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Theorem rlimcn1 14938
 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 6846 . . 3 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
31feqmptd 6729 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
4 rlimcn1.4 . . . 4 (𝜑𝐹:𝑋⟶ℂ)
54feqmptd 6729 . . 3 (𝜑𝐹 = (𝑣𝑋 ↦ (𝐹𝑣)))
6 fveq2 6666 . . 3 (𝑣 = (𝐺𝑤) → (𝐹𝑣) = (𝐹‘(𝐺𝑤)))
72, 3, 5, 6fmptco 6886 . 2 (𝜑 → (𝐹𝐺) = (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))))
8 rlimcn1.5 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
9 fvexd 6681 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ V)
109ralrimiva 3186 . . . . . . . 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 5086 . . . . . . . . 9 (𝜑 → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1413ad2antrr 722 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1510, 11, 14rlimi 14863 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
16 fvoveq1 7174 . . . . . . . . . . . . . 14 (𝑧 = (𝐺𝑤) → (abs‘(𝑧𝐶)) = (abs‘((𝐺𝑤) − 𝐶)))
1716breq1d 5072 . . . . . . . . . . . . 13 (𝑧 = (𝐺𝑤) → ((abs‘(𝑧𝐶)) < 𝑦 ↔ (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
1817imbrov2fvoveq 7176 . . . . . . . . . . . 12 (𝑧 = (𝐺𝑤) → (((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) ↔ ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
19 simplrr 774 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
202ad4ant14 748 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
2118, 19, 20rspcdva 3628 . . . . . . . . . . 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 3181 . . . . . . . . 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 3277 . . . . . . . 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 3288 . . . . 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 3186 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
304ffvelrnda 6846 . . . . . 6 ((𝜑 ∧ (𝐺𝑤) ∈ 𝑋) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
312, 30syldan 591 . . . . 5 ((𝜑𝑤𝐴) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
3231ralrimiva 3186 . . . 4 (𝜑 → ∀𝑤𝐴 (𝐹‘(𝐺𝑤)) ∈ ℂ)
331fdmd 6519 . . . . 5 (𝜑 → dom 𝐺 = 𝐴)
34 rlimss 14852 . . . . . 6 (𝐺𝑟 𝐶 → dom 𝐺 ⊆ ℝ)
3512, 34syl 17 . . . . 5 (𝜑 → dom 𝐺 ⊆ ℝ)
3633, 35eqsstrrd 4009 . . . 4 (𝜑𝐴 ⊆ ℝ)
37 rlimcn1.2 . . . . 5 (𝜑𝐶𝑋)
384, 37ffvelrnd 6847 . . . 4 (𝜑 → (𝐹𝐶) ∈ ℂ)
3932, 36, 38rlim2 14846 . . 3 (𝜑 → ((𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶) ↔ ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
4029, 39mpbird 258 . 2 (𝜑 → (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶))
417, 40eqbrtrd 5084 1 (𝜑 → (𝐹𝐺) ⇝𝑟 (𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107  ∀wral 3142  ∃wrex 3143  Vcvv 3499   ⊆ wss 3939   class class class wbr 5062   ↦ cmpt 5142  dom cdm 5553   ∘ ccom 5557  ⟶wf 6347  ‘cfv 6351  (class class class)co 7151  ℂcc 10527  ℝcr 10528   < clt 10667   ≤ cle 10668   − cmin 10862  ℝ+crp 12382  abscabs 14586   ⇝𝑟 crli 14835 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-pm 8402  df-rlim 14839 This theorem is referenced by:  rlimcn1b  14939  rlimdiv  14995
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