fourierdlem38 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem38 46917

Description: The function 𝐹 is continuous on every interval induced by the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem38.cn	⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ))
fourierdlem38.p	⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem38.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem38.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem38.h	⊢ 𝐻 = (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹))
fourierdlem38.ranq	⊢ (𝜑 → ran 𝑄 = 𝐻)

**Assertion**
Ref	Expression
fourierdlem38	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))

Distinct variable groups: 𝑖,𝐹 𝑖,𝑀,𝑛,𝑝 𝑄,𝑖,𝑝 𝜑,𝑖 Allowed substitution hints: 𝜑(𝑛, 𝑝) 𝐴(𝑖, 𝑛, 𝑝) 𝑃(𝑖, 𝑛, 𝑝) 𝑄(𝑛) 𝐹(𝑛, 𝑝) 𝐻(𝑖, 𝑛, 𝑝)

Proof of Theorem fourierdlem38 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 781	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
2		simplll 787	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝜑)
3		ioossicc 13476	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
4		pire 26670	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
5	4	renegcli 11534	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
6	5	rexri 11282	. . . . . . . . . . 11 ⊢ -π ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
8	4	rexri 11282	. . . . . . . . . . 11 ⊢ π ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ^*)
10		fourierdlem38.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
11		fourierdlem38.m	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℕ)
12		fourierdlem38.q	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
13	10, 11, 12	fourierdlem15 46894	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
14	13	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
15		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
16	7, 9, 14, 15	fourierdlem8 46887	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
17	3, 16	sstrid 3949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
18	17	sselda 3938	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ (-π[,]π))
19	18	adantr 486	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (-π[,]π))
20		simpr 490	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ dom 𝐹)
21		simpllr 788	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑖 ∈ (0..^𝑀))
22	11	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑀 ∈ ℕ)
23	12	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑄 ∈ (𝑃‘𝑀))
24		simp2 1155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (-π[,]π))
25		simp3 1156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ dom 𝐹)
26	24, 25	eldifd 3917	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ((-π[,]π) ∖ dom 𝐹))
27		elun2 4136	. . . . . . . . 9 ⊢ (𝑥 ∈ ((-π[,]π) ∖ dom 𝐹) → 𝑥 ∈ (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)))
28	26, 27	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)))
29		fourierdlem38.ranq	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑄 = 𝐻)
30		fourierdlem38.h	. . . . . . . . . 10 ⊢ 𝐻 = (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹))
31	29, 30	eqtr2di 2817	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)) = ran 𝑄)
32	31	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)) = ran 𝑄)
33	28, 32	eleqtrd 2867	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ran 𝑄)
34	10, 22, 23, 33	fourierdlem12 46891	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
35	2, 19, 20, 21, 34	syl31anc 1400	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
36	1, 35	condan 830	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ dom 𝐹)
37	36	ralrimiva 3159	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 ∈ dom 𝐹)
38		dfss3 3927	. . 3 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 ∈ dom 𝐹)
39	37, 38	sylibr 237	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
40		fourierdlem38.cn	. . 3 ⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ))
41	40	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹 ∈ (dom 𝐹–cn→ℂ))
42		rescncf 25107	. 2 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 → (𝐹 ∈ (dom 𝐹–cn→ℂ) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)))
43	39, 41, 42	sylc 66	1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3081 {crab 3418 ∖ cdif 3903 ∪ cun 3904 ⊆ wss 3906 class class class wbr 5111 ↦ cmpt 5194 dom cdm 5663 ran crn 5664 ↾ cres 5665 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ↑_m cmap 8830 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 + caddc 11118 ℝ^*cxr 11257 < clt 11258 -cneg 11457 ℕcn 12248 (,)cioo 13388 [,]cicc 13391 ...cfz 13551 ..^cfzo 13699 πcpi 16142 –cn→ccncf 25086

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-fl 13843 df-seq 14056 df-exp 14116 df-fac 14328 df-bc 14357 df-hash 14385 df-shft 15128 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-limsup 15546 df-clim 15563 df-rlim 15564 df-sum 15762 df-ef 16143 df-sin 16145 df-cos 16146 df-pi 16148 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-xrs 17578 df-qtop 17583 df-imas 17584 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-mulg 19178 df-cntz 19431 df-cmn 19896 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-fbas 21569 df-fg 21570 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-nei 23305 df-lp 23343 df-perf 23344 df-cn 23434 df-cnp 23435 df-haus 23522 df-tx 23770 df-hmeo 23963 df-fil 24054 df-fm 24146 df-flim 24147 df-flf 24148 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-limc 26076 df-dv 26077

This theorem is used by: fourierdlem102 46980 fourierdlem114 46992

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