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Mirrors > Home > ILE Home > Th. List > cvg1nlemf | GIF version |
Description: Lemma for cvg1n 10751. 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 274 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ⟶ℝ) |
3 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) | |
4 | cvg1nlem.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ ℕ) | |
5 | 4 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ ℕ) |
6 | 3, 5 | nnmulcld 8762 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 · 𝑍) ∈ ℕ) |
7 | 2, 6 | ffvelrnd 5549 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘(𝑗 · 𝑍)) ∈ ℝ) |
8 | cvg1nlem.g | . 2 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍))) | |
9 | 7, 8 | fmptd 5567 | 1 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2414 class class class wbr 3924 ↦ cmpt 3984 ⟶wf 5114 ‘cfv 5118 (class class class)co 5767 ℝcr 7612 + caddc 7616 · cmul 7618 < clt 7793 / cdiv 8425 ℕcn 8713 ℤ≥cuz 9319 ℝ+crp 9434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-1rid 7720 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-inn 8714 |
This theorem is referenced by: cvg1nlemres 10750 |
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