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Theorem fourierdlem71 46192
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 9363 . . . 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 480 . . . . . 6 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝐹:dom 𝐹⟶ℝ)
5 simpl 482 . . . . . . 7 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝜑)
6 simpr 484 . . . . . . . 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 7464 . . . . . . . . . . . . 13 (0...𝑀) ∈ V
98a1i 11 . . . . . . . . . . . 12 (𝜑 → (0...𝑀) ∈ V)
107, 9fexd 7247 . . . . . . . . . . 11 (𝜑𝑄 ∈ V)
11 rnexg 7924 . . . . . . . . . . 11 (𝑄 ∈ V → ran 𝑄 ∈ V)
12 inex1g 5319 . . . . . . . . . . 11 (ran 𝑄 ∈ V → (ran 𝑄 ∩ dom 𝐹) ∈ V)
1310, 11, 123syl 18 . . . . . . . . . 10 (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ V)
1413adantr 480 . . . . . . . . 9 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (ran 𝑄 ∩ dom 𝐹) ∈ V)
15 fourierdlem71.i . . . . . . . . . . . . . 14 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
16 ovex 7464 . . . . . . . . . . . . . . 15 (0..^𝑀) ∈ V
1716mptex 7243 . . . . . . . . . . . . . 14 (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ V
1815, 17eqeltri 2837 . . . . . . . . . . . . 13 𝐼 ∈ V
1918rnex 7932 . . . . . . . . . . . 12 ran 𝐼 ∈ V
2019a1i 11 . . . . . . . . . . 11 (𝜑 → ran 𝐼 ∈ V)
2120uniexd 7762 . . . . . . . . . 10 (𝜑 ran 𝐼 ∈ V)
2221adantr 480 . . . . . . . . 9 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → ran 𝐼 ∈ V)
23 uniprg 4923 . . . . . . . . 9 (((ran 𝑄 ∩ dom 𝐹) ∈ V ∧ ran 𝐼 ∈ V) → {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
2414, 22, 23syl2anc 584 . . . . . . . 8 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
256, 24eleqtrd 2843 . . . . . . 7 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
26 elinel2 4202 . . . . . . . . 9 (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
2726adantl 481 . . . . . . . 8 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
28 simpll 767 . . . . . . . . 9 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
29 elunnel1 4154 . . . . . . . . . 10 ((𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ran 𝐼)
3029adantll 714 . . . . . . . . 9 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ran 𝐼)
3115funmpt2 6605 . . . . . . . . . . . . 13 Fun 𝐼
32 elunirn 7271 . . . . . . . . . . . . 13 (Fun 𝐼 → (𝑥 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖)))
3331, 32ax-mp 5 . . . . . . . . . . . 12 (𝑥 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
3433biimpi 216 . . . . . . . . . . 11 (𝑥 ran 𝐼 → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
3534adantl 481 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
36 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ dom 𝐼𝑖 ∈ dom 𝐼)
37 ovex 7464 . . . . . . . . . . . . . . . . . . . 20 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
3837, 15dmmpti 6712 . . . . . . . . . . . . . . . . . . 19 dom 𝐼 = (0..^𝑀)
3936, 38eleqtrdi 2851 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ dom 𝐼𝑖 ∈ (0..^𝑀))
4039adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
4137a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V)
4215fvmpt2 7027 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
4340, 41, 42syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
44 fourierdlem71.fcn . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
45 cncff 24919 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
46 fdm 6745 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
4744, 45, 463syl 18 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
4839, 47sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
49 ssdmres 6031 . . . . . . . . . . . . . . . . 17 (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5048, 49sylibr 234 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
5143, 50eqsstrd 4018 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ dom 𝐹)
52513adant3 1133 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ dom 𝐹)
53 simp3 1139 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ (𝐼𝑖))
5452, 53sseldd 3984 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ dom 𝐹)
55543exp 1120 . . . . . . . . . . . 12 (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹)))
5655adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹)))
5756rexlimdv 3153 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹))
5835, 57mpd 15 . . . . . . . . 9 ((𝜑𝑥 ran 𝐼) → 𝑥 ∈ dom 𝐹)
5928, 30, 58syl2anc 584 . . . . . . . 8 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
6027, 59pm2.61dan 813 . . . . . . 7 ((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) → 𝑥 ∈ dom 𝐹)
615, 25, 60syl2anc 584 . . . . . 6 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 ∈ dom 𝐹)
624, 61ffvelcdmd 7105 . . . . 5 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (𝐹𝑥) ∈ ℝ)
6362recnd 11289 . . . 4 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (𝐹𝑥) ∈ ℂ)
6463abscld 15475 . . 3 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (abs‘(𝐹𝑥)) ∈ ℝ)
65 simpr 484 . . . . . . 7 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
66 fzfid 14014 . . . . . . . . . 10 (𝜑 → (0...𝑀) ∈ Fin)
67 rnffi 45180 . . . . . . . . . 10 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
687, 66, 67syl2anc 584 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ Fin)
69 infi 9302 . . . . . . . . 9 (ran 𝑄 ∈ Fin → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7068, 69syl 17 . . . . . . . 8 (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7170adantr 480 . . . . . . 7 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7265, 71eqeltrd 2841 . . . . . 6 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 ∈ Fin)
73 simpll 767 . . . . . . . 8 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → 𝜑)
74 simpr 484 . . . . . . . . . 10 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑥𝑤)
75 simpl 482 . . . . . . . . . 10 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
