Theorem cnfnc 30193

Description: Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

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

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

Proof of Theorem cnfnc

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

Step	Hyp	Ref	Expression
1		elcnfn 30145	. . . 4 ⊢ (𝑇 ∈ ContFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) < 𝑤)))
2	1	simprbi 496	. . 3 ⊢ (𝑇 ∈ ContFn → ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) < 𝑤))
3		oveq2 7263	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 −ℎ 𝑧) = (𝑦 −ℎ 𝐴))
4	3	fveq2d 6760	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (normℎ‘(𝑦 −ℎ 𝑧)) = (normℎ‘(𝑦 −ℎ 𝐴)))
5	4	breq1d 5080	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 ↔ (normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥))
6		fveq2 6756	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑇‘𝑧) = (𝑇‘𝐴))
7	6	oveq2d 7271	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝑇‘𝑦) − (𝑇‘𝑧)) = ((𝑇‘𝑦) − (𝑇‘𝐴)))
8	7	fveq2d 6760	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) = (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))))
9	8	breq1d 5080	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) < 𝑤 ↔ (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝑤))
10	5, 9	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝐴 → (((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) < 𝑤) ↔ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝑤)))
11	10	rexralbidv 3229	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝑤)))
12		breq2 5074	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝑤 ↔ (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵))
13	12	imbi2d 340	. . . . 5 ⊢ (𝑤 = 𝐵 → (((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝑤) ↔ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵)))
14	13	rexralbidv 3229	. . . 4 ⊢ (𝑤 = 𝐵 → (∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵)))
15	11, 14	rspc2v 3562	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → (∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) < 𝑤) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵)))
16	2, 15	syl5com 31	. 2 ⊢ (𝑇 ∈ ContFn → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵)))
17	16	3impib 1114	1 ⊢ ((𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   class class class wbr 5070  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ℂcc 10800   < clt 10940   − cmin 11135  ℝ⁺crp 12659  abscabs 14873   ℋchba 29182  norm_ℎcno 29186   −_ℎ cmv 29188  ContFnccnfn 29216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-cnfn 30110

This theorem is referenced by:  nmcfnexi  30314