Theorem nmopval 32059

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 13143	. . 3 ⊢ < Or ℝ*
2	1	supex 9410	. 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ) ∈ V
3		ax-hilex 31202	. 2 ⊢ ℋ ∈ V
4		fveq1 6866	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
5	4	fveq2d 6871	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (normℎ‘(𝑡‘𝑦)) = (normℎ‘(𝑇‘𝑦)))
6	5	eqeq2d 2773	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 = (normℎ‘(𝑡‘𝑦)) ↔ 𝑥 = (normℎ‘(𝑇‘𝑦))))
7	6	anbi2d 639	. . . . 5 ⊢ (𝑡 = 𝑇 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))))
8	7	rexbidv 3186	. . . 4 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))))
9	8	abbidv 2828	. . 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 9392	. 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 32042	. 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 8863	1 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  {cab 2740  ∃wrex 3086   class class class wbr 5100  ⟶wf 6517  ‘cfv 6521  supcsup 9386  1c1 11074  ℝ^*cxr 11215   < clt 11216   ≤ cle 11217   ℋchba 31122  norm_ℎcno 31126  norm_opcnop 31148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-hilex 31202

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-po 5555  df-so 5556  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-sup 9388  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-nmop 32042

This theorem is referenced by:  nmopxr  32069  nmoprepnf  32070  nmoplb  32110  nmopub  32111  nmopnegi  32168  nmop0  32189  nmlnop0iALT  32198  nmopun  32217  nmcopexi  32230  pjnmopi  32351