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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climresdm | Structured version Visualization version GIF version |
Description: A real function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
climresdm.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climresdm.2 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
Ref | Expression |
---|---|
climresdm | ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resexg 6027 | . . . 4 ⊢ (𝐹 ∈ dom ⇝ → (𝐹 ↾ (ℤ≥‘𝑀)) ∈ V) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐹 ↾ (ℤ≥‘𝑀)) ∈ V) |
3 | fvexd 6906 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘𝐹) ∈ V) | |
4 | climdm 15505 | . . . . . 6 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹)) | |
5 | 4 | biimpi 215 | . . . . 5 ⊢ (𝐹 ∈ dom ⇝ → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
6 | 5 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
7 | climresdm.1 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
8 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
9 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) | |
10 | 8, 9 | climresd 44876 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ((𝐹 ↾ (ℤ≥‘𝑀)) ⇝ ( ⇝ ‘𝐹) ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))) |
11 | 6, 10 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐹 ↾ (ℤ≥‘𝑀)) ⇝ ( ⇝ ‘𝐹)) |
12 | 2, 3, 11 | breldmd 5912 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ) |
13 | climresdm.2 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
14 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ) → 𝐹 ∈ 𝑉) |
15 | fvexd 6906 | . . 3 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ) → ( ⇝ ‘(𝐹 ↾ (ℤ≥‘𝑀))) ∈ V) | |
16 | climdm 15505 | . . . . . 6 ⊢ ((𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ↔ (𝐹 ↾ (ℤ≥‘𝑀)) ⇝ ( ⇝ ‘(𝐹 ↾ (ℤ≥‘𝑀)))) | |
17 | 16 | biimpi 215 | . . . . 5 ⊢ ((𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ → (𝐹 ↾ (ℤ≥‘𝑀)) ⇝ ( ⇝ ‘(𝐹 ↾ (ℤ≥‘𝑀)))) |
18 | 17 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ) → (𝐹 ↾ (ℤ≥‘𝑀)) ⇝ ( ⇝ ‘(𝐹 ↾ (ℤ≥‘𝑀)))) |
19 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
20 | 19, 14 | climresd 44876 | . . . 4 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ) → ((𝐹 ↾ (ℤ≥‘𝑀)) ⇝ ( ⇝ ‘(𝐹 ↾ (ℤ≥‘𝑀))) ↔ 𝐹 ⇝ ( ⇝ ‘(𝐹 ↾ (ℤ≥‘𝑀))))) |
21 | 18, 20 | mpbid 231 | . . 3 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘(𝐹 ↾ (ℤ≥‘𝑀)))) |
22 | 14, 15, 21 | breldmd 5912 | . 2 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) |
23 | 12, 22 | impbida 798 | 1 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ⇝ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2105 Vcvv 3473 class class class wbr 5148 dom cdm 5676 ↾ cres 5678 ‘cfv 6543 ℤcz 12565 ℤ≥cuz 12829 ⇝ cli 15435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 |
This theorem is referenced by: (None) |
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