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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 12722 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℝ) | 
| 3 | fourierdlem6.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℤ) | |
| 4 | 3 | zred 12722 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℝ) | 
| 5 | 2, 4 | resubcld 11691 | . . . . . 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 11691 | . . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) | 
| 10 | 6, 9 | eqeltrid 2845 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ) | 
| 11 | 5, 10 | remulcld 11291 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ∈ ℝ) | 
| 12 | fourierdlem6.altb | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵) | |
| 13 | 8, 7 | posdifd 11850 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) | 
| 14 | 12, 13 | mpbid 232 | . . . . . . 7 ⊢ (𝜑 → 0 < (𝐵 − 𝐴)) | 
| 15 | 14, 6 | breqtrrdi 5185 | . . . . . 6 ⊢ (𝜑 → 0 < 𝑇) | 
| 16 | 10, 15 | elrpd 13074 | . . . . 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 45532 | . . . . . 6 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) ≤ (𝐵 − 𝐴)) | 
| 20 | 2 | recnd 11289 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℂ) | 
| 21 | 4 | recnd 11289 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℂ) | 
| 22 | 10 | recnd 11289 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) | 
| 23 | 20, 21, 22 | subdird 11720 | . . . . . . 7 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) | 
| 24 | fourierdlem6.5 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
| 25 | 24 | recnd 11289 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ) | 
| 26 | 2, 10 | remulcld 11291 | . . . . . . . . 9 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℝ) | 
| 27 | 26 | recnd 11289 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℂ) | 
| 28 | 4, 10 | remulcld 11291 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℝ) | 
| 29 | 28 | recnd 11289 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℂ) | 
| 30 | 25, 27, 29 | pnpcand 11657 | . . . . . . 7 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) | 
| 31 | 23, 30 | eqtr4d 2780 | . . . . . 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 5175 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ≤ 𝑇) | 
| 34 | 11, 10, 16, 33 | lediv1dd 13135 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) ≤ (𝑇 / 𝑇)) | 
| 35 | 5 | recnd 11289 | . . . . 5 ⊢ (𝜑 → (𝐽 − 𝐼) ∈ ℂ) | 
| 36 | 15 | gt0ne0d 11827 | . . . . 5 ⊢ (𝜑 → 𝑇 ≠ 0) | 
| 37 | 35, 22, 36 | divcan4d 12049 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) = (𝐽 − 𝐼)) | 
| 38 | 22, 36 | dividd 12041 | . . . 4 ⊢ (𝜑 → (𝑇 / 𝑇) = 1) | 
| 39 | 34, 37, 38 | 3brtr3d 5174 | . . 3 ⊢ (𝜑 → (𝐽 − 𝐼) ≤ 1) | 
| 40 | 1red 11262 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 41 | 2, 4, 40 | lesubadd2d 11862 | . . 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 12667 | . . . 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 11290 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ) | 
| 48 | 2, 47 | letri3d 11403 | . 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 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℤcz 12613 [,]cicc 13390 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-rp 13035 df-icc 13394 | 
| This theorem is referenced by: fourierdlem35 46157 fourierdlem51 46172 | 
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