Theorem nmopun 30277

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 30183	. . . . 5 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		lnopf 30122	. . . . 5 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
3	1, 2	syl 17	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4		nmopval 30119	. . . 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 30127	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
8		ressxr 10950	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
9	7, 8	sstrdi 3929	. . . . . 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 10965	. . . 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 3426	. . . . . . 7 ⊢ 𝑧 ∈ V
15		eqeq1 2742	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ 𝑧 = (normℎ‘(𝑇‘𝑦))))
16	15	anbi2d 628	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦)))))
17	16	rexbidv 3225	. . . . . . 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 3602	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))))
19		unopnorm 30180	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
20	19	eqeq2d 2749	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑧 = (normℎ‘(𝑇‘𝑦)) ↔ 𝑧 = (normℎ‘𝑦)))
21	20	anbi2d 628	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘𝑦))))
22		breq1 5073	. . . . . . . . . 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	syl6bi 252	. . . . . . . 8 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) → 𝑧 ≤ 1))
25	24	rexlimdva 3212	. . . . . . 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 593	. . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}) → 𝑧 ≤ 1)
28	27	ralrimiva 3107	. . . 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 29810	. . . . . . . . . . 11 ⊢ ( ℋ ≠ 0ℋ ↔ ∃𝑦 ∈ ℋ 𝑦 ≠ 0ℎ)
31		norm1hex 29514	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℋ 𝑦 ≠ 0ℎ ↔ ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
32	30, 31	sylbb 218	. . . . . . . . . 10 ⊢ ( ℋ ≠ 0ℋ → ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
33	32	adantr 480	. . . . . . . . 9 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
34		1le1 11533	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 1
35		breq1 5073	. . . . . . . . . . . . . 14 ⊢ ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ↔ 1 ≤ 1))
36	34, 35	mpbiri 257	. . . . . . . . . . . . 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 2750	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → (normℎ‘(𝑇‘𝑦)) = 1)
42	41	eqcomd 2744	. . . . . . . . . . . . 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 710	. . . . . . . . . 10 ⊢ ((( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
46	45	reximdva 3202	. . . . . . . . 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 10902	. . . . . . . . 9 ⊢ 1 ∈ V
49		eqeq1 2742	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ 1 = (normℎ‘(𝑇‘𝑦))))
50	49	anbi2d 628	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
51	50	rexbidv 3225	. . . . . . . . 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 3602	. . . . . . . 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 233	. . . . . . 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 5074	. . . . . . 7 ⊢ (𝑤 = 1 → (𝑧 < 𝑤 ↔ 𝑧 < 1))
56	55	rspcev 3552	. . . . . 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 579	. . . . 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 3107	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 < 𝑤))
60		supxr2 12977	. . 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 833	. 2 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) = 1)
62	6, 61	eqtrd 2778	1 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = 1)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883   class class class wbr 5070  ⟶wf 6414  ‘cfv 6418  supcsup 9129  ℝcr 10801  1c1 10803  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   ℋchba 29182  norm_ℎcno 29186  0_ℎc0v 29187  0_ℋc0h 29198  norm_opcnop 29208  LinOpclo 29210  UniOpcuo 29212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-hilex 29262  ax-hfvadd 29263  ax-hvcom 29264  ax-hvass 29265  ax-hv0cl 29266  ax-hvaddid 29267  ax-hfvmul 29268  ax-hvmulid 29269  ax-hvmulass 29270  ax-hvdistr1 29271  ax-hvdistr2 29272  ax-hvmul0 29273  ax-hfi 29342  ax-his1 29345  ax-his2 29346  ax-his3 29347  ax-his4 29348

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-grpo 28756  df-gid 28757  df-ablo 28808  df-vc 28822  df-nv 28855  df-va 28858  df-ba 28859  df-sm 28860  df-0v 28861  df-nmcv 28863  df-hnorm 29231  df-hba 29232  df-hvsub 29234  df-hlim 29235  df-sh 29470  df-ch 29484  df-ch0 29516  df-nmop 30102  df-lnop 30104  df-unop 30106

This theorem is referenced by:  unopbd  30278  unierri  30367