fourierdlem38 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem38 46716

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 778	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
2		simplll 784	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝜑)
3		ioossicc 13437	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
4		pire 26516	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
5	4	renegcli 11492	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
6	5	rexri 11240	. . . . . . . . . . 11 ⊢ -π ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
8	4	rexri 11240	. . . . . . . . . . 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 46693	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
14	13	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
15		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
16	7, 9, 14, 15	fourierdlem8 46686	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
17	3, 16	sstrid 3947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
18	17	sselda 3936	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ (-π[,]π))
19	18	adantr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (-π[,]π))
20		simpr 488	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ dom 𝐹)
21		simpllr 785	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑖 ∈ (0..^𝑀))
22	11	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑀 ∈ ℕ)
23	12	3ad2ant1 1146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑄 ∈ (𝑃‘𝑀))
24		simp2 1150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (-π[,]π))
25		simp3 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ dom 𝐹)
26	24, 25	eldifd 3915	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ((-π[,]π) ∖ dom 𝐹))
27		elun2 4135	. . . . . . . . 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 2814	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)) = ran 𝑄)
32	31	3ad2ant1 1146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)) = ran 𝑄)
33	28, 32	eleqtrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ran 𝑄)
34	10, 22, 23, 33	fourierdlem12 46690	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (-π[,]π) ∧ ¬ 𝑥 ∈ dom 𝐹) ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
35	2, 19, 20, 21, 34	syl31anc 1392	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ ¬ 𝑥 ∈ dom 𝐹) → ¬ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
36	1, 35	condan 827	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑥 ∈ dom 𝐹)
37	36	ralrimiva 3154	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 ∈ dom 𝐹)
38		dfss3 3925	. . 3 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 ∈ dom 𝐹)
39	37, 38	sylibr 236	. 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 484	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹 ∈ (dom 𝐹–cn→ℂ))
42		rescncf 24956	. 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 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ∀wral 3076 {crab 3414 ∖ cdif 3901 ∪ cun 3902 ⊆ wss 3904 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5647 ran crn 5648 ↾ cres 5649 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ↑_m cmap 8808 ℂcc 11071 ℝcr 11072 0cc0 11073 1c1 11074 + caddc 11076 ℝ^*cxr 11215 < clt 11216 -cneg 11415 ℕcn 12210 (,)cioo 13349 [,]cicc 13352 ...cfz 13512 ..^cfzo 13659 πcpi 16096 –cn→ccncf 24935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-seq 14015 df-exp 14075 df-fac 14287 df-bc 14316 df-hash 14344 df-shft 15080 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-limsup 15498 df-clim 15515 df-rlim 15516 df-sum 15714 df-ef 16097 df-sin 16099 df-cos 16100 df-pi 16102 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21413 df-xmet 21414 df-met 21415 df-bl 21416 df-mopn 21417 df-fbas 21418 df-fg 21419 df-cnfld 21422 df-top 22951 df-topon 22968 df-topsp 22990 df-bases 23003 df-cld 23076 df-ntr 23077 df-cls 23078 df-nei 23155 df-lp 23193 df-perf 23194 df-cn 23284 df-cnp 23285 df-haus 23372 df-tx 23619 df-hmeo 23812 df-fil 23903 df-fm 23995 df-flim 23996 df-flf 23997 df-xms 24377 df-ms 24378 df-tms 24379 df-cncf 24937 df-limc 25925 df-dv 25926

This theorem is referenced by: fourierdlem102 46779 fourierdlem114 46791

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