Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clim0cf | Structured version Visualization version GIF version |
Description: Express the predicate 𝐹 converges to 0. Similar to clim 14853, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
clim0cf.nf | ⊢ Ⅎ𝑘𝐹 |
clim0cf.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
clim0cf.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
clim0cf.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
clim0cf.fv | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) |
clim0cf.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
clim0cf | ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim0cf.nf | . . 3 ⊢ Ⅎ𝑘𝐹 | |
2 | clim0cf.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | clim0cf.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | clim0cf.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
5 | clim0cf.fv | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) | |
6 | 0cnd 10636 | . . 3 ⊢ (𝜑 → 0 ∈ ℂ) | |
7 | clim0cf.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | clim2cf 41938 | . 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 0)) < 𝑥)) |
9 | 2 | uztrn2 12265 | . . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
10 | 7 | subid1d 10988 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐵 − 0) = 𝐵) |
11 | 10 | fveq2d 6676 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐵 − 0)) = (abs‘𝐵)) |
12 | 11 | breq1d 5078 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((abs‘(𝐵 − 0)) < 𝑥 ↔ (abs‘𝐵) < 𝑥)) |
13 | 9, 12 | sylan2 594 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → ((abs‘(𝐵 − 0)) < 𝑥 ↔ (abs‘𝐵) < 𝑥)) |
14 | 13 | anassrs 470 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((abs‘(𝐵 − 0)) < 𝑥 ↔ (abs‘𝐵) < 𝑥)) |
15 | 14 | ralbidva 3198 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 0)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝑥)) |
16 | 15 | rexbidva 3298 | . . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 0)) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝑥)) |
17 | 16 | ralbidv 3199 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 0)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝑥)) |
18 | 8, 17 | bitrd 281 | 1 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 ∀wral 3140 ∃wrex 3141 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 < clt 10677 − cmin 10872 ℤcz 11984 ℤ≥cuz 12246 ℝ+crp 12392 abscabs 14595 ⇝ cli 14843 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-z 11985 df-uz 12247 df-clim 14847 |
This theorem is referenced by: etransclem48 42574 |
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