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Theorem fourierdlem71 45832
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 9356 . . . 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 479 . . . . . 6 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝐹:dom 𝐹⟶ℝ)
5 simpl 481 . . . . . . 7 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝜑)
6 simpr 483 . . . . . . . 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 7447 . . . . . . . . . . . . 13 (0...𝑀) ∈ V
98a1i 11 . . . . . . . . . . . 12 (𝜑 → (0...𝑀) ∈ V)
107, 9fexd 7234 . . . . . . . . . . 11 (𝜑𝑄 ∈ V)
11 rnexg 7905 . . . . . . . . . . 11 (𝑄 ∈ V → ran 𝑄 ∈ V)
12 inex1g 5315 . . . . . . . . . . 11 (ran 𝑄 ∈ V → (ran 𝑄 ∩ dom 𝐹) ∈ V)
1310, 11, 123syl 18 . . . . . . . . . 10 (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ V)
1413adantr 479 . . . . . . . . 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 7447 . . . . . . . . . . . . . . 15 (0..^𝑀) ∈ V
1716mptex 7230 . . . . . . . . . . . . . 14 (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ V
1815, 17eqeltri 2822 . . . . . . . . . . . . 13 𝐼 ∈ V
1918rnex 7913 . . . . . . . . . . . 12 ran 𝐼 ∈ V
2019a1i 11 . . . . . . . . . . 11 (𝜑 → ran 𝐼 ∈ V)
2120uniexd 7743 . . . . . . . . . 10 (𝜑 ran 𝐼 ∈ V)
2221adantr 479 . . . . . . . . 9 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → ran 𝐼 ∈ V)
23 uniprg 4922 . . . . . . . . 9 (((ran 𝑄 ∩ dom 𝐹) ∈ V ∧ ran 𝐼 ∈ V) → {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
2414, 22, 23syl2anc 582 . . . . . . . 8 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
256, 24eleqtrd 2828 . . . . . . 7 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
26 elinel2 4195 . . . . . . . . 9 (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
2726adantl 480 . . . . . . . 8 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
28 simpll 765 . . . . . . . . 9 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
29 elunnel1 4147 . . . . . . . . . 10 ((𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ran 𝐼)
3029adantll 712 . . . . . . . . 9 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ran 𝐼)
3115funmpt2 6588 . . . . . . . . . . . . 13 Fun 𝐼
32 elunirn 7256 . . . . . . . . . . . . 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 215 . . . . . . . . . . 11 (𝑥 ran 𝐼 → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
3534adantl 480 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
36 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ dom 𝐼𝑖 ∈ dom 𝐼)
37 ovex 7447 . . . . . . . . . . . . . . . . . . . 20 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
3837, 15dmmpti 6695 . . . . . . . . . . . . . . . . . . 19 dom 𝐼 = (0..^𝑀)
3936, 38eleqtrdi 2836 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ dom 𝐼𝑖 ∈ (0..^𝑀))
4039adantl 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
4137a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V)
4215fvmpt2 7010 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
4340, 41, 42syl2anc 582 . . . . . . . . . . . . . . . 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 24899 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
46 fdm 6727 . . . . . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
49 ssdmres 6013 . . . . . . . . . . . . . . . . 17 (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5048, 49sylibr 233 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
5143, 50eqsstrd 4018 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ dom 𝐹)
52513adant3 1129 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ dom 𝐹)
53 simp3 1135 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ (𝐼𝑖))
5452, 53sseldd 3980 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ dom 𝐹)
55543exp 1116 . . . . . . . . . . . 12 (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹)))
5655adantr 479 . . . . . . . . . . 11 ((𝜑𝑥 ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹)))
5756rexlimdv 3143 . . . . . . . . . 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 582 . . . . . . . 8 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
6027, 59pm2.61dan 811 . . . . . . 7 ((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) → 𝑥 ∈ dom 𝐹)
615, 25, 60syl2anc 582 . . . . . 6 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 ∈ dom 𝐹)
624, 61ffvelcdmd 7089 . . . . 5 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (𝐹𝑥) ∈ ℝ)
6362recnd 11281 . . . 4 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (𝐹𝑥) ∈ ℂ)
6463abscld 15434 . . 3 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (abs‘(𝐹𝑥)) ∈ ℝ)
65 simpr 483 . . . . . . 7 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
66 fzfid 13985 . . . . . . . . . 10 (𝜑 → (0...𝑀) ∈ Fin)
67 rnffi 44816 . . . . . . . . . 10 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
