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Theorem rlimcn1 14167
 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 6267 . . 3 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
31feqmptd 6159 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
4 rlimcn1.4 . . . 4 (𝜑𝐹:𝑋⟶ℂ)
54feqmptd 6159 . . 3 (𝜑𝐹 = (𝑣𝑋 ↦ (𝐹𝑣)))
6 fveq2 6103 . . 3 (𝑣 = (𝐺𝑤) → (𝐹𝑣) = (𝐹‘(𝐺𝑤)))
72, 3, 5, 6fmptco 6303 . 2 (𝜑 → (𝐹𝐺) = (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))))
8 rlimcn1.5 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
9 fvex 6113 . . . . . . . . . 10 (𝐺𝑤) ∈ V
109a1i 11 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ V)
1110ralrimiva 2949 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ∀𝑤𝐴 (𝐺𝑤) ∈ V)
12 simpr 476 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
13 rlimcn1.3 . . . . . . . . . 10 (𝜑𝐺𝑟 𝐶)
143, 13eqbrtrrd 4607 . . . . . . . . 9 (𝜑 → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1514ad2antrr 758 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1611, 12, 15rlimi 14092 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
17 simpll 786 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) → 𝜑)
1817, 2sylan 487 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
19 simplrr 797 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
20 oveq1 6556 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑤) → (𝑧𝐶) = ((𝐺𝑤) − 𝐶))
2120fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑧 = (𝐺𝑤) → (abs‘(𝑧𝐶)) = (abs‘((𝐺𝑤) − 𝐶)))
2221breq1d 4593 . . . . . . . . . . . . . 14 (𝑧 = (𝐺𝑤) → ((abs‘(𝑧𝐶)) < 𝑦 ↔ (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
23 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑤) → (𝐹𝑧) = (𝐹‘(𝐺𝑤)))
2423oveq1d 6564 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑤) → ((𝐹𝑧) − (𝐹𝐶)) = ((𝐹‘(𝐺𝑤)) − (𝐹𝐶)))
2524fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑧 = (𝐺𝑤) → (abs‘((𝐹𝑧) − (𝐹𝐶))) = (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))))
2625breq1d 4593 . . . . . . . . . . . . . 14 (𝑧 = (𝐺𝑤) → ((abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥 ↔ (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
2722, 26imbi12d 333 . . . . . . . . . . . . 13 (𝑧 = (𝐺𝑤) → (((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) ↔ ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
2827rspcv 3278 . . . . . . . . . . . 12 ((𝐺𝑤) ∈ 𝑋 → (∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
2918, 19, 28sylc 63 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
3029imim2d 55 . . . . . . . . . 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)))
3130ralimdva 2945 . . . . . . . . 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)))
3231reximdv 2999 . . . . . . . 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)))
3332expr 641 . . . . . . 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))))
3416, 33mpid 43 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
3534rexlimdva 3013 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → (∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
368, 35mpd 15 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
3736ralrimiva 2949 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
384ffvelrnda 6267 . . . . . 6 ((𝜑 ∧ (𝐺𝑤) ∈ 𝑋) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
392, 38syldan 486 . . . . 5 ((𝜑𝑤𝐴) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
4039ralrimiva 2949 . . . 4 (𝜑 → ∀𝑤𝐴 (𝐹‘(𝐺𝑤)) ∈ ℂ)
41 fdm 5964 . . . . . 6 (𝐺:𝐴𝑋 → dom 𝐺 = 𝐴)
421, 41syl 17 . . . . 5 (𝜑 → dom 𝐺 = 𝐴)
43 rlimss 14081 . . . . . 6 (𝐺𝑟 𝐶 → dom 𝐺 ⊆ ℝ)
4413, 43syl 17 . . . . 5 (𝜑 → dom 𝐺 ⊆ ℝ)
4542, 44eqsstr3d 3603 . . . 4 (𝜑𝐴 ⊆ ℝ)
46 rlimcn1.2 . . . . 5 (𝜑𝐶𝑋)
474, 46ffvelrnd 6268 . . . 4 (𝜑 → (𝐹𝐶) ∈ ℂ)
4840, 45, 47rlim2 14075 . . 3 (𝜑 → ((𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶) ↔ ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
4937, 48mpbird 246 . 2 (𝜑 → (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶))
507, 49eqbrtrd 4605 1 (𝜑 → (𝐹𝐺) ⇝𝑟 (𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814   < clt 9953   ≤ cle 9954   − cmin 10145  ℝ+crp 11708  abscabs 13822   ⇝𝑟 crli 14064 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-rlim 14068 This theorem is referenced by:  rlimcn1b  14168  rlimdiv  14224
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