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Theorem fourierdlem71 42806
Description: A periodic piecewise continuous function, possibly undefined on a finite set in each periodic interval, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem71.dmf (𝜑 → dom 𝐹 ⊆ ℝ)
fourierdlem71.f (𝜑𝐹:dom 𝐹⟶ℝ)
fourierdlem71.a (𝜑𝐴 ∈ ℝ)
fourierdlem71.b (𝜑𝐵 ∈ ℝ)
fourierdlem71.altb (𝜑𝐴 < 𝐵)
fourierdlem71.t 𝑇 = (𝐵𝐴)
fourierdlem71.7 (𝜑𝑀 ∈ ℕ)
fourierdlem71.q (𝜑𝑄:(0...𝑀)⟶ℝ)
fourierdlem71.q0 (𝜑 → (𝑄‘0) = 𝐴)
fourierdlem71.10 (𝜑 → (𝑄𝑀) = 𝐵)
fourierdlem71.fcn ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem71.r ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
fourierdlem71.l ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
fourierdlem71.xpt (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹)
fourierdlem71.fxpt (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))
fourierdlem71.i 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
fourierdlem71.e 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
Assertion
Ref Expression
fourierdlem71 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦)
Distinct variable groups:   𝑥,𝐴,𝑦   𝐵,𝑘,𝑥   𝑦,𝐵   𝑖,𝐹,𝑥,𝑘   𝑦,𝐹   𝑖,𝐼,𝑥   𝑦,𝐼   𝑥,𝐿   𝑖,𝑀,𝑥,𝑘   𝑄,𝑖,𝑥,𝑘   𝑦,𝑄   𝑥,𝑅   𝑇,𝑘,𝑥   𝑦,𝑇   𝜑,𝑖,𝑥,𝑘   𝜑,𝑦
Allowed substitution hints:   𝐴(𝑖,𝑘)   𝐵(𝑖)   𝑅(𝑦,𝑖,𝑘)   𝑇(𝑖)   𝐸(𝑥,𝑦,𝑖,𝑘)   𝐼(𝑘)   𝐿(𝑦,𝑖,𝑘)   𝑀(𝑦)

Proof of Theorem fourierdlem71
Dummy variables 𝑤 𝑏 𝑡 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfi 8781 . . . 4 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∈ Fin
21a1i 11 . . 3 (𝜑 → {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∈ Fin)
3 fourierdlem71.f . . . . . . 7 (𝜑𝐹:dom 𝐹⟶ℝ)
43adantr 484 . . . . . 6 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝐹:dom 𝐹⟶ℝ)
5 simpl 486 . . . . . . 7 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝜑)
6 simpr 488 . . . . . . . 8 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼})
7 fourierdlem71.q . . . . . . . . . . . 12 (𝜑𝑄:(0...𝑀)⟶ℝ)
8 ovex 7172 . . . . . . . . . . . . 13 (0...𝑀) ∈ V
98a1i 11 . . . . . . . . . . . 12 (𝜑 → (0...𝑀) ∈ V)
10 fex 6970 . . . . . . . . . . . 12 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
117, 9, 10syl2anc 587 . . . . . . . . . . 11 (𝜑𝑄 ∈ V)
12 rnexg 7599 . . . . . . . . . . 11 (𝑄 ∈ V → ran 𝑄 ∈ V)
13 inex1g 5190 . . . . . . . . . . 11 (ran 𝑄 ∈ V → (ran 𝑄 ∩ dom 𝐹) ∈ V)
1411, 12, 133syl 18 . . . . . . . . . 10 (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ V)
1514adantr 484 . . . . . . . . 9 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (ran 𝑄 ∩ dom 𝐹) ∈ V)
16 fourierdlem71.i . . . . . . . . . . . . . 14 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
17 ovex 7172 . . . . . . . . . . . . . . 15 (0..^𝑀) ∈ V
1817mptex 6967 . . . . . . . . . . . . . 14 (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ V
1916, 18eqeltri 2889 . . . . . . . . . . . . 13 𝐼 ∈ V
2019rnex 7603 . . . . . . . . . . . 12 ran 𝐼 ∈ V
2120a1i 11 . . . . . . . . . . 11 (𝜑 → ran 𝐼 ∈ V)
2221uniexd 7452 . . . . . . . . . 10 (𝜑 ran 𝐼 ∈ V)
2322adantr 484 . . . . . . . . 9 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → ran 𝐼 ∈ V)
24 uniprg 4821 . . . . . . . . 9 (((ran 𝑄 ∩ dom 𝐹) ∈ V ∧ ran 𝐼 ∈ V) → {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
2515, 23, 24syl2anc 587 . . . . . . . 8 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
266, 25eleqtrd 2895 . . . . . . 7 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
27 elinel2 4126 . . . . . . . . 9 (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
2827adantl 485 . . . . . . . 8 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
29 simpll 766 . . . . . . . . 9 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
30 elunnel1 4080 . . . . . . . . . 10 ((𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ran 𝐼)
3130adantll 713 . . . . . . . . 9 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ran 𝐼)
3216funmpt2 6367 . . . . . . . . . . . . 13 Fun 𝐼
33 elunirn 6992 . . . . . . . . . . . . 13 (Fun 𝐼 → (𝑥 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖)))
3432, 33ax-mp 5 . . . . . . . . . . . 12 (𝑥 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
3534biimpi 219 . . . . . . . . . . 11 (𝑥 ran 𝐼 → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
3635adantl 485 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
37 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ dom 𝐼𝑖 ∈ dom 𝐼)
38 ovex 7172 . . . . . . . . . . . . . . . . . . . 20 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
3938, 16dmmpti 6468 . . . . . . . . . . . . . . . . . . 19 dom 𝐼 = (0..^𝑀)
4037, 39eleqtrdi 2903 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ dom 𝐼𝑖 ∈ (0..^𝑀))
4140adantl 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
4238a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V)
4316fvmpt2 6760 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
4441, 42, 43syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
45 fourierdlem71.fcn . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
46 cncff 23501 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
47 fdm 6499 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
4845, 46, 473syl 18 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
4940, 48sylan2 595 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
