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Mirrors > Home > MPE Home > Th. List > climmpt2 | Structured version Visualization version GIF version |
Description: Relate an integer limit on a not-quite-function to a real limit. (Contributed by Mario Carneiro, 17-Sep-2014.) |
Ref | Expression |
---|---|
climmpt2.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climmpt2.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climmpt2.3 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
climmpt2.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
Ref | Expression |
---|---|
climmpt2 | ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climmpt2.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | climmpt2.3 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
3 | climmpt2.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | eqid 2737 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) | |
5 | 3, 4 | climmpt 15412 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐴)) |
6 | 1, 2, 5 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐴)) |
7 | climmpt2.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
8 | 7 | ralrimiva 3141 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) |
9 | fveq2 6839 | . . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
10 | 9 | eleq1d 2822 | . . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ)) |
11 | 10 | cbvralvw 3223 | . . . . . . 7 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ∈ ℂ) |
12 | fveq2 6839 | . . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛)) | |
13 | 12 | eleq1d 2822 | . . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ)) |
14 | 13 | cbvralvw 3223 | . . . . . . 7 ⊢ (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ∈ ℂ ↔ ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ) |
15 | 11, 14 | bitri 274 | . . . . . 6 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ) |
16 | 8, 15 | sylib 217 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ) |
17 | 16 | r19.21bi 3232 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ) |
18 | 17 | fmpttd 7059 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)):𝑍⟶ℂ) |
19 | 3, 1, 18 | rlimclim 15387 | . 2 ⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 𝐴 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝ 𝐴)) |
20 | 6, 19 | bitr4d 281 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6493 ℂcc 11007 ℤcz 12457 ℤ≥cuz 12721 ⇝ cli 15325 ⇝𝑟 crli 15326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-pm 8726 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-fl 13651 df-clim 15329 df-rlim 15330 |
This theorem is referenced by: (None) |
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