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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 22504 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
7 | 6 | ffvelrnda 6267 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
8 | cncfrss2 22503 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
9 | 4, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
10 | 9 | sselda 3568 | . . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑧) ∈ 𝐵) → (𝐹‘𝑧) ∈ ℂ) |
11 | 7, 10 | syldan 486 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ) |
12 | climcncf.6 | . 2 ⊢ (𝜑 → 𝐺 ⇝ 𝐷) | |
13 | climcncf.5 | . . . 4 ⊢ (𝜑 → 𝐺:𝑍⟶𝐴) | |
14 | fvex 6113 | . . . . 5 ⊢ (ℤ≥‘𝑀) ∈ V | |
15 | 1, 14 | eqeltri 2684 | . . . 4 ⊢ 𝑍 ∈ V |
16 | fex 6394 | . . . 4 ⊢ ((𝐺:𝑍⟶𝐴 ∧ 𝑍 ∈ V) → 𝐺 ∈ V) | |
17 | 13, 15, 16 | sylancl 693 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
18 | coexg 7010 | . . 3 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V) | |
19 | 4, 17, 18 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
20 | cncfi 22505 | . . . . 5 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐷 ∈ 𝐴 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥)) | |
21 | 20 | 3expia 1259 | . . . 4 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐷 ∈ 𝐴) → (𝑥 ∈ ℝ+ → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥))) |
22 | 4, 3, 21 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ+ → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥))) |
23 | 22 | imp 444 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐴 ((abs‘(𝑧 − 𝐷)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐷))) < 𝑥)) |
24 | 13 | ffvelrnda 6267 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐴) |
25 | fvco3 6185 | . . 3 ⊢ ((𝐺:𝑍⟶𝐴 ∧ 𝑘 ∈ 𝑍) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘))) | |
26 | 13, 25 | sylan 487 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘))) |
27 | 1, 2, 3, 11, 12, 19, 23, 24, 26 | climcn1 14170 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 < clt 9953 − cmin 10145 ℤcz 11254 ℤ≥cuz 11563 ℝ+crp 11708 abscabs 13822 ⇝ cli 14063 –cn→ccncf 22487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-z 11255 df-uz 11564 df-cj 13687 df-re 13688 df-im 13689 df-abs 13824 df-clim 14067 df-cncf 22489 |
This theorem is referenced by: leibpi 24469 lgamcvg2 24581 gamcvg 24582 iprodefisum 30880 climexp 38672 fprodsubrecnncnvlem 38794 fprodaddrecnncnvlem 38796 stirlinglem14 38980 |
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