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 | fvex 6113 | . . . 4 ⊢ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V) |
4 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑧𝑋 | |
5 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑧( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) | |
6 | fveq2 6103 | . . . . . . 7 ⊢ (𝑋 = 𝑧 → ((𝐹‘𝑚)‘𝑋) = ((𝐹‘𝑚)‘𝑧)) | |
7 | 6 | mpteq2dv 4673 | . . . . . 6 ⊢ (𝑋 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
8 | eqcom 2617 | . . . . . . . 8 ⊢ (𝑋 = 𝑧 ↔ 𝑧 = 𝑋) | |
9 | 8 | 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)))) |
10 | eqcom 2617 | . . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) | |
11 | 10 | 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)))) |
12 | 9, 11 | bitri 263 | . . . . . 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)))) |
13 | 7, 12 | mpbi 219 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) |
14 | 13 | fveq2d 6107 | . . . 4 ⊢ (𝑧 = 𝑋 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) |
15 | fnlimfvre2.g | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) | |
16 | fnlimfvre2.d | . . . . . . 7 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } | |
17 | nfrab1 3099 | . . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } | |
18 | 16, 17 | nfcxfr 2749 | . . . . . 6 ⊢ Ⅎ𝑥𝐷 |
19 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑧𝐷 | |
20 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑧( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) | |
21 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥 ⇝ | |
22 | nfcv 2751 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑍 | |
23 | fnlimfvre2.n | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹 | |
24 | nfcv 2751 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑚 | |
25 | 23, 24 | nffv 6110 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑚) |
26 | nfcv 2751 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑧 | |
27 | 25, 26 | nffv 6110 | . . . . . . . 8 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑧) |
28 | 22, 27 | nfmpt 4674 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) |
29 | 21, 28 | nffv 6110 | . . . . . 6 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
30 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧)) | |
31 | 30 | mpteq2dv 4673 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧))) |
32 | 31 | fveq2d 6107 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
33 | 18, 19, 20, 29, 32 | cbvmptf 4676 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
34 | 15, 33 | eqtri 2632 | . . . 4 ⊢ 𝐺 = (𝑧 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))) |
35 | 4, 5, 14, 34 | fvmptf 6209 | . . 3 ⊢ ((𝑋 ∈ 𝐷 ∧ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V) → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) |
36 | 1, 3, 35 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) |
37 | fnlimfvre2.p | . . 3 ⊢ Ⅎ𝑚𝜑 | |
38 | fnlimfvre2.m | . . 3 ⊢ Ⅎ𝑚𝐹 | |
39 | fnlimfvre2.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
40 | fnlimfvre2.f | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) | |
41 | 37, 38, 23, 39, 40, 16, 1 | fnlimfvre 38741 | . 2 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) |
42 | 36, 41 | eqeltrd 2688 | 1 ⊢ (𝜑 → (𝐺‘𝑋) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 {crab 2900 Vcvv 3173 ∪ ciun 4455 ∩ ciin 4456 ↦ cmpt 4643 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 ℝcr 9814 ℤ≥cuz 11563 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 |
This theorem is referenced by: (None) |
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