Theorem elcnop 31786

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 6857	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑤) = (𝑇‘𝑤))
2		fveq1 6857	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
3	1, 2	oveq12d 7405	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑤) −ℎ (𝑡‘𝑥)) = ((𝑇‘𝑤) −ℎ (𝑇‘𝑥)))
4	3	fveq2d 6862	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) = (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))))
5	4	breq1d 5117	. . . . . 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 3203	. . . 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 3201	. . 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 31769	. . 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 3662	. 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 30928	. . . 4 ⊢ ℋ ∈ V
12	11, 11	elmap 8844	. . 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 2109  ∀wral 3044  ∃wrex 3053   class class class wbr 5107  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799   < clt 11208  ℝ⁺crp 12951   ℋchba 30848  norm_ℎcno 30852   −_ℎ cmv 30854  ContOpccop 30875

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-hilex 30928

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-cnop 31769

This theorem is referenced by:  cnopc  31842  0cnop  31908  idcnop  31910  lnopconi  31963