Theorem nmopun 32496

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 32402	. . . . 5 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		lnopf 32341	. . . . 5 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
3	1, 2	syl 18	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4		nmopval 32338	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
5	3, 4	syl 18	. . 3 ⊢ (𝑇 ∈ UniOp → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
6	5	adantl 487	. 2 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
7		nmopsetretHIL 32346	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
8		ressxr 11278	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
9	7, 8	sstrdi 3943	. . . . . 6 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
10	3, 9	syl 18	. . . . 5 ⊢ (𝑇 ∈ UniOp → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
11	10	adantl 487	. . . 4 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
12		1xr 11293	. . . 4 ⊢ 1 ∈ ℝ*
13	11, 12	jctir 530	. . 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 2764	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ 𝑧 = (normℎ‘(𝑇‘𝑦))))
16	15	anbi2d 642	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦)))))
17	16	rexbidv 3186	. . . . . . 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 3633	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))))
19		unopnorm 32399	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
20	19	eqeq2d 2771	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑧 = (normℎ‘(𝑇‘𝑦)) ↔ 𝑧 = (normℎ‘𝑦)))
21	20	anbi2d 642	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘𝑦))))
22		breq1 5106	. . . . . . . . . 10 ⊢ (𝑧 = (normℎ‘𝑦) → (𝑧 ≤ 1 ↔ (normℎ‘𝑦) ≤ 1))
23	22	biimparc 485	. . . . . . . . 9 ⊢ (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘𝑦)) → 𝑧 ≤ 1)
24	21, 23	biimtrdi 256	. . . . . . . 8 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) → 𝑧 ≤ 1))
25	24	rexlimdva 3163	. . . . . . 7 ⊢ (𝑇 ∈ UniOp → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦))) → 𝑧 ≤ 1))
26	25	imp 412	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑧 = (normℎ‘(𝑇‘𝑦)))) → 𝑧 ≤ 1)
27	18, 26	sylan2b 606	. . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}) → 𝑧 ≤ 1)
28	27	ralrimiva 3154	. . . 4 ⊢ (𝑇 ∈ UniOp → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 ≤ 1)
29	28	adantl 487	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 ≤ 1)
30		hne0 32029	. . . . . . . . . . 11 ⊢ ( ℋ ≠ 0ℋ ↔ ∃𝑦 ∈ ℋ 𝑦 ≠ 0ℎ)
31		norm1hex 31733	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ℋ 𝑦 ≠ 0ℎ ↔ ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
32	30, 31	sylbb 222	. . . . . . . . . 10 ⊢ ( ℋ ≠ 0ℋ → ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
33	32	adantr 486	. . . . . . . . 9 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1)
34		1le1 11867	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 1
35		breq1 5106	. . . . . . . . . . . . . 14 ⊢ ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ↔ 1 ≤ 1))
36	34, 35	mpbiri 261	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
39		eqeq2 2772	. . . . . . . . . . . . . . . 16 ⊢ ((normℎ‘𝑦) = 1 → ((normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦) ↔ (normℎ‘(𝑇‘𝑦)) = 1))
40	39	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → ((normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦) ↔ (normℎ‘(𝑇‘𝑦)) = 1))
41	38, 40	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → (normℎ‘(𝑇‘𝑦)) = 1)
42	41	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (normℎ‘𝑦) = 1) → 1 = (normℎ‘(𝑇‘𝑦)))
43	42	ex 418	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → 1 = (normℎ‘(𝑇‘𝑦))))
44	37, 43	jcad 522	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
45	44	adantll 727	. . . . . . . . . 10 ⊢ ((( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑦) = 1 → ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
46	45	reximdva 3175	. . . . . . . . 9 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1 → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
47	33, 46	mpd 16	. . . . . . . 8 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦))))
48		1ex 11228	. . . . . . . . 9 ⊢ 1 ∈ V
49		eqeq1 2764	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ 1 = (normℎ‘(𝑇‘𝑦))))
50	49	anbi2d 642	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 1 = (normℎ‘(𝑇‘𝑦)))))
51	50	rexbidv 3186	. . . . . . . . 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 3633	. . . . . . . 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 237	. . . . . . 7 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
54	53	adantr 486	. . . . . 6 ⊢ ((( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
55		breq2 5107	. . . . . . 7 ⊢ (𝑤 = 1 → (𝑧 < 𝑤 ↔ 𝑧 < 1))
56	55	rspcev 3576	. . . . . 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 592	. . . . 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 418	. . . 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 3154	. . 3 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}𝑧 < 𝑤))
60		supxr2 13367	. . 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 850	. 2 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) = 1)
62	6, 61	eqtrd 2795	1 ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = 1)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899   class class class wbr 5103  ⟶wf 6529  ‘cfv 6533  supcsup 9411  ℝcr 11124  1c1 11126  ℝ^*cxr 11267   < clt 11268   ≤ cle 11269   ℋchba 31401  norm_ℎcno 31405  0_ℎc0v 31406  0_ℋc0h 31417  norm_opcnop 31427  LinOpclo 31429  UniOpcuo 31431

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202  ax-pre-sup 11203  ax-hilex 31481  ax-hfvadd 31482  ax-hvcom 31483  ax-hvass 31484  ax-hv0cl 31485  ax-hvaddid 31486  ax-hfvmul 31487  ax-hvmulid 31488  ax-hvmulass 31489  ax-hvdistr1 31490  ax-hvdistr2 31491  ax-hvmul0 31492  ax-hfi 31561  ax-his1 31564  ax-his2 31565  ax-his3 31566  ax-his4 31567

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8697  df-map 8829  df-en 8954  df-dom 8955  df-sdom 8956  df-sup 9413  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-div 11897  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-n0 12530  df-z 12617  df-uz 12889  df-rp 13044  df-seq 14067  df-exp 14127  df-cj 15187  df-re 15188  df-im 15189  df-sqrt 15323  df-abs 15324  df-grpo 30975  df-gid 30976  df-ablo 31027  df-vc 31041  df-nv 31074  df-va 31077  df-ba 31078  df-sm 31079  df-0v 31080  df-nmcv 31082  df-hnorm 31450  df-hba 31451  df-hvsub 31453  df-hlim 31454  df-sh 31689  df-ch 31703  df-ch0 31735  df-nmop 32321  df-lnop 32323  df-unop 32325

This theorem is used by:  unopbd  32497  unierri  32586