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Theorem cnfnc 30870
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 ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 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.
StepHypRef Expression
1 elcnfn 30822 . . . 4 (𝑇 ∈ ContFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤)))
21simprbi 497 . . 3 (𝑇 ∈ ContFn → ∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤))
3 oveq2 7364 . . . . . . . 8 (𝑧 = 𝐴 → (𝑦 𝑧) = (𝑦 𝐴))
43fveq2d 6846 . . . . . . 7 (𝑧 = 𝐴 → (norm‘(𝑦 𝑧)) = (norm‘(𝑦 𝐴)))
54breq1d 5115 . . . . . 6 (𝑧 = 𝐴 → ((norm‘(𝑦 𝑧)) < 𝑥 ↔ (norm‘(𝑦 𝐴)) < 𝑥))
6 fveq2 6842 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑇𝑧) = (𝑇𝐴))
76oveq2d 7372 . . . . . . . 8 (𝑧 = 𝐴 → ((𝑇𝑦) − (𝑇𝑧)) = ((𝑇𝑦) − (𝑇𝐴)))
87fveq2d 6846 . . . . . . 7 (𝑧 = 𝐴 → (abs‘((𝑇𝑦) − (𝑇𝑧))) = (abs‘((𝑇𝑦) − (𝑇𝐴))))
98breq1d 5115 . . . . . 6 (𝑧 = 𝐴 → ((abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤 ↔ (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤))
105, 9imbi12d 344 . . . . 5 (𝑧 = 𝐴 → (((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) ↔ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤)))
1110rexralbidv 3214 . . . 4 (𝑧 = 𝐴 → (∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤)))
12 breq2 5109 . . . . . 6 (𝑤 = 𝐵 → ((abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤 ↔ (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
1312imbi2d 340 . . . . 5 (𝑤 = 𝐵 → (((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤) ↔ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
1413rexralbidv 3214 . . . 4 (𝑤 = 𝐵 → (∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝑤) ↔ ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
1511, 14rspc2v 3590 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → (∀𝑧 ∈ ℋ ∀𝑤 ∈ ℝ+𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝑧)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝑧))) < 𝑤) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
162, 15syl5com 31 . 2 (𝑇 ∈ ContFn → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵)))
17163impib 1116 1 ((𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073   class class class wbr 5105  wf 6492  cfv 6496  (class class class)co 7356  cc 11048   < clt 11188  cmin 11384  +crp 12914  abscabs 15118  chba 29859  normcno 29863   cmv 29865  ContFnccnfn 29893
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-cnex 11106  ax-hilex 29939
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8766  df-cnfn 30787
This theorem is referenced by:  nmcfnexi  30991
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