687, 66, 67syl2anc 582 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ Fin)
69 infi 9293 . . . . . . . . 9 (ran 𝑄 ∈ Fin → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7068, 69syl 17 . . . . . . . 8 (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7170adantr 479 . . . . . . 7 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7265, 71eqeltrd 2826 . . . . . 6 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 ∈ Fin)
73 simpll 765 . . . . . . . 8 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → 𝜑)
74 simpr 483 . . . . . . . . . 10 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑥𝑤)
75 simpl 481 . . . . . . . . . 10 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
7674, 75eleqtrd 2828 . . . . . . . . 9 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
7776adantll 712 . . . . . . . 8 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
783adantr 479 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝐹:dom 𝐹⟶ℝ)
7926adantl 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
8078, 79ffvelcdmd 7089 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹𝑥) ∈ ℝ)
8180recnd 11281 . . . . . . . . 9 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹𝑥) ∈ ℂ)
8281abscld 15434 . . . . . . . 8 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (abs‘(𝐹𝑥)) ∈ ℝ)
8373, 77, 82syl2anc 582 . . . . . . 7 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → (abs‘(𝐹𝑥)) ∈ ℝ)
8483ralrimiva 3136 . . . . . 6 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∀𝑥𝑤 (abs‘(𝐹𝑥)) ∈ ℝ)
85 fimaxre3 12204 . . . . . 6 ((𝑤 ∈ Fin ∧ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
8672, 84, 85syl2anc 582 . . . . 5 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
8786adantlr 713 . . . 4 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
88 simpll 765 . . . . 5 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
89 neqne 2938 . . . . . . 7 𝑤 = (ran 𝑄 ∩ dom 𝐹) → 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹))
90 elprn1 45288 . . . . . . 7 ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∧ 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
9189, 90sylan2 591 . . . . . 6 ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
9291adantll 712 . . . . 5 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
93 fzofi 13986 . . . . . . . 8 (0..^𝑀) ∈ Fin
9415rnmptfi 44812 . . . . . . . 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 479 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → 𝐹:dom 𝐹⟶ℝ)
9897, 58ffvelcdmd 7089 . . . . . . . . 9 ((𝜑𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℝ)
9998recnd 11281 . . . . . . . 8 ((𝜑𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
10099adantlr 713 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
101100abscld 15434 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (abs‘(𝐹𝑥)) ∈ ℝ)
10237, 15fnmpti 6694 . . . . . . . . . . 11 𝐼 Fn (0..^𝑀)
103 fvelrnb 6953 . . . . . . . . . . 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 215 . . . . . . . . 9 (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
106105adantl 480 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
1077adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
108 elfzofz 13694 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
109108adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
110107, 109ffvelcdmd 7089 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
111 fzofzp1 13776 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
112111adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
113107, 112ffvelcdmd 7089 . . . . . . . . . . . . . 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 45552 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
1171163adant3 1129 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
118 fvres 6910 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹𝑥))
119118fveq2d 6895 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = (abs‘(𝐹𝑥)))
120119breq1d 5154 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹𝑥)) ≤ 𝑏))
121120adantl 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹𝑥)) ≤ 𝑏))
122121ralbidva 3166 . . . . . . . . . . . . . . 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 3169 . . . . . . . . . . . . . 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 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))
12537, 42mpan2 689 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
126 id 22 . . . . . . . . . . . . . . . . 17 ((𝐼𝑖) = 𝑡 → (𝐼𝑖) = 𝑡)
127125, 126sylan9req 2787 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
1281273adant1 1127 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
129128raleqdv 3315 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏 ↔ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
130129rexbidv 3169 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
