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Theorem rlimcn1 15297
Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 17-Sep-2014.)
Hypotheses
Ref Expression
rlimcn1.1 (𝜑𝐺:𝐴𝑋)
rlimcn1.2 (𝜑𝐶𝑋)
rlimcn1.3 (𝜑𝐺𝑟 𝐶)
rlimcn1.4 (𝜑𝐹:𝑋⟶ℂ)
rlimcn1.5 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
Assertion
Ref Expression
rlimcn1 (𝜑 → (𝐹𝐺) ⇝𝑟 (𝐹𝐶))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑧,𝐹,𝑦   𝑥,𝐺,𝑦,𝑧   𝜑,𝑥,𝑦   𝑥,𝐶,𝑦,𝑧   𝑧,𝑋
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)   𝑋(𝑥,𝑦)

Proof of Theorem rlimcn1
Dummy variables 𝑤 𝑐 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimcn1.1 . . . 4 (𝜑𝐺:𝐴𝑋)
21ffvelrnda 6961 . . 3 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
31feqmptd 6837 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
4 rlimcn1.4 . . . 4 (𝜑𝐹:𝑋⟶ℂ)
54feqmptd 6837 . . 3 (𝜑𝐹 = (𝑣𝑋 ↦ (𝐹𝑣)))
6 fveq2 6774 . . 3 (𝑣 = (𝐺𝑤) → (𝐹𝑣) = (𝐹‘(𝐺𝑤)))
72, 3, 5, 6fmptco 7001 . 2 (𝜑 → (𝐹𝐺) = (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))))
8 rlimcn1.5 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
9 fvexd 6789 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ V)
109ralrimiva 3103 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ∀𝑤𝐴 (𝐺𝑤) ∈ V)
11 simpr 485 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
12 rlimcn1.3 . . . . . . . . . 10 (𝜑𝐺𝑟 𝐶)
133, 12eqbrtrrd 5098 . . . . . . . . 9 (𝜑 → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1413ad2antrr 723 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1510, 11, 14rlimi 15222 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
16 fvoveq1 7298 . . . . . . . . . . . . . 14 (𝑧 = (𝐺𝑤) → (abs‘(𝑧𝐶)) = (abs‘((𝐺𝑤) − 𝐶)))
1716breq1d 5084 . . . . . . . . . . . . 13 (𝑧 = (𝐺𝑤) → ((abs‘(𝑧𝐶)) < 𝑦 ↔ (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
1817imbrov2fvoveq 7300 . . . . . . . . . . . 12 (𝑧 = (𝐺𝑤) → (((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) ↔ ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
19 simplrr 775 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
202ad4ant14 749 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
2118, 19, 20rspcdva 3562 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
2221imim2d 57 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ((𝑐𝑤 → (abs‘((𝐺𝑤) − 𝐶)) < 𝑦) → (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
2322ralimdva 3108 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) → (∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐺𝑤) − 𝐶)) < 𝑦) → ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
2423reximdv 3202 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) → (∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐺𝑤) − 𝐶)) < 𝑦) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
2524expr 457 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → (∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐺𝑤) − 𝐶)) < 𝑦) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))))
2615, 25mpid 44 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
2726rexlimdva 3213 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → (∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
288, 27mpd 15 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
2928ralrimiva 3103 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
304ffvelrnda 6961 . . . . . 6 ((𝜑 ∧ (𝐺𝑤) ∈ 𝑋) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
312, 30syldan 591 . . . . 5 ((𝜑𝑤𝐴) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
3231ralrimiva 3103 . . . 4 (𝜑 → ∀𝑤𝐴 (𝐹‘(𝐺𝑤)) ∈ ℂ)
331fdmd 6611 . . . . 5 (𝜑 → dom 𝐺 = 𝐴)
34 rlimss 15211 . . . . . 6 (𝐺𝑟 𝐶 → dom 𝐺 ⊆ ℝ)
3512, 34syl 17 . . . . 5 (𝜑 → dom 𝐺 ⊆ ℝ)
3633, 35eqsstrrd 3960 . . . 4 (𝜑𝐴 ⊆ ℝ)
37 rlimcn1.2 . . . . 5 (𝜑𝐶𝑋)
384, 37ffvelrnd 6962 . . . 4 (𝜑 → (𝐹𝐶) ∈ ℂ)
3932, 36, 38rlim2 15205 . . 3 (𝜑 → ((𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶) ↔ ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
4029, 39mpbird 256 . 2 (𝜑 → (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶))
417, 40eqbrtrd 5096 1 (𝜑 → (𝐹𝐺) ⇝𝑟 (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  wrex 3065  Vcvv 3432  wss 3887   class class class wbr 5074  cmpt 5157  dom cdm 5589  ccom 5593  wf 6429  cfv 6433  (class class class)co 7275  cc 10869  cr 10870   < clt 11009  cle 11010  cmin 11205  +crp 12730  abscabs 14945  𝑟 crli 15194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-pm 8618  df-rlim 15198
This theorem is referenced by:  rlimcn1b  15298  rlimdiv  15357
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