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Theorem fourierdlem71 46133
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 9361 . . . 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 7925 . . . . . . . . . . 11 (𝑄 ∈ V → ran 𝑄 ∈ V)
12 inex1g 5325 . . . . . . . . . . 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 2835 . . . . . . . . . . . . 13 𝐼 ∈ V
1918rnex 7933 . . . . . . . . . . . 12 ran 𝐼 ∈ V
2019a1i 11 . . . . . . . . . . 11 (𝜑 → ran 𝐼 ∈ V)
2120uniexd 7761 . . . . . . . . . 10 (𝜑 ran 𝐼 ∈ V)
2221adantr 480 . . . . . . . . 9 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → ran 𝐼 ∈ V)
23 uniprg 4928 . . . . . . . . 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 2841 . . . . . . 7 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
26 elinel2 4212 . . . . . . . . 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 4164 . . . . . . . . . 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 6607 . . . . . . . . . . . . 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 6713 . . . . . . . . . . . . . . . . . . 19 dom 𝐼 = (0..^𝑀)
3936, 38eleqtrdi 2849 . . . . . . . . . . . . . . . . . 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 24933 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
46 fdm 6746 . . . . . . . . . . . . . . . . . . 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 6033 . . . . . . . . . . . . . . . . 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 4034 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ dom 𝐹)
52513adant3 1131 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ dom 𝐹)
53 simp3 1137 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ (𝐼𝑖))
5452, 53sseldd 3996 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ dom 𝐹)
55543exp 1118 . . . . . . . . . . . 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 3151 . . . . . . . . . 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 11287 . . . 4 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (𝐹𝑥) ∈ ℂ)
6463abscld 15472 . . 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 14011 . . . . . . . . . 10 (𝜑 → (0...𝑀) ∈ Fin)
67 rnffi 45118 . . . . . . . . . 10 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
687, 66, 67syl2anc 584 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ Fin)
69 infi 9300 . . . . . . . . 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 2839 . . . . . 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 2841 . . . . . . . . 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 11287 . . . . . . . . 9 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹𝑥) ∈ ℂ)
8281abscld 15472 . . . . . . . 8 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (abs‘(𝐹𝑥)) ∈ ℝ)
8373, 77, 82syl2anc 584 . . . . . . 7 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → (abs‘(𝐹𝑥)) ∈ ℝ)
8483ralrimiva 3144 . . . . . 6 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∀𝑥𝑤 (abs‘(𝐹𝑥)) ∈ ℝ)
85 fimaxre3 12212 . . . . . 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 2946 . . . . . . 7 𝑤 = (ran 𝑄 ∩ dom 𝐹) → 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹))
90 elprn1 45589 . . . . . . 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 14012 . . . . . . . 8 (0..^𝑀) ∈ Fin
9415rnmptfi 45114 . . . . . . . 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 11287 . . . . . . . 8 ((𝜑𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
10099adantlr 715 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
101100abscld 15472 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (abs‘(𝐹𝑥)) ∈ ℝ)
10237, 15fnmpti 6712 . . . . . . . . . . 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 13712 . . . . . . . . . . . . . . . 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 13800 . . . . . . . . . . . . . . . 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 45853 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
1171163adant3 1131 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
118 fvres 6926 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹𝑥))
119118fveq2d 6911 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = (abs‘(𝐹𝑥)))
120119breq1d 5158 . . . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . . . . 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 3177 . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . 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 2796 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
1281273adant1 1129 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
129128raleqdv 3324 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏 ↔ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
130129rexbidv 3177 . . . . . . . . . . . . 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 1118 . . . . . . . . . 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 3151 . . . . . . . 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 4054 . . . . . . 7 (𝑤 = ran 𝐼𝑤 ran 𝐼)
139138adantl 481 . . . . . 6 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
14096, 101, 137, 139ssfiunibd 45260 . . . . 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 4212 . . . . . . . . . 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 4210 . . . . . . . 8 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
147 elun1 4192 . . . . . . . 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 4211 . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (𝑄‘0))
156155adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐴 = (𝑄‘0))
157 fourierdlem71.10 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄𝑀) = 𝐵)
158157eqcomd 2741 . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
165164breq1d 5158 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑥 ↔ (𝑄𝑗) < 𝑥))