131124, 130bitrd 278 . . . . . . . . . . . 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 231 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
1331323exp 1116 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)))
134133adantr 479 . . . . . . . . 9 ((𝜑𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)))
135134rexlimdv 3143 . . . . . . . 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 713 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
138 eqimss 4038 . . . . . . 7 (𝑤 = ran 𝐼𝑤 ran 𝐼)
139138adantl 480 . . . . . 6 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
14096, 101, 137, 139ssfiunibd 44958 . . . . 5 ((𝜑𝑤 = ran 𝐼) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
14188, 92, 140syl2anc 582 . . . 4 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
14287, 141pm2.61dan 811 . . 3 ((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
143 simpr 483 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄)
144 elinel2 4195 . . . . . . . . . 10 (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
145144ad2antlr 725 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ dom 𝐹)
146143, 145elind 4193 . . . . . . . 8 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
147 elun1 4175 . . . . . . . 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 724 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
1517ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
152 elinel1 4194 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ (𝐴[,]𝐵))
153152adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ (𝐴[,]𝐵))
154 fourierdlem71.q0 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄‘0) = 𝐴)
155154eqcomd 2732 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (𝑄‘0))
156155adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐴 = (𝑄‘0))
157 fourierdlem71.10 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄𝑀) = 𝐵)
158157eqcomd 2732 . . . . . . . . . . . . . . 15 (𝜑𝐵 = (𝑄𝑀))
159158adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐵 = (𝑄𝑀))
160156, 159oveq12d 7432 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
161153, 160eleqtrd 2828 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((𝑄‘0)[,](𝑄𝑀)))
162161adantr 479 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((𝑄‘0)[,](𝑄𝑀)))
163 simpr 483 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ¬ 𝑥 ∈ ran 𝑄)
164 fveq2 6891 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
165164breq1d 5154 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑥 ↔ (𝑄𝑗) < 𝑥))
166165cbvrabv 3431 . . . . . . . . . . . 12 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}
167166supeq1i 9481 . . . . . . . . . . 11 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}, ℝ, < )
168150, 151, 162, 163, 167fourierdlem25 45787 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
16939ad2antrl 726 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑖 ∈ (0..^𝑀))
170 simprr 771 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑥 ∈ (𝐼𝑖))
171169, 125syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
172170, 171eleqtrd 2828 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
173169, 172jca 510 . . . . . . . . . . . . 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 2837 . . . . . . . . . . . . . . 15 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ dom 𝐼)
176175ad2antrl 726 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑖 ∈ dom 𝐼)
177 simprr 771 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
178125eqcomd 2732 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
179178ad2antrl 726 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
180177, 179eleqtrd 2828 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ (𝐼𝑖))
181176, 180jca 510 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)))
182173, 181impbida 799 . . . . . . . . . . . 12 (𝜑 → ((𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) ↔ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))))
183182rexbidv2 3165 . . . . . . . . . . 11 (𝜑 → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
184183ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
185168, 184mpbird 256 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
186185, 33sylibr 233 . . . . . . . 8 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ran 𝐼)
187 elun2 4176 . . . . . . . 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 811 . . . . . 6 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
190189ralrimiva 3136 . . . . 5 (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
191 dfss3 3968 . . . . 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 233 . . . 4 (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
19313, 21, 23syl2anc 582 . . . 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 44958 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
196 nfv 1910 . . . . . 6 𝑥𝜑