50 ssdmres 5845 . . . . . . . . . . . . . . . . 17 (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5149, 50sylibr 237 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
5244, 51eqsstrd 3956 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ dom 𝐹)
53523adant3 1129 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ dom 𝐹)
54 simp3 1135 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ (𝐼𝑖))
5553, 54sseldd 3919 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ dom 𝐹)
56553exp 1116 . . . . . . . . . . . 12 (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹)))
5756adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹)))
5857rexlimdv 3245 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹))
5936, 58mpd 15 . . . . . . . . 9 ((𝜑𝑥 ran 𝐼) → 𝑥 ∈ dom 𝐹)
6029, 31, 59syl2anc 587 . . . . . . . 8 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
6128, 60pm2.61dan 812 . . . . . . 7 ((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) → 𝑥 ∈ dom 𝐹)
625, 26, 61syl2anc 587 . . . . . 6 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 ∈ dom 𝐹)
634, 62ffvelrnd 6833 . . . . 5 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (𝐹𝑥) ∈ ℝ)
6463recnd 10662 . . . 4 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (𝐹𝑥) ∈ ℂ)
6564abscld 14791 . . 3 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (abs‘(𝐹𝑥)) ∈ ℝ)
66 simpr 488 . . . . . . 7 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
67 fzfid 13340 . . . . . . . . . 10 (𝜑 → (0...𝑀) ∈ Fin)
68 rnffi 41786 . . . . . . . . . 10 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
697, 67, 68syl2anc 587 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ Fin)
70 infi 8730 . . . . . . . . 9 (ran 𝑄 ∈ Fin → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7169, 70syl 17 . . . . . . . 8 (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7271adantr 484 . . . . . . 7 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7366, 72eqeltrd 2893 . . . . . 6 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 ∈ Fin)
74 simpll 766 . . . . . . . 8 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → 𝜑)
75 simpr 488 . . . . . . . . . 10 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑥𝑤)
76 simpl 486 . . . . . . . . . 10 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
7775, 76eleqtrd 2895 . . . . . . . . 9 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
7877adantll 713 . . . . . . . 8 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
793adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝐹:dom 𝐹⟶ℝ)
8027adantl 485 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
8179, 80ffvelrnd 6833 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹𝑥) ∈ ℝ)
8281recnd 10662 . . . . . . . . 9 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹𝑥) ∈ ℂ)
8382abscld 14791 . . . . . . . 8 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (abs‘(𝐹𝑥)) ∈ ℝ)
8474, 78, 83syl2anc 587 . . . . . . 7 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → (abs‘(𝐹𝑥)) ∈ ℝ)
8584ralrimiva 3152 . . . . . 6 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∀𝑥𝑤 (abs‘(𝐹𝑥)) ∈ ℝ)
86 fimaxre3 11579 . . . . . 6 ((𝑤 ∈ Fin ∧ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
8773, 85, 86syl2anc 587 . . . . 5 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
8887adantlr 714 . . . 4 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
89 simpll 766 . . . . 5 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
90 neqne 2998 . . . . . . 7 𝑤 = (ran 𝑄 ∩ dom 𝐹) → 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹))
91 elprn1 42262 . . . . . . 7 ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∧ 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
9290, 91sylan2 595 . . . . . 6 ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
9392adantll 713 . . . . 5 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
94 fzofi 13341 . . . . . . . 8 (0..^𝑀) ∈ Fin
9516rnmptfi 41782 . . . . . . . 8 ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
9694, 95ax-mp 5 . . . . . . 7 ran 𝐼 ∈ Fin
9796a1i 11 . . . . . 6 ((𝜑𝑤 = ran 𝐼) → ran 𝐼 ∈ Fin)
983adantr 484 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → 𝐹:dom 𝐹⟶ℝ)
9998, 59ffvelrnd 6833 . . . . . . . . 9 ((𝜑𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℝ)
10099recnd 10662 . . . . . . . 8 ((𝜑𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
101100adantlr 714 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
102101abscld 14791 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (abs‘(𝐹𝑥)) ∈ ℝ)
10338, 16fnmpti 6467 . . . . . . . . . . 11 𝐼 Fn (0..^𝑀)
104 fvelrnb 6705 . . . . . . . . . . 11 (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡))
105103, 104ax-mp 5 . . . . . . . . . 10 (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
106105biimpi 219 . . . . . . . . 9 (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
107106adantl 485 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
1087adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109 elfzofz 13052 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
110109adantl 485 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
111108, 110ffvelrnd 6833 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
112 fzofzp1 13133 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
