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Theorem fourierdlem71 40897
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 8400 . . . 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 472 . . . . . 6 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝐹:dom 𝐹⟶ℝ)
5 simpl 474 . . . . . . 7 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝜑)
6 simpr 479 . . . . . . . 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 6841 . . . . . . . . . . . . 13 (0...𝑀) ∈ V
98a1i 11 . . . . . . . . . . . 12 (𝜑 → (0...𝑀) ∈ V)
10 fex 6653 . . . . . . . . . . . 12 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
117, 9, 10syl2anc 696 . . . . . . . . . . 11 (𝜑𝑄 ∈ V)
12 rnexg 7263 . . . . . . . . . . 11 (𝑄 ∈ V → ran 𝑄 ∈ V)
13 inex1g 4953 . . . . . . . . . . 11 (ran 𝑄 ∈ V → (ran 𝑄 ∩ dom 𝐹) ∈ V)
1411, 12, 133syl 18 . . . . . . . . . 10 (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ V)
1514adantr 472 . . . . . . . . 9 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (ran 𝑄 ∩ dom 𝐹) ∈ V)
16 fourierdlem71.i . . . . . . . . . . . . . 14 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
17 ovex 6841 . . . . . . . . . . . . . . 15 (0..^𝑀) ∈ V
1817mptex 6650 . . . . . . . . . . . . . 14 (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ V
1916, 18eqeltri 2835 . . . . . . . . . . . . 13 𝐼 ∈ V
2019rnex 7265 . . . . . . . . . . . 12 ran 𝐼 ∈ V
2120a1i 11 . . . . . . . . . . 11 (𝜑 → ran 𝐼 ∈ V)
22 uniexg 7120 . . . . . . . . . . 11 (ran 𝐼 ∈ V → ran 𝐼 ∈ V)
2321, 22syl 17 . . . . . . . . . 10 (𝜑 ran 𝐼 ∈ V)
2423adantr 472 . . . . . . . . 9 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → ran 𝐼 ∈ V)
25 uniprg 4602 . . . . . . . . 9 (((ran 𝑄 ∩ dom 𝐹) ∈ V ∧ ran 𝐼 ∈ V) → {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
2615, 24, 25syl2anc 696 . . . . . . . 8 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
276, 26eleqtrd 2841 . . . . . . 7 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
28 elinel2 3943 . . . . . . . . 9 (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
2928adantl 473 . . . . . . . 8 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
30 simpll 807 . . . . . . . . 9 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
31 elunnel1 3897 . . . . . . . . . 10 ((𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ran 𝐼)
3231adantll 752 . . . . . . . . 9 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ran 𝐼)
3316funmpt2 6088 . . . . . . . . . . . . 13 Fun 𝐼
34 elunirn 6672 . . . . . . . . . . . . 13 (Fun 𝐼 → (𝑥 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖)))
3533, 34ax-mp 5 . . . . . . . . . . . 12 (𝑥 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
3635biimpi 206 . . . . . . . . . . 11 (𝑥 ran 𝐼 → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
3736adantl 473 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
38 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ dom 𝐼𝑖 ∈ dom 𝐼)
39 ovex 6841 . . . . . . . . . . . . . . . . . . . 20 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
4039, 16dmmpti 6184 . . . . . . . . . . . . . . . . . . 19 dom 𝐼 = (0..^𝑀)
4138, 40syl6eleq 2849 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ dom 𝐼𝑖 ∈ (0..^𝑀))
4241adantl 473 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
4339a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V)
4416fvmpt2 6453 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
4542, 43, 44syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
46 fourierdlem71.fcn . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
47 cncff 22897 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
48 fdm 6212 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
4946, 47, 483syl 18 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5041, 49sylan2 492 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
51 ssdmres 5578 . . . . . . . . . . . . . . . . 17 (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5250, 51sylibr 224 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
5345, 52eqsstrd 3780 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ dom 𝐹)
54533adant3 1127 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ dom 𝐹)
55 simp3 1133 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ (𝐼𝑖))
5654, 55sseldd 3745 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) → 𝑥 ∈ dom 𝐹)
57563exp 1113 . . . . . . . . . . . 12 (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹)))
5857adantr 472 . . . . . . . . . . 11 ((𝜑𝑥 ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹)))
5958rexlimdv 3168 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) → 𝑥 ∈ dom 𝐹))
