Theorem nmopval 32148

Description: Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
nmopval	⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))

Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem nmopval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltso 13165	. . 3 ⊢ < Or ℝ*
2	1	supex 9423	. 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) ∈ V
3		ax-hilex 31291	. 2 ⊢ ℋ ∈ V
4		fveq1 6881	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
5	4	fveq2d 6886	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (normℎ‘(𝑡‘𝑦)) = (normℎ‘(𝑇‘𝑦)))
6	5	eqeq2d 2780	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 = (normℎ‘(𝑡‘𝑦)) ↔ 𝑥 = (normℎ‘(𝑇‘𝑦))))
7	6	anbi2d 641	. . . . 5 ⊢ (𝑡 = 𝑇 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))))
8	7	rexbidv 3195	. . . 4 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))))
9	8	abbidv 2835	. . 3 ⊢ (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
10	9	supeq1d 9405	. 2 ⊢ (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
11		df-nmop 32131	. 2 ⊢ normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦)))}, ℝ*, < ))
12	2, 3, 3, 10, 11	fvmptmap 8878	1 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  {cab 2747  ∃wrex 3095   class class class wbr 5113  ⟶wf 6533  ‘cfv 6537  supcsup 9399  1c1 11100  ℝ^*cxr 11241   < clt 11242   ≤ cle 11243   ℋchba 31211  norm_ℎcno 31215  norm_opcnop 31237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-hilex 31291

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-po 5570  df-so 5571  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9401  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-nmop 32131

This theorem is referenced by:  nmopxr  32158  nmoprepnf  32159  nmoplb  32199  nmopub  32200  nmopnegi  32257  nmop0  32278  nmlnop0iALT  32287  nmopun  32306  nmcopexi  32319  pjnmopi  32440