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 12090 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℝ) |
3 | fourierdlem6.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℤ) | |
4 | 3 | zred 12090 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℝ) |
5 | 2, 4 | resubcld 11070 | . . . . . 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 11070 | . . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
10 | 6, 9 | eqeltrid 2919 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ) |
11 | 5, 10 | remulcld 10673 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ∈ ℝ) |
12 | fourierdlem6.altb | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵) | |
13 | 8, 7 | posdifd 11229 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
14 | 12, 13 | mpbid 234 | . . . . . . 7 ⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
15 | 14, 6 | breqtrrdi 5110 | . . . . . 6 ⊢ (𝜑 → 0 < 𝑇) |
16 | 10, 15 | elrpd 12431 | . . . . 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 41802 | . . . . . 6 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) ≤ (𝐵 − 𝐴)) |
20 | 2 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℂ) |
21 | 4 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
22 | 10 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
23 | 20, 21, 22 | subdird 11099 | . . . . . . 7 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
24 | fourierdlem6.5 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
25 | 24 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
26 | 2, 10 | remulcld 10673 | . . . . . . . . 9 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℝ) |
27 | 26 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℂ) |
28 | 4, 10 | remulcld 10673 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℝ) |
29 | 28 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℂ) |
30 | 25, 27, 29 | pnpcand 11036 | . . . . . . 7 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
31 | 23, 30 | eqtr4d 2861 | . . . . . 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 5100 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ≤ 𝑇) |
34 | 11, 10, 16, 33 | lediv1dd 12492 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) ≤ (𝑇 / 𝑇)) |
35 | 5 | recnd 10671 | . . . . 5 ⊢ (𝜑 → (𝐽 − 𝐼) ∈ ℂ) |
36 | 15 | gt0ne0d 11206 | . . . . 5 ⊢ (𝜑 → 𝑇 ≠ 0) |
37 | 35, 22, 36 | divcan4d 11424 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) = (𝐽 − 𝐼)) |
38 | 22, 36 | dividd 11416 | . . . 4 ⊢ (𝜑 → (𝑇 / 𝑇) = 1) |
39 | 34, 37, 38 | 3brtr3d 5099 | . . 3 ⊢ (𝜑 → (𝐽 − 𝐼) ≤ 1) |
40 | 1red 10644 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
41 | 2, 4, 40 | lesubadd2d 11241 | . . 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 12035 | . . . 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 10672 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ) |
48 | 2, 47 | letri3d 10784 | . 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 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℤcz 11984 [,]cicc 12744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-rp 12393 df-icc 12748 |
This theorem is referenced by: fourierdlem35 42434 fourierdlem51 42449 |
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