197 nfra1 3272 . . . . . 6 𝑥𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦
198196, 197nfan 1895 . . . . 5 𝑥(𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
199 fourierdlem71.dmf . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℝ)
200199sselda 3979 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝑥 ∈ ℝ)
201 fourierdlem71.b . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
202201adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ dom 𝐹) → 𝐵 ∈ ℝ)
203202, 200resubcld 11681 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → (𝐵𝑥) ∈ ℝ)
204 fourierdlem71.t . . . . . . . . . . . . . . . . . . 19 𝑇 = (𝐵𝐴)
205 fourierdlem71.a . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ∈ ℝ)
206201, 205resubcld 11681 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐵𝐴) ∈ ℝ)
207204, 206eqeltrid 2830 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ∈ ℝ)
208207adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ∈ ℝ)
209 fourierdlem71.altb . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐴 < 𝐵)
210205, 201posdifd 11840 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))
211209, 210mpbid 231 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 < (𝐵𝐴))
212211, 204breqtrrdi 5186 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 0 < 𝑇)
213212gt0ne0d 11817 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ≠ 0)
214213adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ≠ 0)
215203, 208, 214redivcld 12085 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐵𝑥) / 𝑇) ∈ ℝ)
216215flcld 13810 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ)
217216zred 12710 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℝ)
218217, 208remulcld 11283 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇) ∈ ℝ)
219200, 218readdcld 11282 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ ℝ)
220 fourierdlem71.e . . . . . . . . . . . . 13 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
221220fvmpt2 7010 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ ℝ) → (𝐸𝑥) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
222200, 219, 221syl2anc 582 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
223222fveq2d 6895 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹‘(𝐸𝑥)) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
224 fvex 6904 . . . . . . . . . . . 12 (⌊‘((𝐵𝑥) / 𝑇)) ∈ V
225 eleq1 2814 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ))
226225anbi2d 628 . . . . . . . . . . . . 13 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) ↔ ((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ)))
227 oveq1 7421 . . . . . . . . . . . . . . . 16 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))
228227oveq2d 7430 . . . . . . . . . . . . . . 15 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
229228fveq2d 6895 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
230229eqeq1d 2728 . . . . . . . . . . . . 13 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥)))
231226, 230imbi12d 343 . . . . . . . . . . . 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 3538 . . . . . . . . . . 11 (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥))
234216, 233mpdan 685 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥))
235223, 234eqtr2d 2767 . . . . . . . . 9 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹‘(𝐸𝑥)))
236235fveq2d 6895 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) = (abs‘(𝐹‘(𝐸𝑥))))
237236adantlr 713 . . . . . . 7 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) = (abs‘(𝐹‘(𝐸𝑥))))
238 fveq2 6891 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
239238fveq2d 6895 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (abs‘(𝐹𝑥)) = (abs‘(𝐹𝑤)))
240239breq1d 5154 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((abs‘(𝐹𝑥)) ≤ 𝑦 ↔ (abs‘(𝐹𝑤)) ≤ 𝑦))
241240cbvralvw 3225 . . . . . . . . . 10 (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 ↔ ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
242241biimpi 215 . . . . . . . . 9 (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
243242ad2antlr 725 . . . . . . . 8 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
244 iocssicc 13460 . . . . . . . . . . 11 (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
245205adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → 𝐴 ∈ ℝ)
246209adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → 𝐴 < 𝐵)
247 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦𝑥 = 𝑦)
248 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
249248oveq1d 7429 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐵𝑥) / 𝑇) = ((𝐵𝑦) / 𝑇))
250249fveq2d 6895 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (⌊‘((𝐵𝑥) / 𝑇)) = (⌊‘((𝐵𝑦) / 𝑇)))
251250oveq1d 7429 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇))