6037, 59mpd 15 . . . . . . . . 9 ((𝜑𝑥 ran 𝐼) → 𝑥 ∈ dom 𝐹)
6130, 32, 60syl2anc 696 . . . . . . . 8 (((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
6229, 61pm2.61dan 867 . . . . . . 7 ((𝜑𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼)) → 𝑥 ∈ dom 𝐹)
635, 27, 62syl2anc 696 . . . . . 6 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → 𝑥 ∈ dom 𝐹)
644, 63ffvelrnd 6523 . . . . 5 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (𝐹𝑥) ∈ ℝ)
6564recnd 10260 . . . 4 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (𝐹𝑥) ∈ ℂ)
6665abscld 14374 . . 3 ((𝜑𝑥 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → (abs‘(𝐹𝑥)) ∈ ℝ)
67 simpr 479 . . . . . . 7 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
68 fzfid 12966 . . . . . . . . . 10 (𝜑 → (0...𝑀) ∈ Fin)
69 rnffi 39855 . . . . . . . . . 10 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
707, 68, 69syl2anc 696 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ Fin)
71 infi 8349 . . . . . . . . 9 (ran 𝑄 ∈ Fin → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7270, 71syl 17 . . . . . . . 8 (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7372adantr 472 . . . . . . 7 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
7467, 73eqeltrd 2839 . . . . . 6 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 ∈ Fin)
75 simpll 807 . . . . . . . 8 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → 𝜑)
76 simpr 479 . . . . . . . . . 10 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑥𝑤)
77 simpl 474 . . . . . . . . . 10 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
7876, 77eleqtrd 2841 . . . . . . . . 9 ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
7978adantll 752 . . . . . . . 8 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
803adantr 472 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝐹:dom 𝐹⟶ℝ)
8128adantl 473 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
8280, 81ffvelrnd 6523 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹𝑥) ∈ ℝ)
8382recnd 10260 . . . . . . . . 9 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹𝑥) ∈ ℂ)
8483abscld 14374 . . . . . . . 8 ((𝜑𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (abs‘(𝐹𝑥)) ∈ ℝ)
8575, 79, 84syl2anc 696 . . . . . . 7 (((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥𝑤) → (abs‘(𝐹𝑥)) ∈ ℝ)
8685ralrimiva 3104 . . . . . 6 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∀𝑥𝑤 (abs‘(𝐹𝑥)) ∈ ℝ)
87 fimaxre3 11162 . . . . . 6 ((𝑤 ∈ Fin ∧ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
8874, 86, 87syl2anc 696 . . . . 5 ((𝜑𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
8988adantlr 753 . . . 4 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
90 simpll 807 . . . . 5 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
91 neqne 2940 . . . . . . 7 𝑤 = (ran 𝑄 ∩ dom 𝐹) → 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹))
92 elprn1 40368 . . . . . . 7 ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∧ 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
9391, 92sylan2 492 . . . . . 6 ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
9493adantll 752 . . . . 5 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ran 𝐼)
95 fzofi 12967 . . . . . . . 8 (0..^𝑀) ∈ Fin
9616rnmptfi 39850 . . . . . . . 8 ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
9795, 96ax-mp 5 . . . . . . 7 ran 𝐼 ∈ Fin
9897a1i 11 . . . . . 6 ((𝜑𝑤 = ran 𝐼) → ran 𝐼 ∈ Fin)
993adantr 472 . . . . . . . . . 10 ((𝜑𝑥 ran 𝐼) → 𝐹:dom 𝐹⟶ℝ)
10099, 60ffvelrnd 6523 . . . . . . . . 9 ((𝜑𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℝ)
101100recnd 10260 . . . . . . . 8 ((𝜑𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
102101adantlr 753 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (𝐹𝑥) ∈ ℂ)
103102abscld 14374 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑥 ran 𝐼) → (abs‘(𝐹𝑥)) ∈ ℝ)
10439, 16fnmpti 6183 . . . . . . . . . . 11 𝐼 Fn (0..^𝑀)
105 fvelrnb 6405 . . . . . . . . . . 11 (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡))
106104, 105ax-mp 5 . . . . . . . . . 10 (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
107106biimpi 206 . . . . . . . . 9 (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
108107adantl 473 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
1097adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
110 elfzofz 12679 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
111110adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
112109, 111ffvelrnd 6523 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
113 fzofzp1 12759 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
114113adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
