fourierclimd - Mathbox for Glauco Siliprandi

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Theorem fourierclimd 47036

Description: Fourier series convergence, for piecewise smooth functions. See fourierd 47035 for the analogous Σ equation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierclimd.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierclimd.t	⊢ 𝑇 = (2 · π)
fourierclimd.per	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierclimd.g	⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
fourierclimd.dmdv	⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
fourierclimd.dvcn	⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
fourierclimd.rlim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
fourierclimd.llim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
fourierclimd.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierclimd.l	⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
fourierclimd.r	⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierclimd.a	⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fourierclimd.b	⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
fourierclimd.s	⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))

**Assertion**
Ref	Expression
fourierclimd	⊢ (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))

Distinct variable groups: 𝑛,𝐹,𝑥 𝑥,𝐺 𝑥,𝑇 𝑛,𝑋,𝑥 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑛) 𝐴(𝑥, 𝑛) 𝐵(𝑥, 𝑛) 𝑅(𝑥, 𝑛) 𝑆(𝑥, 𝑛) 𝑇(𝑛) 𝐺(𝑛) 𝐿(𝑥, 𝑛)

Proof of Theorem fourierclimd Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierclimd.f	. . 3 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		fourierclimd.t	. . 3 ⊢ 𝑇 = (2 · π)
3		fourierclimd.per	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
4		fourierclimd.g	. . 3 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
5		fourierclimd.dmdv	. . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
6		fourierclimd.dvcn	. . 3 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
7		fourierclimd.rlim	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
8		fourierclimd.llim	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
9		fourierclimd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
10		fourierclimd.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
11		fourierclimd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
12		fourierclimd.a	. . 3 ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
13		fourierclimd.b	. . 3 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
14		fourierclimd.s	. . . 4 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))
15		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑘(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))
16		nfmpt1 5208	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
17	12, 16	nfcxfr 2922	. . . . . . . 8 ⊢ Ⅎ𝑛𝐴
18		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑛𝑘
19	17, 18	nffv 6892	. . . . . . 7 ⊢ Ⅎ𝑛(𝐴‘𝑘)
20		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑛 ·
21		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑛(cos‘(𝑘 · 𝑋))
22	19, 20, 21	nfov 7446	. . . . . 6 ⊢ Ⅎ𝑛((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋)))
23		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑛 +
24		nfmpt1 5208	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
25	13, 24	nfcxfr 2922	. . . . . . . 8 ⊢ Ⅎ𝑛𝐵
26	25, 18	nffv 6892	. . . . . . 7 ⊢ Ⅎ𝑛(𝐵‘𝑘)
27		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑛(sin‘(𝑘 · 𝑋))
28	26, 20, 27	nfov 7446	. . . . . 6 ⊢ Ⅎ𝑛((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))
29	22, 23, 28	nfov 7446	. . . . 5 ⊢ Ⅎ𝑛(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
30		fveq2 6882	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘))
31		oveq1 7423	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
32	31	fveq2d 6886	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
33	30, 32	oveq12d 7434	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))))
34		fveq2 6882	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
35	31	fveq2d 6886	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
36	34, 35	oveq12d 7434	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
37	33, 36	oveq12d 7434	. . . . 5 ⊢ (𝑛 = 𝑘 → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))
38	15, 29, 37	cbvmpt 5211	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑘 ∈ ℕ ↦ (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))
39	14, 38	eqtri 2785	. . 3 ⊢ 𝑆 = (𝑘 ∈ ℕ ↦ (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))
40	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39	fourierdlem115 47034	. 2 ⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))
41	40	simpld 500	1 ⊢ (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∖ cdif 3899 ∅c0 4282 class class class wbr 5107 ↦ cmpt 5190 dom cdm 5659 ↾ cres 5661 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 Fincfn 8955 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 + caddc 11128 · cmul 11130 +∞cpnf 11265 -∞cmnf 11266 − cmin 11466 -cneg 11467 / cdiv 11896 ℕcn 12258 2c2 12320 ℕ₀cn0 12529 (,)cioo 13398 (,]cioc 13399 [,)cico 13400 seqcseq 14065 ⇝ cli 15571 Σcsu 15773 sincsin 16151 cosccos 16152 πcpi 16154 –cn→ccncf 25103 ∫citg 25845 lim_ℂ climc 26089 D cdv 26090

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cc 10440 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-symdif 4202 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5075 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-ofr 7682 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-oadd 8462 df-omul 8463 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-dju 9909 df-card 9947 df-acn 9950 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-xnn0 12603 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-ef 16155 df-sin 16157 df-cos 16158 df-pi 16160 df-struct 17241 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-ress 17325 df-plusg 17357 df-mulr 17358 df-starv 17359 df-sca 17360 df-vsca 17361 df-ip 17362 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-hom 17368 df-cco 17369 df-rest 17509 df-topn 17510 df-0g 17528 df-gsum 17529 df-topgen 17530 df-pt 17531 df-prds 17534 df-xrs 17590 df-qtop 17595 df-imas 17596 df-xps 17598 df-mre 17672 df-mrc 17673 df-acs 17675 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-submnd 18891 df-mulg 19190 df-cntz 19443 df-cmn 19908 df-psmet 21576 df-xmet 21577 df-met 21578 df-bl 21579 df-mopn 21580 df-fbas 21581 df-fg 21582 df-cnfld 21585 df-top 23118 df-topon 23135 df-topsp 23157 df-bases 23170 df-cld 23243 df-ntr 23244 df-cls 23245 df-nei 23322 df-lp 23360 df-perf 23361 df-cn 23451 df-cnp 23452 df-t1 23538 df-haus 23539 df-cmp 23611 df-tx 23787 df-hmeo 23980 df-fil 24071 df-fm 24163 df-flim 24164 df-flf 24165 df-xms 24545 df-ms 24546 df-tms 24547 df-cncf 25105 df-ovol 25691 df-vol 25692 df-mbf 25846 df-itg1 25847 df-itg2 25848 df-ibl 25849 df-itg 25850 df-0p 25897 df-ditg 26074 df-limc 26093 df-dv 26094

This theorem is used by: fourierclim 47037

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