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Theorem rlimcn1 14518
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 6522 . . 3 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
31feqmptd 6411 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
4 rlimcn1.4 . . . 4 (𝜑𝐹:𝑋⟶ℂ)
54feqmptd 6411 . . 3 (𝜑𝐹 = (𝑣𝑋 ↦ (𝐹𝑣)))
6 fveq2 6352 . . 3 (𝑣 = (𝐺𝑤) → (𝐹𝑣) = (𝐹‘(𝐺𝑤)))
72, 3, 5, 6fmptco 6559 . 2 (𝜑 → (𝐹𝐺) = (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))))
8 rlimcn1.5 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
9 fvexd 6364 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ V)
109ralrimiva 3104 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ∀𝑤𝐴 (𝐺𝑤) ∈ V)
11 simpr 479 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
12 rlimcn1.3 . . . . . . . . . 10 (𝜑𝐺𝑟 𝐶)
133, 12eqbrtrrd 4828 . . . . . . . . 9 (𝜑 → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1413ad2antrr 764 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1510, 11, 14rlimi 14443 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
16 oveq1 6820 . . . . . . . . . . . . . . 15 (𝑧 = (𝐺𝑤) → (𝑧𝐶) = ((𝐺𝑤) − 𝐶))
1716fveq2d 6356 . . . . . . . . . . . . . 14 (𝑧 = (𝐺𝑤) → (abs‘(𝑧𝐶)) = (abs‘((𝐺𝑤) − 𝐶)))
1817breq1d 4814 . . . . . . . . . . . . 13 (𝑧 = (𝐺𝑤) → ((abs‘(𝑧𝐶)) < 𝑦 ↔ (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
19 fveq2 6352 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑤) → (𝐹𝑧) = (𝐹‘(𝐺𝑤)))
2019oveq1d 6828 . . . . . . . . . . . . . . 15 (𝑧 = (𝐺𝑤) → ((𝐹𝑧) − (𝐹𝐶)) = ((𝐹‘(𝐺𝑤)) − (𝐹𝐶)))
2120fveq2d 6356 . . . . . . . . . . . . . 14 (𝑧 = (𝐺𝑤) → (abs‘((𝐹𝑧) − (𝐹𝐶))) = (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))))
2221breq1d 4814 . . . . . . . . . . . . 13 (𝑧 = (𝐺𝑤) → ((abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥 ↔ (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
2318, 22imbi12d 333 . . . . . . . . . . . 12 (𝑧 = (𝐺𝑤) → (((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) ↔ ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
24 simplrr 820 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
25 simpll 807 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) → 𝜑)
2625, 2sylan 489 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
2723, 24, 26rspcdva 3455 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
2827imim2d 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)))
2928ralimdva 3100 . . . . . . . . 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)))
3029reximdv 3154 . . . . . . . 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)))
3130expr 644 . . . . . . 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))))
3215, 31mpid 44 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
3332rexlimdva 3169 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → (∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
348, 33mpd 15 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
3534ralrimiva 3104 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
364ffvelrnda 6522 . . . . . 6 ((𝜑 ∧ (𝐺𝑤) ∈ 𝑋) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
372, 36syldan 488 . . . . 5 ((𝜑𝑤𝐴) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
3837ralrimiva 3104 . . . 4 (𝜑 → ∀𝑤𝐴 (𝐹‘(𝐺𝑤)) ∈ ℂ)
39 fdm 6212 . . . . . 6 (𝐺:𝐴𝑋 → dom 𝐺 = 𝐴)
401, 39syl 17 . . . . 5 (𝜑 → dom 𝐺 = 𝐴)
41 rlimss 14432 . . . . . 6 (𝐺𝑟 𝐶 → dom 𝐺 ⊆ ℝ)
4212, 41syl 17 . . . . 5 (𝜑 → dom 𝐺 ⊆ ℝ)
4340, 42eqsstr3d 3781 . . . 4 (𝜑𝐴 ⊆ ℝ)
44 rlimcn1.2 . . . . 5 (𝜑𝐶𝑋)
454, 44ffvelrnd 6523 . . . 4 (𝜑 → (𝐹𝐶) ∈ ℂ)
4638, 43, 45rlim2 14426 . . 3 (𝜑 → ((𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶) ↔ ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
4735, 46mpbird 247 . 2 (𝜑 → (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶))
487, 47eqbrtrd 4826 1 (𝜑 → (𝐹𝐺) ⇝𝑟 (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  wss 3715   class class class wbr 4804  cmpt 4881  dom cdm 5266  ccom 5270  wf 6045  cfv 6049  (class class class)co 6813  cc 10126  cr 10127   < clt 10266  cle 10267  cmin 10458  +crp 12025  abscabs 14173  𝑟 crli 14415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-pm 8026  df-rlim 14419
This theorem is referenced by:  rlimcn1b  14519  rlimdiv  14575
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