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 47124

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 13557	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
4		pire 26776	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
5	4	renegcli 11612	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
6	5	rexri 11360	. . . . . . . . . . 11 ⊢ -π ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
8	4	rexri 11360	. . . . . . . . . . 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 47101	. . . . . . . . . . 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 47094	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
17	3, 16	sstrid 3942	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
18	17	sselda 3931	. . . . . . 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 3910	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ((-π[,]π) ∖ dom 𝐹))
27		elun2 4129	. . . . . . . . 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 2813	. . . . . . . . 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 2863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ran 𝑄)
34	10, 22, 23, 33	fourierdlem12 47098	. . . . . 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 3155	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 ∈ dom 𝐹)
38		dfss3 3920	. . 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 25211	. 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 2145 ∀wral 3077 {crab 3413 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 ran crn 5652 ↾ cres 5653 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ↑_m cmap 8840 ℂcc 11191 ℝcr 11192 0cc0 11193 1c1 11194 + caddc 11196 ℝ^*cxr 11335 < clt 11336 -cneg 11535 ℕcn 12328 (,)cioo 13469 [,]cicc 13472 ...cfz 13632 ..^cfzo 13781 πcpi 16225 –cn→ccncf 25190

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 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 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 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ioc 13474 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-seq 14138 df-exp 14198 df-fac 14411 df-bc 14440 df-hash 14468 df-shft 15213 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-limsup 15631 df-clim 15648 df-rlim 15649 df-sum 15847 df-ef 16226 df-sin 16228 df-cos 16229 df-pi 16231 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-mulg 19271 df-cntz 19524 df-cmn 19989 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cn 23538 df-cnp 23539 df-haus 23626 df-tx 23874 df-hmeo 24067 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-xms 24632 df-ms 24633 df-tms 24634 df-cncf 25192 df-limc 26179 df-dv 26180

This theorem is used by: fourierdlem102 47187 fourierdlem114 47199

W3C validator