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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem5 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 34369. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
Ref | Expression |
---|---|
knoppndvlem5.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
knoppndvlem5.f | ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) |
knoppndvlem5.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
knoppndvlem5.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
knoppndvlem5.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
knoppndvlem5 | ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 13444 | . 2 ⊢ (𝜑 → (0...𝐽) ∈ Fin) | |
2 | knoppndvlem5.t | . . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
3 | knoppndvlem5.f | . . 3 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) | |
4 | knoppndvlem5.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑁 ∈ ℕ) |
6 | knoppndvlem5.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
7 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐶 ∈ ℝ) |
8 | knoppndvlem5.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | 8 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐴 ∈ ℝ) |
10 | elfznn0 13103 | . . . 4 ⊢ (𝑖 ∈ (0...𝐽) → 𝑖 ∈ ℕ0) | |
11 | 10 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑖 ∈ ℕ0) |
12 | 2, 3, 5, 7, 9, 11 | knoppcnlem3 34330 | . 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐴)‘𝑖) ∈ ℝ) |
13 | 1, 12 | fsumrecl 15196 | 1 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5120 ‘cfv 6349 (class class class)co 7182 ℝcr 10626 0cc0 10627 1c1 10628 + caddc 10630 · cmul 10632 − cmin 10960 / cdiv 11387 ℕcn 11728 2c2 11783 ℕ0cn0 11988 ...cfz 12993 ⌊cfl 13263 ↑cexp 13533 abscabs 14695 Σcsu 15147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-sup 8991 df-inf 8992 df-oi 9059 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-rp 12485 df-fz 12994 df-fzo 13137 df-fl 13265 df-seq 13473 df-exp 13534 df-hash 13795 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-clim 14947 df-sum 15148 |
This theorem is referenced by: knoppndvlem6 34352 knoppndvlem14 34360 knoppndvlem15 34361 |
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