115109, 114ffvelrnd 6523 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
116 fourierdlem71.l . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
117 fourierdlem71.r . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
118112, 115, 46, 116, 117cncfioobd 40613 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
1191183adant3 1127 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
120 fvres 6368 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹𝑥))
121120fveq2d 6356 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = (abs‘(𝐹𝑥)))
122121breq1d 4814 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹𝑥)) ≤ 𝑏))
123122adantl 473 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹𝑥)) ≤ 𝑏))
124123ralbidva 3123 . . . . . . . . . . . . . . 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))
125124rexbidv 3190 . . . . . . . . . . . . . 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))
1261253adant3 1127 . . . . . . . . . . . . 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))
12739, 44mpan2 709 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
128 id 22 . . . . . . . . . . . . . . . . 17 ((𝐼𝑖) = 𝑡 → (𝐼𝑖) = 𝑡)
129127, 128sylan9req 2815 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
1301293adant1 1125 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
131130raleqdv 3283 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏 ↔ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
132131rexbidv 3190 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
133126, 132bitrd 268 . . . . . . . . . . . 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))
134119, 133mpbid 222 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
1351343exp 1113 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)))
136135adantr 472 . . . . . . . . 9 ((𝜑𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)))
137136rexlimdv 3168 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏))
138108, 137mpd 15 . . . . . . 7 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
139138adantlr 753 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥𝑡 (abs‘(𝐹𝑥)) ≤ 𝑏)
140 eqimss 3798 . . . . . . 7 (𝑤 = ran 𝐼𝑤 ran 𝐼)
141140adantl 473 . . . . . 6 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
14298, 103, 139, 141ssfiunibd 40022 . . . . 5 ((𝜑𝑤 = ran 𝐼) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
14390, 94, 142syl2anc 696 . . . 4 (((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
14489, 143pm2.61dan 867 . . 3 ((𝜑𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼}) → ∃𝑦 ∈ ℝ ∀𝑥𝑤 (abs‘(𝐹𝑥)) ≤ 𝑦)
145 simpr 479 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄)
146 elinel2 3943 . . . . . . . . . 10 (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
147146ad2antlr 765 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ dom 𝐹)
148145, 147elind 3941 . . . . . . . 8 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
149 elun1 3923 . . . . . . . 8 (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
150148, 149syl 17 . . . . . . 7 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
151 fourierdlem71.7 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℕ)
152151ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
1537ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
154 elinel1 3942 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ (𝐴[,]𝐵))
155154adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ (𝐴[,]𝐵))
156 fourierdlem71.q0 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄‘0) = 𝐴)
157156eqcomd 2766 . . . . . . . . . . . . . . 15 (𝜑𝐴 = (𝑄‘0))
158157adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐴 = (𝑄‘0))
159 fourierdlem71.10 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑄𝑀) = 𝐵)
160159eqcomd 2766 . . . . . . . . . . . . . . 15 (𝜑𝐵 = (𝑄𝑀))
161160adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐵 = (𝑄𝑀))
162158, 161oveq12d 6831 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
163155, 162eleqtrd 2841 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((𝑄‘0)[,](𝑄𝑀)))
164163adantr 472 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((𝑄‘0)[,](𝑄𝑀)))
165 simpr 479 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ¬ 𝑥 ∈ ran 𝑄)
166 fveq2 6352 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
167166breq1d 4814 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑥 ↔ (𝑄𝑗) < 𝑥))
168167cbvrabv 3339 . . . . . . . . . . . 12 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}
169168supeq1i 8518 . . . . . . . . . . 11 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑥}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑥}, ℝ, < )