7674, 75eleqtrd 2843 . . . . . . . . 9 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
7776adantll 714 . . . . . . . 8 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
783adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝐹:dom 𝐹⟶ℝ)
7926adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
8078, 79ffvelcdmd 7105 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹𝑥) ∈ ℝ)
8180recnd 11289 . . . . . . . . 9 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹𝑥) ∈ ℂ)
8281abscld 15475 . . . . . . . 8 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (abs‘(𝐹𝑥)) ∈ ℝ)
8373, 77, 82syl2anc 584 . . . . . . 7 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → (abs‘(𝐹𝑥)) ∈ ℝ)
8483ralrimiva 3146 . . . . . 6 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∀𝑥𝑤 (abs‘(𝐹𝑥)) ∈ ℝ)
85 fimaxre3 12214 . . . . . 6 ((𝑤 ∈ Fin ∧ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
8672, 84, 85syl2anc 584 . . . . 5 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
8786adantlr 715 . . . 4 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
88 simpll 767 . . . . 5 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
89 neqne 2948 . . . . . . 7 𝑤 = (ran 𝑄 ∩ dom 𝐹) → 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹))
90 elprn1 45648 . . . . . . 7 ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∧ 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
9189, 90sylan2 593 . . . . . 6 ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
9291adantll 714 . . . . 5 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
93 fzofi 14015 . . . . . . . 8 (0..^𝑀) ∈ Fin
9415rnmptfi 45176 . . . . . . . 8 ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
9593, 94ax-mp 5 . . . . . . 7 ran 𝐼 ∈ Fin
9695a1i 11 . . . . . 6 ((𝜑𝑤 = ran 𝐼) → ran 𝐼 ∈ Fin)
973adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → 𝐹:dom 𝐹⟶ℝ)
9897, 58ffvelcdmd 7105 . . . . . . . . 9 ((𝜑𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℝ)
9998recnd 11289 . . . . . . . 8 ((𝜑𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
10099adantlr 715 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
101100abscld 15475 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (abs‘(𝐹𝑥)) ∈ ℝ)
10237, 15fnmpti 6711 . . . . . . . . . . 11 𝐼 Fn (0..^𝑀)
103 fvelrnb 6969 . . . . . . . . . . 11 (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡))
104102, 103ax-mp 5 . . . . . . . . . 10 (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
105104biimpi 216 . . . . . . . . 9 (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
106105adantl 481 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
1077adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
108 elfzofz 13715 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
109108adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
110107, 109ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
111 fzofzp1 13803 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
112111adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
113107, 112ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
114 fourierdlem71.l . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
115 fourierdlem71.r . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
116110, 113, 44, 114, 115cncfioobd 45912 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
1171163adant3 1133 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
118 fvres 6925 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹𝑥))
119118fveq2d 6910 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = (abs‘(𝐹𝑥)))
120119breq1d 5153 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹𝑥)) ≤ 𝑏))
121120adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹𝑥)) ≤ 𝑏))
122121ralbidva 3176 . . . . . . . . . . . . . . 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))
123122rexbidv 3179 . . . . . . . . . . . . . 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))
1241233adant3 1133 . . . . . . . . . . . . 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))
12537, 42mpan2 691 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
126 id 22 . . . . . . . . . . . . . . . . 17 ((𝐼𝑖) = 𝑡 → (𝐼𝑖) = 𝑡)
127125, 126sylan9req 2798 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
1281273adant1 1131 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
129128raleqdv 3326 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏 ↔ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
130129rexbidv 3179 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
131124, 130bitrd 279 . . . . . . . . . . . 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))
132117, 131mpbid 232 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
1331323exp 1120 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)))
134133adantr 480 . . . . . . . . 9 ((𝜑𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)))
135134rexlimdv 3153 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
136106, 135mpd 15 . . . . . . 7 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
137136adantlr 715 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
138 eqimss 4042 . . . . . . 7 (𝑤 = ran 𝐼𝑤 ran 𝐼)
139138adantl 481 . . . . . 6 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
14096, 101, 137, 139ssfiunibd 45321 . . . . 5 ((𝜑𝑤 = ran 𝐼) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
14188, 92, 140syl2anc 584 . . . 4 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
14287, 141pm2.61dan 813 . . 3 ((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
143 simpr 484 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄)
144 elinel2 4202 . . . . . . . . . 10 (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
145144ad2antlr 727 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ dom 𝐹)
146143, 145elind 4200 . . . . . . . 8 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
147 elun1 4182 . . . . . . . 8 (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
