Theorem elcnop 31876

Description: Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
elcnop	⊢ (𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦)))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝑇

Proof of Theorem elcnop

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6905	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑤) = (𝑇‘𝑤))
2		fveq1 6905	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
3	1, 2	oveq12d 7449	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑤) −ℎ (𝑡‘𝑥)) = ((𝑇‘𝑤) −ℎ (𝑇‘𝑥)))
4	3	fveq2d 6910	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) = (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))))
5	4	breq1d 5153	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦 ↔ (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦))
6	5	imbi2d 340	. . . . 5 ⊢ (𝑡 = 𝑇 → (((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦) ↔ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦)))
7	6	rexralbidv 3223	. . . 4 ⊢ (𝑡 = 𝑇 → (∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦)))
8	7	2ralbidv 3221	. . 3 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦)))
9		df-cnop 31859	. . 3 ⊢ ContOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦)}
10	8, 9	elrab2 3695	. 2 ⊢ (𝑇 ∈ ContOp ↔ (𝑇 ∈ ( ℋ ↑m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦)))
11		ax-hilex 31018	. . . 4 ⊢ ℋ ∈ V
12	11, 11	elmap 8911	. . 3 ⊢ (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
13	12	anbi1i 624	. 2 ⊢ ((𝑇 ∈ ( ℋ ↑m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦)))
14	10, 13	bitri 275	1 ⊢ (𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   class class class wbr 5143  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866   < clt 11295  ℝ⁺crp 13034   ℋchba 30938  norm_ℎcno 30942   −_ℎ cmv 30944  ContOpccop 30965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-hilex 31018

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-cnop 31859

This theorem is referenced by:  cnopc  31932  0cnop  31998  idcnop  32000  lnopconi  32053