170152, 153, 164, 165, 169fourierdlem25 40852 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
17141ad2antrl 766 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑖 ∈ (0..^𝑀))
172 simprr 813 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑥 ∈ (𝐼𝑖))
173171, 127syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
174172, 173eleqtrd 2841 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
175171, 174jca 555 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖))) → (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
176 id 22 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0..^𝑀))
177176, 40syl6eleqr 2850 . . . . . . . . . . . . . . 15 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ dom 𝐼)
178177ad2antrl 766 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑖 ∈ dom 𝐼)
179 simprr 813 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
180127eqcomd 2766 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
181180ad2antrl 766 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
182179, 181eleqtrd 2841 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ (𝐼𝑖))
183178, 182jca 555 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) → (𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)))
184175, 183impbida 913 . . . . . . . . . . . 12 (𝜑 → ((𝑖 ∈ dom 𝐼𝑥 ∈ (𝐼𝑖)) ↔ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))))
185184rexbidv2 3186 . . . . . . . . . . 11 (𝜑 → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
186185ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
187170, 186mpbird 247 . . . . . . . . 9 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼𝑖))
188187, 35sylibr 224 . . . . . . . 8 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ran 𝐼)
189 elun2 3924 . . . . . . . 8 (𝑥 ran 𝐼𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
190188, 189syl 17 . . . . . . 7 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
191150, 190pm2.61dan 867 . . . . . 6 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
192191ralrimiva 3104 . . . . 5 (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
193 dfss3 3733 . . . . 5 (((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼) ↔ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
194192, 193sylibr 224 . . . 4 (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
19514, 23, 25syl2anc 696 . . . 4 (𝜑 {(ran 𝑄 ∩ dom 𝐹), ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ran 𝐼))
196194, 195sseqtr4d 3783 . . 3 (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ {(ran 𝑄 ∩ dom 𝐹), ran 𝐼})
1972, 66, 144, 196ssfiunibd 40022 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
198 nfv 1992 . . . . . 6 𝑥𝜑
199 nfra1 3079 . . . . . 6 𝑥𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦
200198, 199nfan 1977 . . . . 5 𝑥(𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦)
201 fourierdlem71.dmf . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℝ)
202201sselda 3744 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝑥 ∈ ℝ)
203 fourierdlem71.b . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
204203adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ dom 𝐹) → 𝐵 ∈ ℝ)
205204, 202resubcld 10650 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → (𝐵𝑥) ∈ ℝ)
206 fourierdlem71.t . . . . . . . . . . . . . . . . . . 19 𝑇 = (𝐵𝐴)
207 fourierdlem71.a . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ∈ ℝ)
208203, 207resubcld 10650 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐵𝐴) ∈ ℝ)
209206, 208syl5eqel 2843 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ∈ ℝ)
210209adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ∈ ℝ)
211 fourierdlem71.altb . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐴 < 𝐵)
212207, 203posdifd 10806 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))
213211, 212mpbid 222 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 < (𝐵𝐴))
214213, 206syl6breqr 4846 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 0 < 𝑇)
215214gt0ne0d 10784 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇 ≠ 0)
216215adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ dom 𝐹) → 𝑇 ≠ 0)
217205, 210, 216redivcld 11045 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐵𝑥) / 𝑇) ∈ ℝ)
218217flcld 12793 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ)
219218zred 11674 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ dom 𝐹) → (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℝ)
220219, 210remulcld 10262 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇) ∈ ℝ)
221202, 220readdcld 10261 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ ℝ)
