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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 12720 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℝ) |
3 | fourierdlem6.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℤ) | |
4 | 3 | zred 12720 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℝ) |
5 | 2, 4 | resubcld 11689 | . . . . . 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 11689 | . . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
10 | 6, 9 | eqeltrid 2843 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ) |
11 | 5, 10 | remulcld 11289 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ∈ ℝ) |
12 | fourierdlem6.altb | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵) | |
13 | 8, 7 | posdifd 11848 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
14 | 12, 13 | mpbid 232 | . . . . . . 7 ⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
15 | 14, 6 | breqtrrdi 5190 | . . . . . 6 ⊢ (𝜑 → 0 < 𝑇) |
16 | 10, 15 | elrpd 13072 | . . . . 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 45472 | . . . . . 6 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) ≤ (𝐵 − 𝐴)) |
20 | 2 | recnd 11287 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℂ) |
21 | 4 | recnd 11287 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
22 | 10 | recnd 11287 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
23 | 20, 21, 22 | subdird 11718 | . . . . . . 7 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
24 | fourierdlem6.5 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
25 | 24 | recnd 11287 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
26 | 2, 10 | remulcld 11289 | . . . . . . . . 9 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℝ) |
27 | 26 | recnd 11287 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℂ) |
28 | 4, 10 | remulcld 11289 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℝ) |
29 | 28 | recnd 11287 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℂ) |
30 | 25, 27, 29 | pnpcand 11655 | . . . . . . 7 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
31 | 23, 30 | eqtr4d 2778 | . . . . . 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 5180 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ≤ 𝑇) |
34 | 11, 10, 16, 33 | lediv1dd 13133 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) ≤ (𝑇 / 𝑇)) |
35 | 5 | recnd 11287 | . . . . 5 ⊢ (𝜑 → (𝐽 − 𝐼) ∈ ℂ) |
36 | 15 | gt0ne0d 11825 | . . . . 5 ⊢ (𝜑 → 𝑇 ≠ 0) |
37 | 35, 22, 36 | divcan4d 12047 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) = (𝐽 − 𝐼)) |
38 | 22, 36 | dividd 12039 | . . . 4 ⊢ (𝜑 → (𝑇 / 𝑇) = 1) |
39 | 34, 37, 38 | 3brtr3d 5179 | . . 3 ⊢ (𝜑 → (𝐽 − 𝐼) ≤ 1) |
40 | 1red 11260 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
41 | 2, 4, 40 | lesubadd2d 11860 | . . 3 ⊢ (𝜑 → ((𝐽 − 𝐼) ≤ 1 ↔ 𝐽 ≤ (𝐼 + 1))) |
42 | 39, 41 | mpbid 232 | . 2 ⊢ (𝜑 → 𝐽 ≤ (𝐼 + 1)) |
43 | fourierdlem6.iltj | . . 3 ⊢ (𝜑 → 𝐼 < 𝐽) | |
44 | zltp1le 12665 | . . . 4 ⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽)) | |
45 | 3, 1, 44 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽)) |
46 | 43, 45 | mpbid 232 | . 2 ⊢ (𝜑 → (𝐼 + 1) ≤ 𝐽) |
47 | 4, 40 | readdcld 11288 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ) |
48 | 2, 47 | letri3d 11401 | . 2 ⊢ (𝜑 → (𝐽 = (𝐼 + 1) ↔ (𝐽 ≤ (𝐼 + 1) ∧ (𝐼 + 1) ≤ 𝐽))) |
49 | 42, 46, 48 | mpbir2and 713 | 1 ⊢ (𝜑 → 𝐽 = (𝐼 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℤcz 12611 [,]cicc 13387 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-rp 13033 df-icc 13391 |
This theorem is referenced by: fourierdlem35 46098 fourierdlem51 46113 |
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