Theorem nmfnval 31965

Description: Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem nmfnval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltso 13083	. . 3 ⊢ < Or ℝ*
2	1	supex 9367	. 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ) ∈ V
3		ax-hilex 31088	. 2 ⊢ ℋ ∈ V
4		cnex 11110	. 2 ⊢ ℂ ∈ V
5		fveq1 6826	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
6	5	fveq2d 6831	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (abs‘(𝑡‘𝑦)) = (abs‘(𝑇‘𝑦)))
7	6	eqeq2d 2750	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 = (abs‘(𝑡‘𝑦)) ↔ 𝑥 = (abs‘(𝑇‘𝑦))))
8	7	anbi2d 636	. . . . 5 ⊢ (𝑡 = 𝑇 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))))
9	8	rexbidv 3163	. . . 4 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))))
10	9	abbidv 2805	. . 3 ⊢ (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))})
11	10	supeq1d 9349	. 2 ⊢ (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ))
12		df-nmfn 31934	. 2 ⊢ normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦)))}, ℝ*, < ))
13	2, 3, 4, 11, 12	fvmptmap 8819	1 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  {cab 2717  ∃wrex 3063   class class class wbr 5072  ⟶wf 6481  ‘cfv 6485  supcsup 9343  ℂcc 11027  1c1 11030  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171  abscabs 15187   ℋchba 31008  norm_ℎcno 31012  norm_fncnmf 31040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-hilex 31088

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-nmfn 31934

This theorem is referenced by:  nmfnxr  31968  nmfnrepnf  31969  nmfnlb  32013  nmfnleub  32014  nmfn0  32076  nmcfnexi  32140  branmfn  32194