Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 23428 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
7 | 6 | ffvelrnda 6843 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
8 | cncfrss2 23427 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
9 | 4, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
10 | 9 | sselda 3964 | . . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑧) ∈ 𝐵) → (𝐹‘𝑧) ∈ ℂ) |
11 | 7, 10 | syldan 591 | . 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 6677 | . . . 4 ⊢ 𝑍 ∈ V |
15 | fex 6980 | . . . 4 ⊢ ((𝐺:𝑍⟶𝐴 ∧ 𝑍 ∈ V) → 𝐺 ∈ V) | |
16 | 13, 14, 15 | sylancl 586 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
17 | coexg 7623 | . . 3 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V) | |
18 | 4, 16, 17 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
19 | cncfi 23429 | . . . . 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 1113 | . . . 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 584 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ+ → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥))) |
22 | 21 | imp 407 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥)) |
23 | 13 | ffvelrnda 6843 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐴) |
24 | fvco3 6753 | . . 3 ⊢ ((𝐺:𝑍⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘))) | |
25 | 13, 24 | sylan 580 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘))) |
26 | 1, 2, 3, 11, 12, 18, 22, 23, 25 | climcn1 14936 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 Vcvv 3492 ⊆ wss 3933 class class class wbr 5057 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 < clt 10663 − cmin 10858 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 abscabs 14581 ⇝ cli 14829 –cn→ccncf 23411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-2 11688 df-z 11970 df-uz 12232 df-cj 14446 df-re 14447 df-im 14448 df-abs 14583 df-clim 14833 df-cncf 23413 |
This theorem is referenced by: leibpi 25447 lgamcvg2 25559 gamcvg 25560 iprodefisum 32870 climexp 41762 fprodsubrecnncnvlem 42067 fprodaddrecnncnvlem 42069 stirlinglem14 42249 |
Copyright terms: Public domain | W3C validator |