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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnlimfvre2 | Structured version Visualization version GIF version |
Description: The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
fnlimfvre2.p | ⊢ Ⅎ𝑚𝜑 |
fnlimfvre2.m | ⊢ Ⅎ𝑚𝐹 |
fnlimfvre2.n | ⊢ Ⅎ𝑥𝐹 |
fnlimfvre2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
fnlimfvre2.f | ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) |
fnlimfvre2.d | ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } |
fnlimfvre2.g | ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) |
fnlimfvre2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
fnlimfvre2 | ⊢ (𝜑 → (𝐺‘𝑋) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnlimfvre2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
2 | fvexd 6346 | . . 3 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V) | |
3 | nfcv 2913 | . . . 4 ⊢ Ⅎ𝑧𝑋 | |
4 | nfcv 2913 | . . . 4 ⊢ Ⅎ𝑧( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) | |
5 | fveq2 6333 | . . . . . . 7 ⊢ (𝑋 = 𝑧 → ((𝐹‘𝑚)‘𝑋) = ((𝐹‘𝑚)‘𝑧)) | |
6 | 5 | mpteq2dv 4880 | . . . . . 6 ⊢ (𝑋 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
7 | eqcom 2778 | . . . . . . . 8 ⊢ (𝑋 = 𝑧 ↔ 𝑧 = 𝑋) | |
8 | 7 | imbi1i 338 | . . . . . . 7 ⊢ ((𝑋 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) ↔ (𝑧 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
9 | eqcom 2778 | . . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) | |
10 | 9 | imbi2i 325 | . . . . . . 7 ⊢ ((𝑧 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) ↔ (𝑧 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) |
11 | 8, 10 | bitri 264 | . . . . . 6 ⊢ ((𝑋 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) ↔ (𝑧 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) |
12 | 6, 11 | mpbi 220 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) |
13 | 12 | fveq2d 6337 | . . . 4 ⊢ (𝑧 = 𝑋 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) |
14 | fnlimfvre2.g | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) | |
15 | fnlimfvre2.d | . . . . . . 7 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } | |
16 | nfrab1 3271 | . . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } | |
17 | 15, 16 | nfcxfr 2911 | . . . . . 6 ⊢ Ⅎ𝑥𝐷 |
18 | nfcv 2913 | . . . . . 6 ⊢ Ⅎ𝑧𝐷 | |
19 | nfcv 2913 | . . . . . 6 ⊢ Ⅎ𝑧( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) | |
20 | nfcv 2913 | . . . . . . 7 ⊢ Ⅎ𝑥 ⇝ | |
21 | nfcv 2913 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑍 | |
22 | fnlimfvre2.n | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹 | |
23 | nfcv 2913 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑚 | |
24 | 22, 23 | nffv 6341 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑚) |
25 | nfcv 2913 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑧 | |
26 | 24, 25 | nffv 6341 | . . . . . . . 8 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑧) |
27 | 21, 26 | nfmpt 4881 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) |
28 | 20, 27 | nffv 6341 | . . . . . 6 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
29 | fveq2 6333 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧)) | |
30 | 29 | mpteq2dv 4880 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
31 | 30 | fveq2d 6337 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
32 | 17, 18, 19, 28, 31 | cbvmptf 4883 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
33 | 14, 32 | eqtri 2793 | . . . 4 ⊢ 𝐺 = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
34 | 3, 4, 13, 33 | fvmptf 6445 | . . 3 ⊢ ((𝑋 ∈ 𝐷 ∧ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V) → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) |
35 | 1, 2, 34 | syl2anc 573 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) |
36 | fnlimfvre2.p | . . 3 ⊢ Ⅎ𝑚𝜑 | |
37 | fnlimfvre2.m | . . 3 ⊢ Ⅎ𝑚𝐹 | |
38 | fnlimfvre2.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
39 | fnlimfvre2.f | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) | |
40 | 36, 37, 22, 38, 39, 15, 1 | fnlimfvre 40419 | . 2 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) |
41 | 35, 40 | eqeltrd 2850 | 1 ⊢ (𝜑 → (𝐺‘𝑋) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 Ⅎwnf 1856 ∈ wcel 2145 Ⅎwnfc 2900 {crab 3065 Vcvv 3351 ∪ ciun 4655 ∩ ciin 4656 ↦ cmpt 4864 dom cdm 5250 ⟶wf 6026 ‘cfv 6030 ℝcr 10141 ℤ≥cuz 11893 ⇝ cli 14423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-pm 8016 df-en 8114 df-dom 8115 df-sdom 8116 df-sup 8508 df-inf 8509 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-n0 11500 df-z 11585 df-uz 11894 df-rp 12036 df-fl 12801 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-rlim 14428 |
This theorem is referenced by: (None) |
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