166165cbvrabv 3444 . . . . . . . . . . . 12 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}
167166supeq1i 9485 . . . . . . . . . . 11 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}, ℝ, < )
168150, 151, 162, 163, 167fourierdlem25 46088 . . . . . . . . . 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 2841 . . . . . . . . . . . . . 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 2850 . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . 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 3173 . . . . . . . . . . 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 4193 . . . . . . . 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 3144 . . . . 5 (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
191 dfss3 3984 . . . . 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 4037 . . 3 (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼})
1952, 64, 142, 194ssfiunibd 45260 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
196 nfv 1912 . . . . . 6 𝑥𝜑
197 nfra1 3282 . . . . . 6 𝑥𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦
198196, 197nfan 1897 . . . . 5 𝑥(𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
199 fourierdlem71.dmf . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℝ)
200199sselda 3995 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝑥 ∈ ℝ)
201 fourierdlem71.b . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
202201adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ dom 𝐹) → 𝐵 ∈ ℝ)
203202, 200resubcld 11689 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → (𝐵𝑥) ∈ ℝ)
204 fourierdlem71.t . . . . . . . . . . . . . . . . . . 19 𝑇 = (𝐵𝐴)
205 fourierdlem71.a . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ∈ ℝ)
206201, 205resubcld 11689 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐵𝐴) ∈ ℝ)
207204, 206eqeltrid 2843 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ∈ ℝ)
208207adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ∈ ℝ)
209 fourierdlem71.altb . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐴 < 𝐵)
210205, 201posdifd 11848 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))
211209, 210mpbid 232 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 < (𝐵𝐴))
212211, 204breqtrrdi 5190 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 0 < 𝑇)
213212gt0ne0d 11825 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ≠ 0)
214213adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ≠ 0)
215203, 208, 214redivcld 12093 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐵𝑥) / 𝑇) ∈ ℝ)
216215flcld 13835 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ)
217216zred 12720 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℝ)
218217, 208remulcld 11289 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇) ∈ ℝ)
219200, 218readdcld 11288 . . . . . . . . . . . 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 6911 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹‘(𝐸𝑥)) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
224 fvex 6920 . . . . . . . . . . . 12 (⌊‘((𝐵𝑥) / 𝑇)) ∈ V
225 eleq1 2827 . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
230229eqeq1d 2737 . . . . . . . . . . . . 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 2776 . . . . . . . . 9 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹‘(𝐸𝑥)))
236235fveq2d 6911 . . . . . . . 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 6907 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
239238fveq2d 6911 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (abs‘(𝐹𝑥)) = (abs‘(𝐹𝑤)))
240239breq1d 5158 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((abs‘(𝐹𝑥)) ≤ 𝑦 ↔ (abs‘(𝐹𝑤)) ≤ 𝑦))
241240cbvralvw 3235 . . . . . . . . . 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 13474 . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . 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 5261 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
254220, 253eqtri 2763 . . . . . . . . . . . . 13 𝐸 = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
255245, 202, 246, 204, 254fourierdlem4 46067 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝐸:ℝ⟶(𝐴(,]𝐵))
256255, 200ffvelcdmd 7105 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴(,]𝐵))
257244, 256sselid 3993 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴[,]𝐵))
258228eleq1d 2824 . . . . . . . . . . . . . 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 2839 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ dom 𝐹)
264257, 263elind 4210 . . . . . . . . 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 6907 . . . . . . . . . . 11 (𝑤 = (𝐸𝑥) → (𝐹𝑤) = (𝐹‘(𝐸𝑥)))
267266fveq2d 6911 . . . . . . . . . 10 (𝑤 = (𝐸𝑥) → (abs‘(𝐹𝑤)) = (abs‘(𝐹‘(𝐸𝑥))))
268267breq1d 5158 . . . . . . . . 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 5170 . . . . . 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 3255 . . . 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 3168 . 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 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cun 3961  cin 3962  wss 3963  {cpr 4633   cuni 4912   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  cres 5691  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Fincfn 8984  supcsup 9478  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  cz 12611  (,)cioo 13384  (,]cioc 13385  [,]cicc 13387  ...cfz 13544  ..^cfzo 13691  cfl 13827  abscabs 15270  cnccncf 24916   lim climc 25912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ioc 13389  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-cn 23251  df-cnp 23252  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916
This theorem is referenced by:  fourierdlem94  46156  fourierdlem113  46175
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