 2	syl 17	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4		nmopval 29430	. . . 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 474	. 2 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
7		nmopsetretHIL 29438	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
8		ressxr 10483	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
9	7, 8	syl6ss 3865	. . . . . 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 474	. . . 4 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
12		1xr 10499	. . . 4 ⊢ 1 ∈ ℝ*
13	11, 12	jctir 513	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ* ∧ 1 ∈ ℝ*))
14		vex 3413	. . . . . . 7 ⊢ 𝑧 ∈ V
15		eqeq1 2777	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ 𝑧 = (normℎ‘(𝑇‘𝑦))))
16	15	anbi2d 620	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦)))))
17	16	rexbidv 3237	. . . . . . 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 3577	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))))
19		unopnorm 29491	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
20	19	eqeq2d 2783	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑧 = (normℎ‘(𝑇‘𝑦)) ↔ 𝑧 = (normℎ‘𝑦)))
21	20	anbi2d 620	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘𝑦))))
22		breq1 4929	. . . . . . . . . 10 ⊢ (𝑧 = (normℎ‘𝑦) → (𝑧 ≤ 1 ↔ (normℎ‘𝑦) ≤ 1))
23	22	biimparc 472	. . . . . . . . 9 ⊢ (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘𝑦)) → 𝑧 ≤ 1)
24	21, 23	syl6bi 245	. . . . . . . 8 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) → 𝑧 ≤ 1))
25	24	rexlimdva 3224	. . . . . . 7 ⊢ (𝑇 ∈ UniOp → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) → 𝑧 ≤ 1))
26	25	imp 398	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦)))) → 𝑧 ≤ 1)
27	18, 26	sylan2b 585	. . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}) → 𝑧 ≤ 1)
28	27	ralrimiva 3127	. . . 4 ⊢ (𝑇 ∈ UniOp → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 ≤ 1)
29	28	adantl 474	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 ≤ 1)
30		hne0 29121	. . . . . . . . . . 11 ⊢ ( ℋ ≠ 0ℋ ↔ ∃𝑦 ∈ ℋ 𝑦 ≠ 0ℎ)
31		norm1hex 28823	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℋ 𝑦 ≠ 0ℎ ↔ ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
32	30, 31	sylbb 211	. . . . . . . . . 10 ⊢ ( ℋ ≠ 0ℋ → ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
33	32	adantr 473	. . . . . . . . 9 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
34		1le1 11068	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 1
35		breq1 4929	. . . . . . . . . . . . . 14 ⊢ ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ↔ 1 ≤ 1))
36	34, 35	mpbiri 250	. . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
39		eqeq2 2784	. . . . . . . . . . . . . . . 16 ⊢ ((normℎ‘𝑦) = 1 → ((normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦) ↔ (normℎ‘(𝑇‘𝑦)) = 1))
40	39	adantl 474	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → ((normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦) ↔ (normℎ‘(𝑇‘𝑦)) = 1))
41	38, 40	mpbid 224	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → (normℎ‘(𝑇‘𝑦)) = 1)
42	41	eqcomd 2779	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → 1 = (normℎ‘(𝑇‘𝑦)))
43	42	ex 405	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → 1 = (normℎ‘(𝑇‘𝑦))))
44	37, 43	jcad 505	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
45	44	adantll 702	. . . . . . . . . 10 ⊢ ((( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
46	45	reximdva 3214	. . . . . . . . 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 10434	. . . . . . . . 9 ⊢ 1 ∈ V
49		eqeq1 2777	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ 1 = (normℎ‘(𝑇‘𝑦))))
50	49	anbi2d 620	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
51	50	rexbidv 3237	. . . . . . . . 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 3577	. . . . . . . 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 226	. . . . . . 7 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
54	53	adantr 473	. . . . . 6 ⊢ ((( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
55		breq2 4930	. . . . . . 7 ⊢ (𝑤 = 1 → (𝑧 < 𝑤 ↔ 𝑧 < 1))
56	55	rspcev 3530	. . . . . 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 572	. . . . 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 405	. . . 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 3127	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 < 𝑤))
60		supxr2 12522	. . 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 825	. 2 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) = 1)
62	6, 61	eqtrd 2809	1 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = 1)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051  {cab 2753   ≠ wne 2962  ∀wral 3083  ∃wrex 3084   ⊆ wss 3824   class class class wbr 4926  ⟶wf 6182  ‘cfv 6186  supcsup 8698  ℝcr 10333  1c1 10335  ℝ^*cxr 10472   < clt 10473   ≤ cle 10474   ℋchba 28491  norm_ℎcno 28495  0_ℎc0v 28496  0_ℋc0h 28507  norm_opcnop 28517  LinOpclo 28519  UniOpcuo 28521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-pre-sup 10412  ax-hilex 28571  ax-hfvadd 28572  ax-hvcom 28573  ax-hvass 28574  ax-hv0cl 28575  ax-hvaddid 28576  ax-hfvmul 28577  ax-hvmulid 28578  ax-hvmulass 28579  ax-hvdistr1 28580  ax-hvdistr2 28581  ax-hvmul0 28582  ax-hfi 28651  ax-his1 28654  ax-his2 28655  ax-his3 28656  ax-his4 28657

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-er 8088  df-map 8207  df-en 8306  df-dom 8307  df-sdom 8308  df-sup 8700  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-div 11098  df-nn 11439  df-2 11502  df-3 11503  df-4 11504  df-n0 11707  df-z 11793  df-uz 12058  df-rp 12204  df-seq 13184  df-exp 13244  df-cj 14318  df-re 14319  df-im 14320  df-sqrt 14454  df-abs 14455  df-grpo 28063  df-gid 28064  df-ablo 28115  df-vc 28129  df-nv 28162  df-va 28165  df-ba 28166  df-sm 28167  df-0v 28168  df-nmcv 28170  df-hnorm 28540  df-hba 28541  df-hvsub 28543  df-hlim 28544  df-sh 28779  df-ch 28793  df-ch0 28825  df-nmop 29413  df-lnop 29415  df-unop 29417

This theorem is referenced by:  unopbd  29589  unierri  29678