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 12081 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℝ) |
3 | fourierdlem6.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℤ) | |
4 | 3 | zred 12081 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℝ) |
5 | 2, 4 | resubcld 11062 | . . . . . 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 11062 | . . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
10 | 6, 9 | eqeltrid 2917 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ) |
11 | 5, 10 | remulcld 10665 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ∈ ℝ) |
12 | fourierdlem6.altb | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵) | |
13 | 8, 7 | posdifd 11221 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
14 | 12, 13 | mpbid 234 | . . . . . . 7 ⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
15 | 14, 6 | breqtrrdi 5100 | . . . . . 6 ⊢ (𝜑 → 0 < 𝑇) |
16 | 10, 15 | elrpd 12422 | . . . . 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 41788 | . . . . . 6 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) ≤ (𝐵 − 𝐴)) |
20 | 2 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℂ) |
21 | 4 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
22 | 10 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
23 | 20, 21, 22 | subdird 11091 | . . . . . . 7 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
24 | fourierdlem6.5 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
25 | 24 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
26 | 2, 10 | remulcld 10665 | . . . . . . . . 9 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℝ) |
27 | 26 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℂ) |
28 | 4, 10 | remulcld 10665 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℝ) |
29 | 28 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℂ) |
30 | 25, 27, 29 | pnpcand 11028 | . . . . . . 7 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
31 | 23, 30 | eqtr4d 2859 | . . . . . 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 5090 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ≤ 𝑇) |
34 | 11, 10, 16, 33 | lediv1dd 12483 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) ≤ (𝑇 / 𝑇)) |
35 | 5 | recnd 10663 | . . . . 5 ⊢ (𝜑 → (𝐽 − 𝐼) ∈ ℂ) |
36 | 15 | gt0ne0d 11198 | . . . . 5 ⊢ (𝜑 → 𝑇 ≠ 0) |
37 | 35, 22, 36 | divcan4d 11416 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) = (𝐽 − 𝐼)) |
38 | 22, 36 | dividd 11408 | . . . 4 ⊢ (𝜑 → (𝑇 / 𝑇) = 1) |
39 | 34, 37, 38 | 3brtr3d 5089 | . . 3 ⊢ (𝜑 → (𝐽 − 𝐼) ≤ 1) |
40 | 1red 10636 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
41 | 2, 4, 40 | lesubadd2d 11233 | . . 3 ⊢ (𝜑 → ((𝐽 − 𝐼) ≤ 1 ↔ 𝐽 ≤ (𝐼 + 1))) |
42 | 39, 41 | mpbid 234 | . 2 ⊢ (𝜑 → 𝐽 ≤ (𝐼 + 1)) |
43 | fourierdlem6.iltj | . . 3 ⊢ (𝜑 → 𝐼 < 𝐽) | |
44 | zltp1le 12026 | . . . 4 ⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽)) | |
45 | 3, 1, 44 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽)) |
46 | 43, 45 | mpbid 234 | . 2 ⊢ (𝜑 → (𝐼 + 1) ≤ 𝐽) |
47 | 4, 40 | readdcld 10664 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ) |
48 | 2, 47 | letri3d 10776 | . 2 ⊢ (𝜑 → (𝐽 = (𝐼 + 1) ↔ (𝐽 ≤ (𝐼 + 1) ∧ (𝐼 + 1) ≤ 𝐽))) |
49 | 42, 46, 48 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐽 = (𝐼 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 − cmin 10864 / cdiv 11291 ℤcz 11975 [,]cicc 12735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-rp 12384 df-icc 12739 |
This theorem is referenced by: fourierdlem35 42421 fourierdlem51 42436 |
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