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 1915 | . . . 4 ⊢ Ⅎ𝑚 𝑧 ∈ 𝐷 | |
3 | 1, 2 | nfan 1900 | . . 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 487 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ 𝐷) | |
11 | 3, 4, 5, 6, 8, 9, 10 | fnlimfvre 41975 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) ∈ ℝ) |
12 | fnlimf.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) | |
13 | nfrab1 3384 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } | |
14 | 9, 13 | nfcxfr 2975 | . . . 4 ⊢ Ⅎ𝑥𝐷 |
15 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑧𝐷 | |
16 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑧( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) | |
17 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥 ⇝ | |
18 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑥𝑍 | |
19 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑚 | |
20 | 5, 19 | nffv 6680 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚) |
21 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
22 | 20, 21 | nffv 6680 | . . . . . 6 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑧) |
23 | 18, 22 | nfmpt 5163 | . . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) |
24 | 17, 23 | nffv 6680 | . . . 4 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
25 | fveq2 6670 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧)) | |
26 | 25 | mpteq2dv 5162 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
27 | 26 | fveq2d 6674 | . . . 4 ⊢ (𝑥 = 𝑧 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
28 | 14, 15, 16, 24, 27 | cbvmptf 5165 | . . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
29 | 12, 28 | eqtri 2844 | . 2 ⊢ 𝐺 = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
30 | 11, 29 | fmptd 6878 | 1 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 {crab 3142 ∪ ciun 4919 ∩ ciin 4920 ↦ cmpt 5146 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 ℝcr 10536 ℤ≥cuz 12244 ⇝ cli 14841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fl 13163 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 |
This theorem is referenced by: (None) |
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