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Mirrors > Home > ILE Home > Th. List > cvg1nlemf | GIF version |
Description: Lemma for cvg1n 9872. 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 270 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ⟶ℝ) |
3 | simpr 108 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) | |
4 | cvg1nlem.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ ℕ) | |
5 | 4 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ ℕ) |
6 | 3, 5 | nnmulcld 8087 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 · 𝑍) ∈ ℕ) |
7 | 2, 6 | ffvelrnd 5324 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘(𝑗 · 𝑍)) ∈ ℝ) |
8 | cvg1nlem.g | . 2 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍))) | |
9 | 7, 8 | fmptd 5343 | 1 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∀wral 2348 class class class wbr 3785 ↦ cmpt 3839 ⟶wf 4918 ‘cfv 4922 (class class class)co 5532 ℝcr 6980 + caddc 6984 · cmul 6986 < clt 7153 / cdiv 7760 ℕcn 8039 ℤ≥cuz 8619 ℝ+crp 8734 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-1rid 7083 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-ov 5535 df-inn 8040 |
This theorem is referenced by: cvg1nlemres 9871 |
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