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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 24787 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
7 | 6 | ffvelcdmda 7088 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
8 | cncfrss2 24786 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
9 | 4, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
10 | 9 | sselda 3978 | . . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑧) ∈ 𝐵) → (𝐹‘𝑧) ∈ ℂ) |
11 | 7, 10 | syldan 590 | . 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 6905 | . . . 4 ⊢ 𝑍 ∈ V |
15 | fex 7232 | . . . 4 ⊢ ((𝐺:𝑍⟶𝐴 ∧ 𝑍 ∈ V) → 𝐺 ∈ V) | |
16 | 13, 14, 15 | sylancl 585 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
17 | coexg 7929 | . . 3 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V) | |
18 | 4, 16, 17 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
19 | cncfi 24788 | . . . . 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 1119 | . . . 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 583 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ+ → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥))) |
22 | 21 | imp 406 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥)) |
23 | 13 | ffvelcdmda 7088 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐴) |
24 | fvco3 6991 | . . 3 ⊢ ((𝐺:𝑍⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘))) | |
25 | 13, 24 | sylan 579 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘))) |
26 | 1, 2, 3, 11, 12, 18, 22, 23, 25 | climcn1 15554 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 Vcvv 3469 ⊆ wss 3944 class class class wbr 5142 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℂcc 11122 < clt 11264 − cmin 11460 ℤcz 12574 ℤ≥cuz 12838 ℝ+crp 12992 abscabs 15199 ⇝ cli 15446 –cn→ccncf 24770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-2 12291 df-z 12575 df-uz 12839 df-cj 15064 df-re 15065 df-im 15066 df-abs 15201 df-clim 15450 df-cncf 24772 |
This theorem is referenced by: leibpi 26848 lgamcvg2 26961 gamcvg 26962 iprodefisum 35258 climexp 44906 fprodsubrecnncnvlem 45208 fprodaddrecnncnvlem 45210 stirlinglem14 45388 |
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