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 6203 | . . 3 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V) | |
3 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑧𝑋 | |
4 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑧( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) | |
5 | fveq2 6191 | . . . . . . 7 ⊢ (𝑋 = 𝑧 → ((𝐹‘𝑚)‘𝑋) = ((𝐹‘𝑚)‘𝑧)) | |
6 | 5 | mpteq2dv 4745 | . . . . . 6 ⊢ (𝑋 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
7 | eqcom 2629 | . . . . . . . 8 ⊢ (𝑋 = 𝑧 ↔ 𝑧 = 𝑋) | |
8 | 7 | imbi1i 339 | . . . . . . 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 2629 | . . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) | |
10 | 9 | imbi2i 326 | . . . . . . 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 6195 | . . . 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 3122 | . . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } | |
17 | 15, 16 | nfcxfr 2762 | . . . . . 6 ⊢ Ⅎ𝑥𝐷 |
18 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑧𝐷 | |
19 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑧( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) | |
20 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑥 ⇝ | |
21 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑍 | |
22 | fnlimfvre2.n | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹 | |
23 | nfcv 2764 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑚 | |
24 | 22, 23 | nffv 6198 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑚) |
25 | nfcv 2764 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑧 | |
26 | 24, 25 | nffv 6198 | . . . . . . . 8 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑧) |
27 | 21, 26 | nfmpt 4746 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) |
28 | 20, 27 | nffv 6198 | . . . . . 6 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
29 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧)) | |
30 | 29 | mpteq2dv 4745 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
31 | 30 | fveq2d 6195 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
32 | 17, 18, 19, 28, 31 | cbvmptf 4748 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
33 | 14, 32 | eqtri 2644 | . . . 4 ⊢ 𝐺 = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
34 | 3, 4, 13, 33 | fvmptf 6301 | . . 3 ⊢ ((𝑋 ∈ 𝐷 ∧ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V) → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) |
35 | 1, 2, 34 | syl2anc 693 | . 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 39906 | . 2 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) |
41 | 35, 40 | eqeltrd 2701 | 1 ⊢ (𝜑 → (𝐺‘𝑋) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 {crab 2916 Vcvv 3200 ∪ ciun 4520 ∩ ciin 4521 ↦ cmpt 4729 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 ℝcr 9935 ℤ≥cuz 11687 ⇝ cli 14215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 |
This theorem is referenced by: (None) |
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