Theorem nmopval 27888

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 11718	. . 3 ⊢ < Or ℝ*
2	1	supex 8127	. 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) ∈ V
3		ax-hilex 27029	. 2 ⊢ ℋ ∈ V
4		fveq1 5985	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
5	4	fveq2d 5990	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (normℎ‘(𝑡‘𝑦)) = (normℎ‘(𝑇‘𝑦)))
6	5	eqeq2d 2524	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 = (normℎ‘(𝑡‘𝑦)) ↔ 𝑥 = (normℎ‘(𝑇‘𝑦))))
7	6	anbi2d 735	. . . . 5 ⊢ (𝑡 = 𝑇 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))))
8	7	rexbidv 2938	. . . 4 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))))
9	8	abbidv 2632	. . 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 8110	. 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 27871	. 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 7655	1 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1474  {cab 2500  ∃wrex 2801   class class class wbr 4481  ⟶wf 5685  ‘cfv 5689  supcsup 8104  1c1 9691  ℝ^*cxr 9827   < clt 9828   ≤ cle 9829   ℋchil 26949  norm_ℎcno 26953  norm_opcnop 26975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722  ax-cnex 9746  ax-resscn 9747  ax-pre-lttri 9764  ax-pre-lttrn 9765  ax-hilex 27029

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-po 4853  df-so 4854  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-er 7504  df-map 7621  df-en 7717  df-dom 7718  df-sdom 7719  df-sup 8106  df-pnf 9830  df-mnf 9831  df-xr 9832  df-ltxr 9833  df-nmop 27871

This theorem is referenced by:  nmopxr  27898  nmoprepnf  27899  nmoplb  27939  nmopub  27940  nmopnegi  27997  nmop0  28018  nmlnop0iALT  28027  nmopun  28046  nmcopexi  28059  pjnmopi  28180