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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem6 | Structured version Visualization version GIF version |
Description: 𝑋 is in the periodic partition, when the considered interval is centered at 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierdlem6.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
fourierdlem6.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
fourierdlem6.altb | ⊢ (𝜑 → 𝐴 < 𝐵) |
fourierdlem6.t | ⊢ 𝑇 = (𝐵 − 𝐴) |
fourierdlem6.5 | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
fourierdlem6.i | ⊢ (𝜑 → 𝐼 ∈ ℤ) |
fourierdlem6.j | ⊢ (𝜑 → 𝐽 ∈ ℤ) |
fourierdlem6.iltj | ⊢ (𝜑 → 𝐼 < 𝐽) |
fourierdlem6.iel | ⊢ (𝜑 → (𝑋 + (𝐼 · 𝑇)) ∈ (𝐴[,]𝐵)) |
fourierdlem6.jel | ⊢ (𝜑 → (𝑋 + (𝐽 · 𝑇)) ∈ (𝐴[,]𝐵)) |
Ref | Expression |
---|---|
fourierdlem6 | ⊢ (𝜑 → 𝐽 = (𝐼 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fourierdlem6.j | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℤ) | |
2 | 1 | zred 11817 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℝ) |
3 | fourierdlem6.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℤ) | |
4 | 3 | zred 11817 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℝ) |
5 | 2, 4 | resubcld 10789 | . . . . . 6 ⊢ (𝜑 → (𝐽 − 𝐼) ∈ ℝ) |
6 | fourierdlem6.t | . . . . . . 7 ⊢ 𝑇 = (𝐵 − 𝐴) | |
7 | fourierdlem6.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
8 | fourierdlem6.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | 7, 8 | resubcld 10789 | . . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
10 | 6, 9 | syl5eqel 2910 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ) |
11 | 5, 10 | remulcld 10394 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ∈ ℝ) |
12 | fourierdlem6.altb | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵) | |
13 | 8, 7 | posdifd 10946 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
14 | 12, 13 | mpbid 224 | . . . . . . 7 ⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
15 | 14, 6 | syl6breqr 4917 | . . . . . 6 ⊢ (𝜑 → 0 < 𝑇) |
16 | 10, 15 | elrpd 12160 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ+) |
17 | fourierdlem6.jel | . . . . . . 7 ⊢ (𝜑 → (𝑋 + (𝐽 · 𝑇)) ∈ (𝐴[,]𝐵)) | |
18 | fourierdlem6.iel | . . . . . . 7 ⊢ (𝜑 → (𝑋 + (𝐼 · 𝑇)) ∈ (𝐴[,]𝐵)) | |
19 | 8, 7, 17, 18 | iccsuble 40539 | . . . . . 6 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) ≤ (𝐵 − 𝐴)) |
20 | 2 | recnd 10392 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℂ) |
21 | 4 | recnd 10392 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
22 | 10 | recnd 10392 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
23 | 20, 21, 22 | subdird 10818 | . . . . . . 7 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
24 | fourierdlem6.5 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
25 | 24 | recnd 10392 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
26 | 2, 10 | remulcld 10394 | . . . . . . . . 9 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℝ) |
27 | 26 | recnd 10392 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℂ) |
28 | 4, 10 | remulcld 10394 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℝ) |
29 | 28 | recnd 10392 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℂ) |
30 | 25, 27, 29 | pnpcand 10757 | . . . . . . 7 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
31 | 23, 30 | eqtr4d 2864 | . . . . . 6 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) = ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇)))) |
32 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑇 = (𝐵 − 𝐴)) |
33 | 19, 31, 32 | 3brtr4d 4907 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ≤ 𝑇) |
34 | 11, 10, 16, 33 | lediv1dd 12221 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) ≤ (𝑇 / 𝑇)) |
35 | 5 | recnd 10392 | . . . . 5 ⊢ (𝜑 → (𝐽 − 𝐼) ∈ ℂ) |
36 | 15 | gt0ne0d 10923 | . . . . 5 ⊢ (𝜑 → 𝑇 ≠ 0) |
37 | 35, 22, 36 | divcan4d 11140 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) = (𝐽 − 𝐼)) |
38 | 22, 36 | dividd 11132 | . . . 4 ⊢ (𝜑 → (𝑇 / 𝑇) = 1) |
39 | 34, 37, 38 | 3brtr3d 4906 | . . 3 ⊢ (𝜑 → (𝐽 − 𝐼) ≤ 1) |
40 | 1red 10364 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
41 | 2, 4, 40 | lesubadd2d 10958 | . . 3 ⊢ (𝜑 → ((𝐽 − 𝐼) ≤ 1 ↔ 𝐽 ≤ (𝐼 + 1))) |
42 | 39, 41 | mpbid 224 | . 2 ⊢ (𝜑 → 𝐽 ≤ (𝐼 + 1)) |
43 | fourierdlem6.iltj | . . 3 ⊢ (𝜑 → 𝐼 < 𝐽) | |
44 | zltp1le 11762 | . . . 4 ⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽)) | |
45 | 3, 1, 44 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽)) |
46 | 43, 45 | mpbid 224 | . 2 ⊢ (𝜑 → (𝐼 + 1) ≤ 𝐽) |
47 | 4, 40 | readdcld 10393 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ) |
48 | 2, 47 | letri3d 10505 | . 2 ⊢ (𝜑 → (𝐽 = (𝐼 + 1) ↔ (𝐽 ≤ (𝐼 + 1) ∧ (𝐼 + 1) ≤ 𝐽))) |
49 | 42, 46, 48 | mpbir2and 704 | 1 ⊢ (𝜑 → 𝐽 = (𝐼 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 (class class class)co 6910 ℝcr 10258 0cc0 10259 1c1 10260 + caddc 10262 · cmul 10264 < clt 10398 ≤ cle 10399 − cmin 10592 / cdiv 11016 ℤcz 11711 [,]cicc 12473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-n0 11626 df-z 11712 df-rp 12120 df-icc 12477 |
This theorem is referenced by: fourierdlem35 41151 fourierdlem51 41166 |
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