Theorem cnopc 31942

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 31886	. . . 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 496	. . 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 7439	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 −ℎ 𝑧) = (𝑦 −ℎ 𝐴))
4	3	fveq2d 6911	. . . . . . 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 6907	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑇‘𝑧) = (𝑇‘𝐴))
7	6	oveq2d 7447	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝑇‘𝑦) −ℎ (𝑇‘𝑧)) = ((𝑇‘𝑦) −ℎ (𝑇‘𝐴)))
8	7	fveq2d 6911	. . . . . . 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 3221	. . . 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 3221	. . . 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 3633	. . 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 1115	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 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   class class class wbr 5148  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   < clt 11293  ℝ⁺crp 13032   ℋchba 30948  norm_ℎcno 30952   −_ℎ cmv 30954  ContOpccop 30975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-hilex 31028

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-cnop 31869

This theorem is referenced by:  nmcopexi  32056