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Theorem rlimcn1 14261
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 6320 . . 3 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
31feqmptd 6211 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
4 rlimcn1.4 . . . 4 (𝜑𝐹:𝑋⟶ℂ)
54feqmptd 6211 . . 3 (𝜑𝐹 = (𝑣𝑋 ↦ (𝐹𝑣)))
6 fveq2 6153 . . 3 (𝑣 = (𝐺𝑤) → (𝐹𝑣) = (𝐹‘(𝐺𝑤)))
72, 3, 5, 6fmptco 6357 . 2 (𝜑 → (𝐹𝐺) = (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))))
8 rlimcn1.5 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
9 fvex 6163 . . . . . . . . . 10 (𝐺𝑤) ∈ V
109a1i 11 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ V)
1110ralrimiva 2961 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ∀𝑤𝐴 (𝐺𝑤) ∈ V)
12 simpr 477 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
13 rlimcn1.3 . . . . . . . . . 10 (𝜑𝐺𝑟 𝐶)
143, 13eqbrtrrd 4642 . . . . . . . . 9 (𝜑 → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1514ad2antrr 761 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (𝑤𝐴 ↦ (𝐺𝑤)) ⇝𝑟 𝐶)
1611, 12, 15rlimi 14186 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
17 simpll 789 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) → 𝜑)
1817, 2sylan 488 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → (𝐺𝑤) ∈ 𝑋)
19 simplrr 800 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
20 oveq1 6617 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑤) → (𝑧𝐶) = ((𝐺𝑤) − 𝐶))
2120fveq2d 6157 . . . . . . . . . . . . . . 15 (𝑧 = (𝐺𝑤) → (abs‘(𝑧𝐶)) = (abs‘((𝐺𝑤) − 𝐶)))
2221breq1d 4628 . . . . . . . . . . . . . 14 (𝑧 = (𝐺𝑤) → ((abs‘(𝑧𝐶)) < 𝑦 ↔ (abs‘((𝐺𝑤) − 𝐶)) < 𝑦))
23 fveq2 6153 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑤) → (𝐹𝑧) = (𝐹‘(𝐺𝑤)))
2423oveq1d 6625 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑤) → ((𝐹𝑧) − (𝐹𝐶)) = ((𝐹‘(𝐺𝑤)) − (𝐹𝐶)))
2524fveq2d 6157 . . . . . . . . . . . . . . 15 (𝑧 = (𝐺𝑤) → (abs‘((𝐹𝑧) − (𝐹𝐶))) = (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))))
2625breq1d 4628 . . . . . . . . . . . . . 14 (𝑧 = (𝐺𝑤) → ((abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥 ↔ (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
2722, 26imbi12d 334 . . . . . . . . . . . . 13 (𝑧 = (𝐺𝑤) → (((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) ↔ ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
2827rspcv 3294 . . . . . . . . . . . 12 ((𝐺𝑤) ∈ 𝑋 → (∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
2918, 19, 28sylc 65 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑦 ∈ ℝ+ ∧ ∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))) ∧ 𝑤𝐴) → ((abs‘((𝐺𝑤) − 𝐶)) < 𝑦 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
3029imim2d 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)))
3130ralimdva 2957 . . . . . . . . 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 3011 . . . . . . . 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 642 . . . . . . 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 44 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) → (∀𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥) → ∃𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
3534rexlimdva 3025 . . . . 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 2961 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥))
384ffvelrnda 6320 . . . . . 6 ((𝜑 ∧ (𝐺𝑤) ∈ 𝑋) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
392, 38syldan 487 . . . . 5 ((𝜑𝑤𝐴) → (𝐹‘(𝐺𝑤)) ∈ ℂ)
4039ralrimiva 2961 . . . 4 (𝜑 → ∀𝑤𝐴 (𝐹‘(𝐺𝑤)) ∈ ℂ)
41 fdm 6013 . . . . . 6 (𝐺:𝐴𝑋 → dom 𝐺 = 𝐴)
421, 41syl 17 . . . . 5 (𝜑 → dom 𝐺 = 𝐴)
43 rlimss 14175 . . . . . 6 (𝐺𝑟 𝐶 → dom 𝐺 ⊆ ℝ)
4413, 43syl 17 . . . . 5 (𝜑 → dom 𝐺 ⊆ ℝ)
4542, 44eqsstr3d 3624 . . . 4 (𝜑𝐴 ⊆ ℝ)
46 rlimcn1.2 . . . . 5 (𝜑𝐶𝑋)
474, 46ffvelrnd 6321 . . . 4 (𝜑 → (𝐹𝐶) ∈ ℂ)
4840, 45, 47rlim2 14169 . . 3 (𝜑 → ((𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶) ↔ ∀𝑥 ∈ ℝ+𝑐 ∈ ℝ ∀𝑤𝐴 (𝑐𝑤 → (abs‘((𝐹‘(𝐺𝑤)) − (𝐹𝐶))) < 𝑥)))
4937, 48mpbird 247 . 2 (𝜑 → (𝑤𝐴 ↦ (𝐹‘(𝐺𝑤))) ⇝𝑟 (𝐹𝐶))
507, 49eqbrtrd 4640 1 (𝜑 → (𝐹𝐺) ⇝𝑟 (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  wss 3559   class class class wbr 4618  cmpt 4678  dom cdm 5079  ccom 5083  wf 5848  cfv 5852  (class class class)co 6610  cc 9886  cr 9887   < clt 10026  cle 10027  cmin 10218  +crp 11784  abscabs 13916  𝑟 crli 14158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-pm 7812  df-rlim 14162
This theorem is referenced by:  rlimcn1b  14262  rlimdiv  14318
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