Theorem cnfnc 32017

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 31969	. . . 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 7376	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 −ℎ 𝑧) = (𝑦 −ℎ 𝐴))
4	3	fveq2d 6846	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (normℎ‘(𝑦 −ℎ 𝑧)) = (normℎ‘(𝑦 −ℎ 𝐴)))
5	4	breq1d 5110	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((normℎ‘(𝑦 −ℎ 𝑧)) < 𝑥 ↔ (normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥))
6		fveq2 6842	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑇‘𝑧) = (𝑇‘𝐴))
7	6	oveq2d 7384	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝑇‘𝑦) − (𝑇‘𝑧)) = ((𝑇‘𝑦) − (𝑇‘𝐴)))
8	7	fveq2d 6846	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (abs‘((𝑇‘𝑦) − (𝑇‘𝑧))) = (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))))
9	8	breq1d 5110	. . . . . 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 3204	. . . 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 5104	. . . . . 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 3204	. . . 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 3589	. . 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 1117	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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   class class class wbr 5100  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  ℂcc 11036   < clt 11178   − cmin 11376  ℝ⁺crp 12917  abscabs 15169   ℋchba 31006  norm_ℎcno 31010   −_ℎ cmv 31012  ContFnccnfn 31040

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-hilex 31086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-cnfn 31934

This theorem is referenced by:  nmcfnexi  32138