148146, 147syl 17 . . . . . . 7 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
149 fourierdlem71.7 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℕ)
150149ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
1517ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
152 elinel1 4201 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ (𝐴[,]𝐵))
153152adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ (𝐴[,]𝐵))
154 fourierdlem71.q0 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄‘0) = 𝐴)
155154eqcomd 2743 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (𝑄‘0))
156155adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐴 = (𝑄‘0))
157 fourierdlem71.10 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄𝑀) = 𝐵)
158157eqcomd 2743 . . . . . . . . . . . . . . 15 (𝜑𝐵 = (𝑄𝑀))
159158adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐵 = (𝑄𝑀))
160156, 159oveq12d 7449 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
161153, 160eleqtrd 2843 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((𝑄‘0)[,](𝑄𝑀)))
162161adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((𝑄‘0)[,](𝑄𝑀)))
163 simpr 484 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ¬ 𝑥 ∈ ran 𝑄)
164 fveq2 6906 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
165164breq1d 5153 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑥 ↔ (𝑄𝑗) < 𝑥))
166165cbvrabv 3447 . . . . . . . . . . . 12 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}
167166supeq1i 9487 . . . . . . . . . . 11 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}, ℝ, < )
168150, 151, 162, 163, 167fourierdlem25 46147 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
16939ad2antrl 728 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑖 ∈ (0..^𝑀))
170 simprr 773 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑥 ∈ (𝐼𝑖))
171169, 125syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
172170, 171eleqtrd 2843 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
173169, 172jca 511 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
174 id 22 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0..^𝑀))
175174, 38eleqtrrdi 2852 . . . . . . . . . . . . . . 15 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ dom 𝐼)
176175ad2antrl 728 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑖 ∈ dom 𝐼)
177 simprr 773 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
178125eqcomd 2743 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
179178ad2antrl 728 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
180177, 179eleqtrd 2843 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ (𝐼𝑖))
181176, 180jca 511 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)))
182173, 181impbida 801 . . . . . . . . . . . 12 (𝜑 → ((𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) ↔ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))))
183182rexbidv2 3175 . . . . . . . . . . 11 (𝜑 → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
184183ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
185168, 184mpbird 257 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
186185, 33sylibr 234 . . . . . . . 8 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ran 𝐼)
187 elun2 4183 . . . . . . . 8 (𝑥 ran 𝐼𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
188186, 187syl 17 . . . . . . 7 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
189148, 188pm2.61dan 813 . . . . . 6 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
190189ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
191 dfss3 3972 . . . . 5 (((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼) ↔ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
192190, 191sylibr 234 . . . 4 (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
19313, 21, 23syl2anc 584 . . . 4 (𝜑 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
194192, 193sseqtrrd 4021 . . 3 (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼})
1952, 64, 142, 194ssfiunibd 45321 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
196 nfv 1914 . . . . . 6 𝑥𝜑
197 nfra1 3284 . . . . . 6 𝑥𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦
198196, 197nfan 1899 . . . . 5 𝑥(𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
199 fourierdlem71.dmf . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℝ)
200199sselda 3983 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝑥 ∈ ℝ)
201 fourierdlem71.b . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
202201adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ dom 𝐹) → 𝐵 ∈ ℝ)
203202, 200resubcld 11691 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → (𝐵𝑥) ∈ ℝ)
204 fourierdlem71.t . . . . . . . . . . . . . . . . . . 19 𝑇 = (𝐵𝐴)
205 fourierdlem71.a . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ∈ ℝ)
206201, 205resubcld 11691 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐵𝐴) ∈ ℝ)
207204, 206eqeltrid 2845 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ∈ ℝ)
208207adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ∈ ℝ)
209 fourierdlem71.altb . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐴 < 𝐵)
210205, 201posdifd 11850 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))
211209, 210mpbid 232 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 < (𝐵𝐴))
212211, 204breqtrrdi 5185 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 0 < 𝑇)
213212gt0ne0d 11827 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ≠ 0)
214213adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ≠ 0)
215203, 208, 214redivcld 12095 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐵𝑥) / 𝑇) ∈ ℝ)
216215flcld 13838 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ)
217216zred 12722 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℝ)
218217, 208remulcld 11291 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇) ∈ ℝ)
219200, 218readdcld 11290 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ ℝ)
220 fourierdlem71.e . . . . . . . . . . . . 13 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