113112adantl 485 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
114108, 113ffvelrnd 6833 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
115 fourierdlem71.l . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
116 fourierdlem71.r . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
117111, 114, 45, 115, 116cncfioobd 42526 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
1181173adant3 1129 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
119 fvres 6668 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹𝑥))
120119fveq2d 6653 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = (abs‘(𝐹𝑥)))
121120breq1d 5043 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹𝑥)) ≤ 𝑏))
122121adantl 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹𝑥)) ≤ 𝑏))
123122ralbidva 3164 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏))
124123rexbidv 3259 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏))
1251243adant3 1129 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏))
12638, 43mpan2 690 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
127 id 22 . . . . . . . . . . . . . . . . 17 ((𝐼𝑖) = 𝑡 → (𝐼𝑖) = 𝑡)
128126, 127sylan9req 2857 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
1291283adant1 1127 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
130129raleqdv 3367 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏 ↔ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
131130rexbidv 3259 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
132125, 131bitrd 282 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
133118, 132mpbid 235 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
1341333exp 1116 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)))
135134adantr 484 . . . . . . . . 9 ((𝜑𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)))
136135rexlimdv 3245 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
137107, 136mpd 15 . . . . . . 7 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
138137adantlr 714 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
139 eqimss 3974 . . . . . . 7 (𝑤 = ran 𝐼𝑤 ran 𝐼)
140139adantl 485 . . . . . 6 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
14197, 102, 138, 140ssfiunibd 41928 . . . . 5 ((𝜑𝑤 = ran 𝐼) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
14289, 93, 141syl2anc 587 . . . 4 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
14388, 142pm2.61dan 812 . . 3 ((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
144 simpr 488 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄)
145 elinel2 4126 . . . . . . . . . 10 (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
146145ad2antlr 726 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ dom 𝐹)
147144, 146elind 4124 . . . . . . . 8 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
148 elun1 4106 . . . . . . . 8 (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
149147, 148syl 17 . . . . . . 7 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
150 fourierdlem71.7 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℕ)
151150ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
1527ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
153 elinel1 4125 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ (𝐴[,]𝐵))
154153adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ (𝐴[,]𝐵))
155 fourierdlem71.q0 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄‘0) = 𝐴)
156155eqcomd 2807 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (𝑄‘0))
157156adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐴 = (𝑄‘0))
158 fourierdlem71.10 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄𝑀) = 𝐵)
159158eqcomd 2807 . . . . . . . . . . . . . . 15 (𝜑𝐵 = (𝑄𝑀))
160159adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐵 = (𝑄𝑀))
161157, 160oveq12d 7157 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
162154, 161eleqtrd 2895 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((𝑄‘0)[,](𝑄𝑀)))
163162adantr 484 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((𝑄‘0)[,](𝑄𝑀)))
164 simpr 488 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ¬ 𝑥 ∈ ran 𝑄)
165 fveq2 6649 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
166165breq1d 5043 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑥 ↔ (𝑄𝑗) < 𝑥))
167166cbvrabv 3442 . . . . . . . . . . . 12 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}
168167supeq1i 8899 . . . . . . . . . . 11 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}, ℝ, < )
169151, 152, 163, 164, 168fourierdlem25 42761 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
17040ad2antrl 727 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑖 ∈ (0..^𝑀))
171 simprr 772 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑥 ∈ (𝐼𝑖))
172170, 126syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
173171, 172eleqtrd 2895 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
174170, 173jca 515 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
175 id 22 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0..^𝑀))
176175, 39eleqtrrdi 2904 . . . . . . . . . . . . . . 15 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ dom 𝐼)
177176ad2antrl 727 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑖 ∈ dom 𝐼)
