Theorem cnfnc 29403

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 29355	. . . 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 497	. . 3 ⊢ (𝑇 ∈ ContFn → ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+ ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) < 𝑤))
3		oveq2 7029	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 −ℎ 𝑧) = (𝑦 −ℎ 𝐴))
4	3	fveq2d 6547	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (normℎ‘(𝑦 −ℎ 𝑧)) = (normℎ‘(𝑦 −ℎ 𝐴)))
5	4	breq1d 4976	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 ↔ (normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥))
6		fveq2 6543	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑇‘𝑧) = (𝑇‘𝐴))
7	6	oveq2d 7037	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝑇‘𝑦) − (𝑇‘𝑧)) = ((𝑇‘𝑦) − (𝑇‘𝐴)))
8	7	fveq2d 6547	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) = (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))))
9	8	breq1d 4976	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) < 𝑤 ↔ (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝑤))
10	5, 9	imbi12d 346	. . . . 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 3264	. . . 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 4970	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝑤 ↔ (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵))
13	12	imbi2d 342	. . . . 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 3264	. . . 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 3572	. . 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 1109	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 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106   class class class wbr 4966  ⟶wf 6226  ‘cfv 6230  (class class class)co 7021  ℂcc 10386   < clt 10526   − cmin 10722  ℝ⁺crp 12244  abscabs 14432   ℋchba 28392  norm_ℎcno 28396   −_ℎ cmv 28398  ContFnccnfn 28426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-hilex 28472

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-fv 6238  df-ov 7024  df-oprab 7025  df-mpo 7026  df-map 8263  df-cnfn 29320

This theorem is referenced by:  nmcfnexi  29524