fourierdlem67 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem67 46079

Description: 𝐺 is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem67.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem67.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem67.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
fourierdlem67.w	⊢ (𝜑 → 𝑊 ∈ ℝ)
fourierdlem67.h	⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
fourierdlem67.k	⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
fourierdlem67.u	⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
fourierdlem67.n	⊢ (𝜑 → 𝑁 ∈ ℝ)
fourierdlem67.s	⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))
fourierdlem67.g	⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))

**Assertion**
Ref	Expression
fourierdlem67	⊢ (𝜑 → 𝐺:(-π[,]π)⟶ℝ)

Distinct variable groups: 𝑁,𝑠 𝜑,𝑠 Allowed substitution hints: 𝑆(𝑠) 𝑈(𝑠) 𝐹(𝑠) 𝐺(𝑠) 𝐻(𝑠) 𝐾(𝑠) 𝑊(𝑠) 𝑋(𝑠) 𝑌(𝑠)

**Proof of Theorem fourierdlem67**
Step	Hyp	Ref	Expression
1		fourierdlem67.f	. . . . 5 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		fourierdlem67.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ)
3		fourierdlem67.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℝ)
4		fourierdlem67.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℝ)
5		fourierdlem67.h	. . . . 5 ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
6		fourierdlem67.k	. . . . 5 ⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
7		fourierdlem67.u	. . . . 5 ⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
8	1, 2, 3, 4, 5, 6, 7	fourierdlem55 46067	. . . 4 ⊢ (𝜑 → 𝑈:(-π[,]π)⟶ℝ)
9	8	ffvelcdmda 7098	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) ∈ ℝ)
10		fourierdlem67.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
11		fourierdlem67.s	. . . . . 6 ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))
12	11	fourierdlem5 46018	. . . . 5 ⊢ (𝑁 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ)
13	10, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝑆:(-π[,]π)⟶ℝ)
14	13	ffvelcdmda 7098	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑆‘𝑠) ∈ ℝ)
15	9, 14	remulcld 11282	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ)
16		fourierdlem67.g	. 2 ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
17	15, 16	fmptd 7128	1 ⊢ (𝜑 → 𝐺:(-π[,]π)⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ifcif 4530 class class class wbr 5149 ↦ cmpt 5232 ⟶wf 6554 ‘cfv 6558 (class class class)co 7425 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 < clt 11286 − cmin 11483 -cneg 11484 / cdiv 11911 2c2 12312 [,]cicc 13380 sincsin 16085 πcpi 16088

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 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 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-of 7691 df-om 7881 df-1st 8007 df-2nd 8008 df-supp 8179 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-1o 8499 df-2o 8500 df-er 8738 df-map 8861 df-pm 8862 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12518 df-z 12605 df-dec 12725 df-uz 12870 df-q 12982 df-rp 13026 df-xneg 13145 df-xadd 13146 df-xmul 13147 df-ioo 13381 df-ioc 13382 df-ico 13383 df-icc 13384 df-fz 13538 df-fzo 13682 df-fl 13818 df-mod 13896 df-seq 14029 df-exp 14089 df-fac 14299 df-bc 14328 df-hash 14356 df-shft 15092 df-cj 15124 df-re 15125 df-im 15126 df-sqrt 15260 df-abs 15261 df-limsup 15493 df-clim 15510 df-rlim 15511 df-sum 15709 df-ef 16089 df-sin 16091 df-cos 16092 df-pi 16094 df-struct 17170 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-ress 17264 df-plusg 17300 df-mulr 17301 df-starv 17302 df-sca 17303 df-vsca 17304 df-ip 17305 df-tset 17306 df-ple 17307 df-ds 17309 df-unif 17310 df-hom 17311 df-cco 17312 df-rest 17458 df-topn 17459 df-0g 17477 df-gsum 17478 df-topgen 17479 df-pt 17480 df-prds 17483 df-xrs 17538 df-qtop 17543 df-imas 17544 df-xps 17546 df-mre 17620 df-mrc 17621 df-acs 17623 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18795 df-mulg 19084 df-cntz 19333 df-cmn 19800 df-psmet 21355 df-xmet 21356 df-met 21357 df-bl 21358 df-mopn 21359 df-fbas 21360 df-fg 21361 df-cnfld 21364 df-top 22897 df-topon 22914 df-topsp 22936 df-bases 22950 df-cld 23024 df-ntr 23025 df-cls 23026 df-nei 23103 df-lp 23141 df-perf 23142 df-cn 23232 df-cnp 23233 df-haus 23320 df-tx 23567 df-hmeo 23760 df-fil 23851 df-fm 23943 df-flim 23944 df-flf 23945 df-xms 24327 df-ms 24328 df-tms 24329 df-cncf 24899 df-limc 25897 df-dv 25898

This theorem is referenced by: fourierdlem87 46099 fourierdlem95 46107 fourierdlem103 46115 fourierdlem104 46116

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