fourierdlem38 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem38		Structured version Visualization version GIF version

Theorem fourierdlem38 46573

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 769	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
2		simplll 775	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝜑)
3		ioossicc 13386	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
4		pire 26421	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
5	4	renegcli 11455	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
6	5	rexri 11203	. . . . . . . . . . 11 ⊢ -π ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
8	4	rexri 11203	. . . . . . . . . . 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 46550	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
14	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
15		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
16	7, 9, 14, 15	fourierdlem8 46543	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
17	3, 16	sstrid 3933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
18	17	sselda 3921	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ (-π[,]π))
19	18	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (-π[,]π))
20		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ dom 𝐹)
21		simpllr 776	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑖 ∈ (0..^𝑀))
22	11	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑀 ∈ ℕ)
23	12	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑄 ∈ (𝑃‘𝑀))
24		simp2 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (-π[,]π))
25		simp3 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ dom 𝐹)
26	24, 25	eldifd 3900	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ((-π[,]π) ∖ dom 𝐹))
27		elun2 4123	. . . . . . . . 9 ⊢ (𝑥 ∈ ((-π[,]π) ∖ dom 𝐹) → 𝑥 ∈ (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)))
28	26, 27	syl 17	. . . . . . . 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 2788	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)) = ran 𝑄)
32	31	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)) = ran 𝑄)
33	28, 32	eleqtrd 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ran 𝑄)
34	10, 22, 23, 33	fourierdlem12 46547	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
35	2, 19, 20, 21, 34	syl31anc 1376	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
36	1, 35	condan 818	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ dom 𝐹)
37	36	ralrimiva 3129	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 ∈ dom 𝐹)
38		dfss3 3910	. . 3 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 ∈ dom 𝐹)
39	37, 38	sylibr 234	. 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 480	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹 ∈ (dom 𝐹–cn→ℂ))
42		rescncf 24864	. 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 65	1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 {crab 3389 ∖ cdif 3886 ∪ cun 3887 ⊆ wss 3889 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ran crn 5632 ↾ cres 5633 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 ℝ^*cxr 11178 < clt 11179 -cneg 11378 ℕcn 12174 (,)cioo 13298 [,]cicc 13301 ...cfz 13461 ..^cfzo 13608 πcpi 16031 –cn→ccncf 24843

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834

This theorem is referenced by: fourierdlem102 46636 fourierdlem114 46648

W3C validator