178 simprr 772 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
179126eqcomd 2807 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
180179ad2antrl 727 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
181178, 180eleqtrd 2895 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ (𝐼𝑖))
182177, 181jca 515 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)))
183174, 182impbida 800 . . . . . . . . . . . 12 (𝜑 → ((𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) ↔ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))))
184183rexbidv2 3257 . . . . . . . . . . 11 (𝜑 → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
185184ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
186169, 185mpbird 260 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
187186, 34sylibr 237 . . . . . . . 8 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ran 𝐼)
188 elun2 4107 . . . . . . . 8 (𝑥 ran 𝐼𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
189187, 188syl 17 . . . . . . 7 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
190149, 189pm2.61dan 812 . . . . . 6 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
191190ralrimiva 3152 . . . . 5 (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
192 dfss3 3906 . . . . 5 (((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼) ↔ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
193191, 192sylibr 237 . . . 4 (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
19414, 22, 24syl2anc 587 . . . 4 (𝜑 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
195193, 194sseqtrrd 3959 . . 3 (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼})
1962, 65, 143, 195ssfiunibd 41928 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
197 nfv 1915 . . . . . 6 𝑥𝜑
198 nfra1 3186 . . . . . 6 𝑥𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦
199197, 198nfan 1900 . . . . 5 𝑥(𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
200 fourierdlem71.dmf . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℝ)
201200sselda 3918 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝑥 ∈ ℝ)
202 fourierdlem71.b . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
203202adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ dom 𝐹) → 𝐵 ∈ ℝ)
204203, 201resubcld 11061 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → (𝐵𝑥) ∈ ℝ)
205 fourierdlem71.t . . . . . . . . . . . . . . . . . . 19 𝑇 = (𝐵𝐴)
206 fourierdlem71.a . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ∈ ℝ)
207202, 206resubcld 11061 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐵𝐴) ∈ ℝ)
208205, 207eqeltrid 2897 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ∈ ℝ)
209208adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ∈ ℝ)
210 fourierdlem71.altb . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐴 < 𝐵)
211206, 202posdifd 11220 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))
212210, 211mpbid 235 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 < (𝐵𝐴))
213212, 205breqtrrdi 5075 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 0 < 𝑇)
214213gt0ne0d 11197 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ≠ 0)
215214adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ≠ 0)
216204, 209, 215redivcld 11461 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐵𝑥) / 𝑇) ∈ ℝ)
217216flcld 13167 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ)
218217zred 12079 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℝ)
219218, 209remulcld 10664 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇) ∈ ℝ)
220201, 219readdcld 10663 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ ℝ)
221 fourierdlem71.e . . . . . . . . . . . . 13 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
222221fvmpt2 6760 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ ℝ) → (𝐸𝑥) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
223201, 220, 222syl2anc 587 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
224223fveq2d 6653 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹‘(𝐸𝑥)) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
225 fvex 6662 . . . . . . . . . . . 12 (⌊‘((𝐵𝑥) / 𝑇)) ∈ V
226 eleq1 2880 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ))
227226anbi2d 631 . . . . . . . . . . . . 13 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) ↔ ((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ)))
228 oveq1 7146 . . . . . . . . . . . . . . . 16 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))
229228oveq2d 7155 . . . . . . . . . . . . . . 15 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
230229fveq2d 6653 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
231230eqeq1d 2803 . . . . . . . . . . . . 13 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥)))
232227, 231imbi12d 348 . . . . . . . . . . . 12 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥)) ↔ (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥))))
233 fourierdlem71.fxpt . . . . . . . . . . . 12 (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))
234225, 232, 233vtocl 3510 . . . . . . . . . . 11 (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥))
235217, 234mpdan 686 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥))
236224, 235eqtr2d 2837 . . . . . . . . 9 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹‘(𝐸𝑥)))
