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 11358 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℝ) |
3 | fourierdlem6.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℤ) | |
4 | 3 | zred 11358 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ ℝ) |
5 | 2, 4 | resubcld 10337 | . . . . . 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 10337 | . . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
10 | 6, 9 | syl5eqel 2692 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ) |
11 | 5, 10 | remulcld 9949 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ∈ ℝ) |
12 | fourierdlem6.altb | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵) | |
13 | 8, 7 | posdifd 10493 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
14 | 12, 13 | mpbid 221 | . . . . . . 7 ⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
15 | 14, 6 | syl6breqr 4625 | . . . . . 6 ⊢ (𝜑 → 0 < 𝑇) |
16 | 10, 15 | elrpd 11745 | . . . . 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 38592 | . . . . . 6 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) ≤ (𝐵 − 𝐴)) |
20 | 2 | recnd 9947 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℂ) |
21 | 4 | recnd 9947 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
22 | 10 | recnd 9947 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
23 | 20, 21, 22 | subdird 10366 | . . . . . . 7 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
24 | fourierdlem6.5 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
25 | 24 | recnd 9947 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
26 | 2, 10 | remulcld 9949 | . . . . . . . . 9 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℝ) |
27 | 26 | recnd 9947 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 · 𝑇) ∈ ℂ) |
28 | 4, 10 | remulcld 9949 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℝ) |
29 | 28 | recnd 9947 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 · 𝑇) ∈ ℂ) |
30 | 25, 27, 29 | pnpcand 10308 | . . . . . . 7 ⊢ (𝜑 → ((𝑋 + (𝐽 · 𝑇)) − (𝑋 + (𝐼 · 𝑇))) = ((𝐽 · 𝑇) − (𝐼 · 𝑇))) |
31 | 23, 30 | eqtr4d 2647 | . . . . . 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 4615 | . . . . 5 ⊢ (𝜑 → ((𝐽 − 𝐼) · 𝑇) ≤ 𝑇) |
34 | 11, 10, 16, 33 | lediv1dd 11806 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) ≤ (𝑇 / 𝑇)) |
35 | 5 | recnd 9947 | . . . . 5 ⊢ (𝜑 → (𝐽 − 𝐼) ∈ ℂ) |
36 | 15 | gt0ne0d 10471 | . . . . 5 ⊢ (𝜑 → 𝑇 ≠ 0) |
37 | 35, 22, 36 | divcan4d 10686 | . . . 4 ⊢ (𝜑 → (((𝐽 − 𝐼) · 𝑇) / 𝑇) = (𝐽 − 𝐼)) |
38 | 22, 36 | dividd 10678 | . . . 4 ⊢ (𝜑 → (𝑇 / 𝑇) = 1) |
39 | 34, 37, 38 | 3brtr3d 4614 | . . 3 ⊢ (𝜑 → (𝐽 − 𝐼) ≤ 1) |
40 | 1red 9934 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
41 | 2, 4, 40 | lesubadd2d 10505 | . . 3 ⊢ (𝜑 → ((𝐽 − 𝐼) ≤ 1 ↔ 𝐽 ≤ (𝐼 + 1))) |
42 | 39, 41 | mpbid 221 | . 2 ⊢ (𝜑 → 𝐽 ≤ (𝐼 + 1)) |
43 | fourierdlem6.iltj | . . 3 ⊢ (𝜑 → 𝐼 < 𝐽) | |
44 | zltp1le 11304 | . . . 4 ⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽)) | |
45 | 3, 1, 44 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽)) |
46 | 43, 45 | mpbid 221 | . 2 ⊢ (𝜑 → (𝐼 + 1) ≤ 𝐽) |
47 | 4, 40 | readdcld 9948 | . . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ) |
48 | 2, 47 | letri3d 10058 | . 2 ⊢ (𝜑 → (𝐽 = (𝐼 + 1) ↔ (𝐽 ≤ (𝐼 + 1) ∧ (𝐼 + 1) ≤ 𝐽))) |
49 | 42, 46, 48 | mpbir2and 959 | 1 ⊢ (𝜑 → 𝐽 = (𝐼 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℤcz 11254 [,]cicc 12049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-rp 11709 df-icc 12053 |
This theorem is referenced by: fourierdlem35 39035 fourierdlem51 39050 |
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