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Theorem rlimres 15501
Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimres (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ 𝐡) β‡π‘Ÿ 𝐴)
Proof of Theorem rlimres
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 4228 . . . . . . . 8 (dom 𝐹 ∩ 𝐡) βŠ† dom 𝐹
2 ssralv 4050 . . . . . . . 8 ((dom 𝐹 ∩ 𝐡) βŠ† dom 𝐹 β†’ (βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯) β†’ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯)))
31, 2ax-mp 5 . . . . . . 7 (βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯) β†’ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))
43reximi 3084 . . . . . 6 (βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))
54ralimi 3083 . . . . 5 (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯) β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))
65anim2i 617 . . . 4 ((𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯)) β†’ (𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯)))
76a1i 11 . . 3 (𝐹 β‡π‘Ÿ 𝐴 β†’ ((𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯)) β†’ (𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))))
8 rlimf 15444 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ 𝐹:dom πΉβŸΆβ„‚)
9 rlimss 15445 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ dom 𝐹 βŠ† ℝ)
10 eqidd 2733 . . . 4 ((𝐹 β‡π‘Ÿ 𝐴 ∧ 𝑧 ∈ dom 𝐹) β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘§))
118, 9, 10rlim 15438 . . 3 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β‡π‘Ÿ 𝐴 ↔ (𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))))
12 fssres 6757 . . . . . 6 ((𝐹:dom πΉβŸΆβ„‚ ∧ (dom 𝐹 ∩ 𝐡) βŠ† dom 𝐹) β†’ (𝐹 β†Ύ (dom 𝐹 ∩ 𝐡)):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚)
138, 1, 12sylancl 586 . . . . 5 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ (dom 𝐹 ∩ 𝐡)):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚)
14 resres 5994 . . . . . . 7 ((𝐹 β†Ύ dom 𝐹) β†Ύ 𝐡) = (𝐹 β†Ύ (dom 𝐹 ∩ 𝐡))
15 ffn 6717 . . . . . . . . 9 (𝐹:dom πΉβŸΆβ„‚ β†’ 𝐹 Fn dom 𝐹)
16 fnresdm 6669 . . . . . . . . 9 (𝐹 Fn dom 𝐹 β†’ (𝐹 β†Ύ dom 𝐹) = 𝐹)
178, 15, 163syl 18 . . . . . . . 8 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ dom 𝐹) = 𝐹)
1817reseq1d 5980 . . . . . . 7 (𝐹 β‡π‘Ÿ 𝐴 β†’ ((𝐹 β†Ύ dom 𝐹) β†Ύ 𝐡) = (𝐹 β†Ύ 𝐡))
1914, 18eqtr3id 2786 . . . . . 6 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ (dom 𝐹 ∩ 𝐡)) = (𝐹 β†Ύ 𝐡))
2019feq1d 6702 . . . . 5 (𝐹 β‡π‘Ÿ 𝐴 β†’ ((𝐹 β†Ύ (dom 𝐹 ∩ 𝐡)):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚ ↔ (𝐹 β†Ύ 𝐡):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚))
2113, 20mpbid 231 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ 𝐡):(dom 𝐹 ∩ 𝐡)βŸΆβ„‚)
221, 9sstrid 3993 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ (dom 𝐹 ∩ 𝐡) βŠ† ℝ)
23 elinel2 4196 . . . . . 6 (𝑧 ∈ (dom 𝐹 ∩ 𝐡) β†’ 𝑧 ∈ 𝐡)
2423fvresd 6911 . . . . 5 (𝑧 ∈ (dom 𝐹 ∩ 𝐡) β†’ ((𝐹 β†Ύ 𝐡)β€˜π‘§) = (πΉβ€˜π‘§))
2524adantl 482 . . . 4 ((𝐹 β‡π‘Ÿ 𝐴 ∧ 𝑧 ∈ (dom 𝐹 ∩ 𝐡)) β†’ ((𝐹 β†Ύ 𝐡)β€˜π‘§) = (πΉβ€˜π‘§))
2621, 22, 25rlim 15438 . . 3 (𝐹 β‡π‘Ÿ 𝐴 β†’ ((𝐹 β†Ύ 𝐡) β‡π‘Ÿ 𝐴 ↔ (𝐴 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ (dom 𝐹 ∩ 𝐡)(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐴)) < π‘₯))))
277, 11, 263imtr4d 293 . 2 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ 𝐡) β‡π‘Ÿ 𝐴))
2827pm2.43i 52 1 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β†Ύ 𝐡) β‡π‘Ÿ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948   class class class wbr 5148  dom cdm 5676   β†Ύ cres 5678   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  β„‚cc 11107  β„cr 11108   < clt 11247   ≀ cle 11248   βˆ’ cmin 11443  β„+crp 12973  abscabs 15180   β‡π‘Ÿ crli 15428
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-pm 8822  df-rlim 15432
This theorem is referenced by:  rlimres2  15504  pnt  27114
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