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Mirrors > Home > MPE Home > Th. List > climcncf | Structured version Visualization version GIF version |
Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.) |
Ref | Expression |
---|---|
climcncf.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climcncf.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climcncf.4 | ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) |
climcncf.5 | ⊢ (𝜑 → 𝐺:𝑍⟶𝐴) |
climcncf.6 | ⊢ (𝜑 → 𝐺 ⇝ 𝐷) |
climcncf.7 | ⊢ (𝜑 → 𝐷 ∈ 𝐴) |
Ref | Expression |
---|---|
climcncf | ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climcncf.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climcncf.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climcncf.7 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝐴) | |
4 | climcncf.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) | |
5 | cncff 24279 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
7 | 6 | ffvelcdmda 7039 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
8 | cncfrss2 24278 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
9 | 4, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
10 | 9 | sselda 3948 | . . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑧) ∈ 𝐵) → (𝐹‘𝑧) ∈ ℂ) |
11 | 7, 10 | syldan 592 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ) |
12 | climcncf.6 | . 2 ⊢ (𝜑 → 𝐺 ⇝ 𝐷) | |
13 | climcncf.5 | . . . 4 ⊢ (𝜑 → 𝐺:𝑍⟶𝐴) | |
14 | 1 | fvexi 6860 | . . . 4 ⊢ 𝑍 ∈ V |
15 | fex 7180 | . . . 4 ⊢ ((𝐺:𝑍⟶𝐴 ∧ 𝑍 ∈ V) → 𝐺 ∈ V) | |
16 | 13, 14, 15 | sylancl 587 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
17 | coexg 7870 | . . 3 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V) | |
18 | 4, 16, 17 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
19 | cncfi 24280 | . . . . 5 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐷 ∈ 𝐴 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥)) | |
20 | 19 | 3expia 1122 | . . . 4 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑥 ∈ ℝ+ → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥))) |
21 | 4, 3, 20 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ+ → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥))) |
22 | 21 | imp 408 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥)) |
23 | 13 | ffvelcdmda 7039 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐴) |
24 | fvco3 6944 | . . 3 ⊢ ((𝐺:𝑍⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘))) | |
25 | 13, 24 | sylan 581 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘))) |
26 | 1, 2, 3, 11, 12, 18, 22, 23, 25 | climcn1 15483 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 Vcvv 3447 ⊆ wss 3914 class class class wbr 5109 ∘ ccom 5641 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 ℂcc 11057 < clt 11197 − cmin 11393 ℤcz 12507 ℤ≥cuz 12771 ℝ+crp 12923 abscabs 15128 ⇝ cli 15375 –cn→ccncf 24262 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-2 12224 df-z 12508 df-uz 12772 df-cj 14993 df-re 14994 df-im 14995 df-abs 15130 df-clim 15379 df-cncf 24264 |
This theorem is referenced by: leibpi 26315 lgamcvg2 26427 gamcvg 26428 iprodefisum 34377 climexp 43936 fprodsubrecnncnvlem 44238 fprodaddrecnncnvlem 44240 stirlinglem14 44418 |
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