252247, 251oveq12d 7432 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) = (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
253252cbvmptv 5257 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
254220, 253eqtri 2754 . . . . . . . . . . . . 13 𝐸 = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
255245, 202, 246, 204, 254fourierdlem4 45766 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝐸:ℝ⟶(𝐴(,]𝐵))
256255, 200ffvelcdmd 7089 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴(,]𝐵))
257244, 256sselid 3977 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴[,]𝐵))
258228eleq1d 2811 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹 ↔ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹))
259226, 258imbi12d 343 . . . . . . . . . . . . 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 3538 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
262216, 261mpdan 685 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
263222, 262eqeltrd 2826 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ dom 𝐹)
264257, 263elind 4193 . . . . . . . . 9 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
265264adantlr 713 . . . . . . . 8 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
266 fveq2 6891 . . . . . . . . . . 11 (𝑤 = (𝐸𝑥) → (𝐹𝑤) = (𝐹‘(𝐸𝑥)))
267266fveq2d 6895 . . . . . . . . . 10 (𝑤 = (𝐸𝑥) → (abs‘(𝐹𝑤)) = (abs‘(𝐹‘(𝐸𝑥))))
268267breq1d 5154 . . . . . . . . 9 (𝑤 = (𝐸𝑥) → ((abs‘(𝐹𝑤)) ≤ 𝑦 ↔ (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦))
269268rspccva 3607 . . . . . . . 8 ((∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦 ∧ (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦)
270243, 265, 269syl2anc 582 . . . . . . 7 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦)
271237, 270eqbrtrd 5166 . . . . . 6 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) ≤ 𝑦)
272271ex 411 . . . . 5 ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) → (𝑥 ∈ dom 𝐹 → (abs‘(𝐹𝑥)) ≤ 𝑦))
273198, 272ralrimi 3245 . . . 4 ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦)
274273ex 411 . . 3 (𝜑 → (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦))
275274reximdv 3160 . 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 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  {crab 3420  Vcvv 3463  cun 3945  cin 3946  wss 3947  {cpr 4626   cuni 4906   class class class wbr 5144  cmpt 5227  dom cdm 5673  ran crn 5674  cres 5675  Fun wfun 6538   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7414  Fincfn 8964  supcsup 9474  cc 11145  cr 11146  0cc0 11147  1c1 11148   + caddc 11150   · cmul 11152   < clt 11287  cle 11288  cmin 11483   / cdiv 11910  cn 12256  cz 12602  (,)cioo 13370  (,]cioc 13371  [,]cicc 13373  ...cfz 13530  ..^cfzo 13673  cfl 13802  abscabs 15232  cnccncf 24882   lim climc 25877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-cnex 11203  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223  ax-pre-mulgt0 11224  ax-pre-sup 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-iin 4997  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7680  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-er 8724  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9397  df-fi 9445  df-sup 9476  df-inf 9477  df-oi 9544  df-card 9973  df-pnf 11289  df-mnf 11290  df-xr 11291  df-ltxr 11292  df-le 11293  df-sub 11485  df-neg 11486  df-div 11911  df-nn 12257  df-2 12319  df-3 12320  df-4 12321  df-5 12322  df-6 12323  df-7 12324  df-8 12325  df-9 12326  df-n0 12517  df-z 12603  df-dec 12722  df-uz 12867  df-q 12977  df-rp 13021  df-xneg 13138  df-xadd 13139  df-xmul 13140  df-ioo 13374  df-ioc 13375  df-ico 13376  df-icc 13377  df-fz 13531  df-fzo 13674  df-fl 13804  df-seq 14014  df-exp 14074  df-hash 14341  df-cj 15097  df-re 15098  df-im 15099  df-sqrt 15233  df-abs 15234  df-struct 17142  df-sets 17159  df-slot 17177  df-ndx 17189  df-base 17207  df-ress 17236  df-plusg 17272  df-mulr 17273  df-starv 17274  df-sca 17275  df-vsca 17276  df-ip 17277  df-tset 17278  df-ple 17279  df-ds 17281  df-unif 17282  df-hom 17283  df-cco 17284  df-rest 17430  df-topn 17431  df-0g 17449  df-gsum 17450  df-topgen 17451  df-pt 17452  df-prds 17455  df-xrs 17510  df-qtop 17515  df-imas 17516  df-xps 17518  df-mre 17592  df-mrc 17593  df-acs 17595  df-mgm 18626  df-sgrp 18705  df-mnd 18721  df-submnd 18767  df-mulg 19056  df-cntz 19305  df-cmn 19774  df-psmet 21329  df-xmet 21330  df-met 21331  df-bl 21332  df-mopn 21333  df-cnfld 21338  df-top 22882  df-topon 22899  df-topsp 22921  df-bases 22935  df-cld 23009  df-ntr 23010  df-cls 23011  df-cn 23217  df-cnp 23218  df-cmp 23377  df-tx 23552  df-hmeo 23745  df-xms 24312  df-ms 24313  df-tms 24314  df-cncf 24884  df-limc 25881
This theorem is referenced by:  fourierdlem94  45855  fourierdlem113  45874
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