221220fvmpt2 7027 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ ℝ) → (𝐸𝑥) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
222200, 219, 221syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
223222fveq2d 6910 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹‘(𝐸𝑥)) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
224 fvex 6919 . . . . . . . . . . . 12 (⌊‘((𝐵𝑥) / 𝑇)) ∈ V
225 eleq1 2829 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ))
226225anbi2d 630 . . . . . . . . . . . . 13 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) ↔ ((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ)))
227 oveq1 7438 . . . . . . . . . . . . . . . 16 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))
228227oveq2d 7447 . . . . . . . . . . . . . . 15 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
229228fveq2d 6910 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
230229eqeq1d 2739 . . . . . . . . . . . . 13 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥)))
231226, 230imbi12d 344 . . . . . . . . . . . 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))))
232 fourierdlem71.fxpt . . . . . . . . . . . 12 (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))
233224, 231, 232vtocl 3558 . . . . . . . . . . 11 (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥))
234216, 233mpdan 687 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥))
235223, 234eqtr2d 2778 . . . . . . . . 9 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹‘(𝐸𝑥)))
236235fveq2d 6910 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) = (abs‘(𝐹‘(𝐸𝑥))))
237236adantlr 715 . . . . . . 7 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) = (abs‘(𝐹‘(𝐸𝑥))))
238 fveq2 6906 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
239238fveq2d 6910 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (abs‘(𝐹𝑥)) = (abs‘(𝐹𝑤)))
240239breq1d 5153 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((abs‘(𝐹𝑥)) ≤ 𝑦 ↔ (abs‘(𝐹𝑤)) ≤ 𝑦))
241240cbvralvw 3237 . . . . . . . . . 10 (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 ↔ ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
242241biimpi 216 . . . . . . . . 9 (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
243242ad2antlr 727 . . . . . . . 8 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
244 iocssicc 13477 . . . . . . . . . . 11 (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
245205adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → 𝐴 ∈ ℝ)
246209adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → 𝐴 < 𝐵)
247 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦𝑥 = 𝑦)
248 oveq2 7439 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
249248oveq1d 7446 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐵𝑥) / 𝑇) = ((𝐵𝑦) / 𝑇))
250249fveq2d 6910 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (⌊‘((𝐵𝑥) / 𝑇)) = (⌊‘((𝐵𝑦) / 𝑇)))
251250oveq1d 7446 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇))
252247, 251oveq12d 7449 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) = (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
253252cbvmptv 5255 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
254220, 253eqtri 2765 . . . . . . . . . . . . 13 𝐸 = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
255245, 202, 246, 204, 254fourierdlem4 46126 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝐸:ℝ⟶(𝐴(,]𝐵))
256255, 200ffvelcdmd 7105 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴(,]𝐵))
257244, 256sselid 3981 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴[,]𝐵))
258228eleq1d 2826 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹 ↔ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹))
259226, 258imbi12d 344 . . . . . . . . . . . . 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)))
260 fourierdlem71.xpt . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹)
261224, 259, 260vtocl 3558 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
262216, 261mpdan 687 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
263222, 262eqeltrd 2841 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ dom 𝐹)
264257, 263elind 4200 . . . . . . . . 9 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
265264adantlr 715 . . . . . . . 8 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
266 fveq2 6906 . . . . . . . . . . 11 (𝑤 = (𝐸𝑥) → (𝐹𝑤) = (𝐹‘(𝐸𝑥)))
267266fveq2d 6910 . . . . . . . . . 10 (𝑤 = (𝐸𝑥) → (abs‘(𝐹𝑤)) = (abs‘(𝐹‘(𝐸𝑥))))
268267breq1d 5153 . . . . . . . . 9 (𝑤 = (𝐸𝑥) → ((abs‘(𝐹𝑤)) ≤ 𝑦 ↔ (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦))
269268rspccva 3621 . . . . . . . 8 ((∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦 ∧ (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦)
270243, 265, 269syl2anc 584 . . . . . . 7 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦)
271237, 270eqbrtrd 5165 . . . . . 6 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) ≤ 𝑦)
272271ex 412 . . . . 5 ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) → (𝑥 ∈ dom 𝐹 → (abs‘(𝐹𝑥)) ≤ 𝑦))
273198, 272ralrimi 3257 . . . 4 ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦)
274273ex 412 . . 3 (𝜑 → (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦))
275274reximdv 3170 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦))
276195, 275mpd 15 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cun 3949  cin 3950  wss 3951  {cpr 4628   cuni 4907   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  supcsup 9480  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  cz 12613  (,)cioo 13387  (,]cioc 13388  [,]cicc 13390  ...cfz 13547  ..^cfzo 13694  cfl 13830  abscabs 15273  cnccncf 24902   lim climc 25897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-cn 23235  df-cnp 23236  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901
This theorem is referenced by:  fourierdlem94  46215  fourierdlem113  46234
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