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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnlimf | Structured version Visualization version GIF version |
Description: The limit function of real functions, is a real-valued function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
fnlimf.p | ⊢ Ⅎ𝑚𝜑 |
fnlimf.m | ⊢ Ⅎ𝑚𝐹 |
fnlimf.n | ⊢ Ⅎ𝑥𝐹 |
fnlimf.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
fnlimf.f | ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) |
fnlimf.d | ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } |
fnlimf.g | ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) |
Ref | Expression |
---|---|
fnlimf | ⊢ (𝜑 → 𝐺:𝐷⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnlimf.p | . . . 4 ⊢ Ⅎ𝑚𝜑 | |
2 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑚 𝑧 ∈ 𝐷 | |
3 | 1, 2 | nfan 1902 | . . 3 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑧 ∈ 𝐷) |
4 | fnlimf.m | . . 3 ⊢ Ⅎ𝑚𝐹 | |
5 | fnlimf.n | . . 3 ⊢ Ⅎ𝑥𝐹 | |
6 | fnlimf.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
7 | fnlimf.f | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) | |
8 | 7 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) |
9 | fnlimf.d | . . 3 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } | |
10 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ 𝐷) | |
11 | 3, 4, 5, 6, 8, 9, 10 | fnlimfvre 44376 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) ∈ ℝ) |
12 | fnlimf.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) | |
13 | nfrab1 3451 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } | |
14 | 9, 13 | nfcxfr 2901 | . . . 4 ⊢ Ⅎ𝑥𝐷 |
15 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑧𝐷 | |
16 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑧( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) | |
17 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥 ⇝ | |
18 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑥𝑍 | |
19 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑚 | |
20 | 5, 19 | nffv 6898 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚) |
21 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
22 | 20, 21 | nffv 6898 | . . . . . 6 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑧) |
23 | 18, 22 | nfmpt 5254 | . . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) |
24 | 17, 23 | nffv 6898 | . . . 4 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
25 | fveq2 6888 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧)) | |
26 | 25 | mpteq2dv 5249 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
27 | 26 | fveq2d 6892 | . . . 4 ⊢ (𝑥 = 𝑧 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
28 | 14, 15, 16, 24, 27 | cbvmptf 5256 | . . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
29 | 12, 28 | eqtri 2760 | . 2 ⊢ 𝐺 = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
30 | 11, 29 | fmptd 7110 | 1 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 {crab 3432 ∪ ciun 4996 ∩ ciin 4997 ↦ cmpt 5230 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 ℝcr 11105 ℤ≥cuz 12818 ⇝ cli 15424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fl 13753 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 |
This theorem is referenced by: (None) |
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