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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 1914 | . . . 4 ⊢ Ⅎ𝑚 𝑧 ∈ 𝐷 | |
| 3 | 1, 2 | nfan 1899 | . . 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 715 | . . 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 484 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ 𝐷) | |
| 11 | 3, 4, 5, 6, 8, 9, 10 | fnlimfvre 45689 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) ∈ ℝ) | 
| 12 | fnlimf.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) | |
| 13 | nfrab1 3457 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } | |
| 14 | 9, 13 | nfcxfr 2903 | . . . 4 ⊢ Ⅎ𝑥𝐷 | 
| 15 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑧𝐷 | |
| 16 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑧( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) | |
| 17 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥 ⇝ | |
| 18 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥𝑍 | |
| 19 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑚 | |
| 20 | 5, 19 | nffv 6916 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚) | 
| 21 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
| 22 | 20, 21 | nffv 6916 | . . . . . 6 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑧) | 
| 23 | 18, 22 | nfmpt 5249 | . . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) | 
| 24 | 17, 23 | nffv 6916 | . . . 4 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) | 
| 25 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧)) | |
| 26 | 25 | mpteq2dv 5244 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) | 
| 27 | 26 | fveq2d 6910 | . . . 4 ⊢ (𝑥 = 𝑧 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) | 
| 28 | 14, 15, 16, 24, 27 | cbvmptf 5251 | . . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) | 
| 29 | 12, 28 | eqtri 2765 | . 2 ⊢ 𝐺 = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) | 
| 30 | 11, 29 | fmptd 7134 | 1 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 {crab 3436 ∪ ciun 4991 ∩ ciin 4992 ↦ cmpt 5225 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 ℝcr 11154 ℤ≥cuz 12878 ⇝ cli 15520 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fl 13832 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 | 
| This theorem is referenced by: (None) | 
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