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Mirrors > Home > ILE Home > Th. List > cvg1nlemf | GIF version |
Description: Lemma for cvg1n 10758. 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 8769 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 · 𝑍) ∈ ℕ) |
7 | 2, 6 | ffvelrnd 5556 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘(𝑗 · 𝑍)) ∈ ℝ) |
8 | cvg1nlem.g | . 2 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍))) | |
9 | 7, 8 | fmptd 5574 | 1 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 class class class wbr 3929 ↦ cmpt 3989 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ℝcr 7619 + caddc 7623 · cmul 7625 < clt 7800 / cdiv 8432 ℕcn 8720 ℤ≥cuz 9326 ℝ+crp 9441 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-1rid 7727 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-inn 8721 |
This theorem is referenced by: cvg1nlemres 10757 |
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