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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 24933 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
7 | 6 | ffvelcdmda 7104 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
8 | cncfrss2 24932 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
9 | 4, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
10 | 9 | sselda 3995 | . . 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 6921 | . . . 4 ⊢ 𝑍 ∈ V |
15 | fex 7246 | . . . 4 ⊢ ((𝐺:𝑍⟶𝐴 ∧ 𝑍 ∈ V) → 𝐺 ∈ V) | |
16 | 13, 14, 15 | sylancl 586 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
17 | coexg 7952 | . . 3 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V) | |
18 | 4, 16, 17 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
19 | cncfi 24934 | . . . . 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 1120 | . . . 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 406 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥)) |
23 | 13 | ffvelcdmda 7104 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐴) |
24 | fvco3 7008 | . . 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 15625 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 < clt 11293 − cmin 11490 ℤcz 12611 ℤ≥cuz 12876 ℝ+crp 13032 abscabs 15270 ⇝ cli 15517 –cn→ccncf 24916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-z 12612 df-uz 12877 df-cj 15135 df-re 15136 df-im 15137 df-abs 15272 df-clim 15521 df-cncf 24918 |
This theorem is referenced by: leibpi 27000 lgamcvg2 27113 gamcvg 27114 iprodefisum 35721 climexp 45561 fprodsubrecnncnvlem 45863 fprodaddrecnncnvlem 45865 stirlinglem14 46043 |
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