Theorem cnopc 31421

Description: Basic continuity property of a continuous Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
cnopc	⊢ ((𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝐵))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑇,𝑦

Proof of Theorem cnopc

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elcnop 31365	. . . 4 ⊢ (𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝑧))) < 𝑤)))
2	1	simprbi 497	. . 3 ⊢ (𝑇 ∈ ContOp → ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝑧))) < 𝑤))
3		oveq2 7419	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 −ℎ 𝑧) = (𝑦 −ℎ 𝐴))
4	3	fveq2d 6895	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (normℎ‘(𝑦 −ℎ 𝑧)) = (normℎ‘(𝑦 −ℎ 𝐴)))
5	4	breq1d 5158	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 ↔ (normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥))
6		fveq2 6891	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑇‘𝑧) = (𝑇‘𝐴))
7	6	oveq2d 7427	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝑇‘𝑦) −ℎ (𝑇‘𝑧)) = ((𝑇‘𝑦) −ℎ (𝑇‘𝐴)))
8	7	fveq2d 6895	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝑧))) = (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))))
9	8	breq1d 5158	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝑧))) < 𝑤 ↔ (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝑤))
10	5, 9	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝐴 → (((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝑧))) < 𝑤) ↔ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝑤)))
11	10	rexralbidv 3220	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝑧))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝑤)))
12		breq2 5152	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝑤 ↔ (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝐵))
13	12	imbi2d 340	. . . . 5 ⊢ (𝑤 = 𝐵 → (((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝑤) ↔ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝐵)))
14	13	rexralbidv 3220	. . . 4 ⊢ (𝑤 = 𝐵 → (∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝐵)))
15	11, 14	rspc2v 3622	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → (∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝑧))) < 𝑤) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝐵)))
16	2, 15	syl5com 31	. 2 ⊢ (𝑇 ∈ ContOp → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝐵)))
17	16	3impib 1116	1 ⊢ ((𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   class class class wbr 5148  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411   < clt 11252  ℝ⁺crp 12978   ℋchba 30427  norm_ℎcno 30431   −_ℎ cmv 30433  ContOpccop 30454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-hilex 30507

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-cnop 31348

This theorem is referenced by:  nmcopexi  31535