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Mirrors > Home > ILE Home > Th. List > cvg1nlemf | GIF version |
Description: Lemma for cvg1n 9585. The modified sequence 𝐺 is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.) |
Ref | Expression |
---|---|
cvg1n.f | ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) |
cvg1n.c | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
cvg1n.cau | ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛)))) |
cvg1nlem.g | ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍))) |
cvg1nlem.z | ⊢ (𝜑 → 𝑍 ∈ ℕ) |
cvg1nlem.start | ⊢ (𝜑 → 𝐶 < 𝑍) |
Ref | Expression |
---|---|
cvg1nlemf | ⊢ (𝜑 → 𝐺:ℕ⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvg1n.f | . . . 4 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) | |
2 | 1 | adantr 261 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ⟶ℝ) |
3 | simpr 103 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) | |
4 | cvg1nlem.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ ℕ) | |
5 | 4 | adantr 261 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ ℕ) |
6 | 3, 5 | nnmulcld 7962 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 · 𝑍) ∈ ℕ) |
7 | 2, 6 | ffvelrnd 5303 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘(𝑗 · 𝑍)) ∈ ℝ) |
8 | cvg1nlem.g | . 2 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍))) | |
9 | 7, 8 | fmptd 5322 | 1 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∀wral 2306 class class class wbr 3764 ↦ cmpt 3818 ⟶wf 4898 ‘cfv 4902 (class class class)co 5512 ℝcr 6888 + caddc 6892 · cmul 6894 < clt 7060 / cdiv 7651 ℕcn 7914 ℤ≥cuz 8473 ℝ+crp 8583 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-1rid 6991 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-ov 5515 df-inn 7915 |
This theorem is referenced by: cvg1nlemres 9584 |
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