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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem8 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 34370. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
Ref | Expression |
---|---|
knoppndvlem8.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
knoppndvlem8.f | ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) |
knoppndvlem8.a | ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) |
knoppndvlem8.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppndvlem8.j | ⊢ (𝜑 → 𝐽 ∈ ℕ0) |
knoppndvlem8.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
knoppndvlem8.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndvlem8.1 | ⊢ (𝜑 → 2 ∥ 𝑀) |
Ref | Expression |
---|---|
knoppndvlem8 | ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppndvlem8.t | . . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
2 | knoppndvlem8.f | . . 3 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) | |
3 | knoppndvlem8.a | . . 3 ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) | |
4 | knoppndvlem8.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ ℕ0) | |
5 | knoppndvlem8.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | knoppndvlem8.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
7 | 1, 2, 3, 4, 5, 6 | knoppndvlem7 34354 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) · (𝑇‘(𝑀 / 2)))) |
8 | knoppndvlem8.1 | . . . . 5 ⊢ (𝜑 → 2 ∥ 𝑀) | |
9 | 2z 12108 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ) |
11 | 2ne0 11833 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
13 | 10, 12, 5 | 3jca 1129 | . . . . . 6 ⊢ (𝜑 → (2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑀 ∈ ℤ)) |
14 | dvdsval2 15715 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑀 ∈ ℤ) → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ)) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ)) |
16 | 8, 15 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝑀 / 2) ∈ ℤ) |
17 | 1, 16 | dnizeq0 34311 | . . 3 ⊢ (𝜑 → (𝑇‘(𝑀 / 2)) = 0) |
18 | 17 | oveq2d 7199 | . 2 ⊢ (𝜑 → ((𝐶↑𝐽) · (𝑇‘(𝑀 / 2))) = ((𝐶↑𝐽) · 0)) |
19 | knoppndvlem8.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
20 | 19 | knoppndvlem3 34350 | . . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
21 | 20 | simpld 498 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
22 | 21 | recnd 10760 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
23 | 22, 4 | expcld 13615 | . . 3 ⊢ (𝜑 → (𝐶↑𝐽) ∈ ℂ) |
24 | 23 | mul01d 10930 | . 2 ⊢ (𝜑 → ((𝐶↑𝐽) · 0) = 0) |
25 | 7, 18, 24 | 3eqtrd 2778 | 1 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 class class class wbr 5040 ↦ cmpt 5120 ‘cfv 6350 (class class class)co 7183 ℝcr 10627 0cc0 10628 1c1 10629 + caddc 10631 · cmul 10633 < clt 10766 − cmin 10961 -cneg 10962 / cdiv 11388 ℕcn 11729 2c2 11784 ℕ0cn0 11989 ℤcz 12075 (,)cioo 12834 ⌊cfl 13264 ↑cexp 13534 abscabs 14696 ∥ cdvds 15712 |
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 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 |
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-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-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 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-sup 8992 df-inf 8993 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-n0 11990 df-z 12076 df-uz 12338 df-rp 12486 df-ioo 12838 df-ico 12840 df-fl 13266 df-seq 13474 df-exp 13535 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-dvds 15713 |
This theorem is referenced by: knoppndvlem10 34357 |
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