237236fveq2d 6653 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) = (abs‘(𝐹‘(𝐸𝑥))))
238237adantlr 714 . . . . . . 7 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) = (abs‘(𝐹‘(𝐸𝑥))))
239 fveq2 6649 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
240239fveq2d 6653 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (abs‘(𝐹𝑥)) = (abs‘(𝐹𝑤)))
241240breq1d 5043 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((abs‘(𝐹𝑥)) ≤ 𝑦 ↔ (abs‘(𝐹𝑤)) ≤ 𝑦))
242241cbvralvw 3399 . . . . . . . . . 10 (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 ↔ ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
243242biimpi 219 . . . . . . . . 9 (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
244243ad2antlr 726 . . . . . . . 8 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
245 iocssicc 12819 . . . . . . . . . . 11 (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
246206adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → 𝐴 ∈ ℝ)
247210adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → 𝐴 < 𝐵)
248 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦𝑥 = 𝑦)
249 oveq2 7147 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
250249oveq1d 7154 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐵𝑥) / 𝑇) = ((𝐵𝑦) / 𝑇))
251250fveq2d 6653 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (⌊‘((𝐵𝑥) / 𝑇)) = (⌊‘((𝐵𝑦) / 𝑇)))
252251oveq1d 7154 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇))
253248, 252oveq12d 7157 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) = (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
254253cbvmptv 5136 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
255221, 254eqtri 2824 . . . . . . . . . . . . 13 𝐸 = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
256246, 203, 247, 205, 255fourierdlem4 42740 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝐸:ℝ⟶(𝐴(,]𝐵))
257256, 201ffvelrnd 6833 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴(,]𝐵))
258245, 257sseldi 3916 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴[,]𝐵))
259229eleq1d 2877 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹 ↔ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹))
260227, 259imbi12d 348 . . . . . . . . . . . . 13 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹) ↔ (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)))
261 fourierdlem71.xpt . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹)
262225, 260, 261vtocl 3510 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
263217, 262mpdan 686 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
264223, 263eqeltrd 2893 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ dom 𝐹)
265258, 264elind 4124 . . . . . . . . 9 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
266265adantlr 714 . . . . . . . 8 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
267 fveq2 6649 . . . . . . . . . . 11 (𝑤 = (𝐸𝑥) → (𝐹𝑤) = (𝐹‘(𝐸𝑥)))
268267fveq2d 6653 . . . . . . . . . 10 (𝑤 = (𝐸𝑥) → (abs‘(𝐹𝑤)) = (abs‘(𝐹‘(𝐸𝑥))))
269268breq1d 5043 . . . . . . . . 9 (𝑤 = (𝐸𝑥) → ((abs‘(𝐹𝑤)) ≤ 𝑦 ↔ (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦))
270269rspccva 3573 . . . . . . . 8 ((∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦 ∧ (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦)
271244, 266, 270syl2anc 587 . . . . . . 7 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦)
272238, 271eqbrtrd 5055 . . . . . 6 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) ≤ 𝑦)
273272ex 416 . . . . 5 ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) → (𝑥 ∈ dom 𝐹 → (abs‘(𝐹𝑥)) ≤ 𝑦))
274199, 273ralrimi 3183 . . . 4 ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦)
275274ex 416 . . 3 (𝜑 → (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦))
276275reximdv 3235 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦))
277196, 276mpd 15 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  cun 3882  cin 3883  wss 3884  {cpr 4530   cuni 4803   class class class wbr 5033  cmpt 5113  dom cdm 5523  ran crn 5524  cres 5525  Fun wfun 6322   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  Fincfn 8496  supcsup 8892  cc 10528  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   · cmul 10535   < clt 10668  cle 10669  cmin 10863   / cdiv 11290  cn 11629  cz 11973  (,)cioo 12730  (,]cioc 12731  [,]cicc 12733  ...cfz 12889  ..^cfzo 13032  cfl 13159  abscabs 14588  cnccncf 23484   lim climc 24468
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-mulf 10610
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ioc 12735  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-hom 16584  df-cco 16585  df-rest 16691  df-topn 16692  df-0g 16710  df-gsum 16711  df-topgen 16712  df-pt 16713  df-prds 16716  df-xrs 16770  df-qtop 16775  df-imas 16776  df-xps 16778  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-mulg 18220  df-cntz 18442  df-cmn 18903  df-psmet 20086  df-xmet 20087  df-met 20088  df-bl 20089  df-mopn 20090  df-cnfld 20095  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629  df-cn 21835  df-cnp 21836  df-cmp 21995  df-tx 22170  df-hmeo 22363  df-xms 22930  df-ms 22931  df-tms 22932  df-cncf 23486  df-limc 24472
This theorem is referenced by:  fourierdlem94  42829  fourierdlem113  42848
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