222 fourierdlem71.e . . . . . . . . . . . . 13 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
223222fvmpt2 6453 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ ℝ) → (𝐸𝑥) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
224202, 221, 223syl2anc 696 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
225224fveq2d 6356 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹‘(𝐸𝑥)) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
226 fvex 6362 . . . . . . . . . . . 12 (⌊‘((𝐵𝑥) / 𝑇)) ∈ V
227 eleq1 2827 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ))
228227anbi2d 742 . . . . . . . . . . . . 13 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) ↔ ((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ)))
229 oveq1 6820 . . . . . . . . . . . . . . . 16 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))
230229oveq2d 6829 . . . . . . . . . . . . . . 15 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))
231230fveq2d 6356 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))))
232231eqeq1d 2762 . . . . . . . . . . . . 13 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥)))
233228, 232imbi12d 333 . . . . . . . . . . . 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))))
234 fourierdlem71.fxpt . . . . . . . . . . . 12 (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))
235226, 233, 234vtocl 3399 . . . . . . . . . . 11 (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥))
236218, 235mpdan 705 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹‘(𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝐹𝑥))
237225, 236eqtr2d 2795 . . . . . . . . 9 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹‘(𝐸𝑥)))
238237fveq2d 6356 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) = (abs‘(𝐹‘(𝐸𝑥))))
239238adantlr 753 . . . . . . 7 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) = (abs‘(𝐹‘(𝐸𝑥))))
240 fveq2 6352 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
241240fveq2d 6356 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (abs‘(𝐹𝑥)) = (abs‘(𝐹𝑤)))
242241breq1d 4814 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((abs‘(𝐹𝑥)) ≤ 𝑦 ↔ (abs‘(𝐹𝑤)) ≤ 𝑦))
243242cbvralv 3310 . . . . . . . . . 10 (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 ↔ ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
244243biimpi 206 . . . . . . . . 9 (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
245244ad2antlr 765 . . . . . . . 8 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦)
246 iocssicc 12454 . . . . . . . . . . 11 (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
247207adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → 𝐴 ∈ ℝ)
248211adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐹) → 𝐴 < 𝐵)
249 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦𝑥 = 𝑦)
250 oveq2 6821 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
251250oveq1d 6828 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐵𝑥) / 𝑇) = ((𝐵𝑦) / 𝑇))
252251fveq2d 6356 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (⌊‘((𝐵𝑥) / 𝑇)) = (⌊‘((𝐵𝑦) / 𝑇)))
253252oveq1d 6828 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇))
254249, 253oveq12d 6831 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) = (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
255254cbvmptv 4902 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))) = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
256222, 255eqtri 2782 . . . . . . . . . . . . 13 𝐸 = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵𝑦) / 𝑇)) · 𝑇)))
257247, 204, 248, 206, 256fourierdlem4 40831 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐹) → 𝐸:ℝ⟶(𝐴(,]𝐵))
258257, 202ffvelrnd 6523 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴(,]𝐵))
259246, 258sseldi 3742 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ (𝐴[,]𝐵))
260230eleq1d 2824 . . . . . . . . . . . . . 14 (𝑘 = (⌊‘((𝐵𝑥) / 𝑇)) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹 ↔ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹))
261228, 260imbi12d 333 . . . . . . . . . . . . 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)))
262 fourierdlem71.xpt . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹)
263226, 261, 262vtocl 3399 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
264218, 263mpdan 705 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
265224, 264eqeltrd 2839 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ dom 𝐹)
266259, 265elind 3941 . . . . . . . . 9 ((𝜑𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
267266adantlr 753 . . . . . . . 8 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
268 fveq2 6352 . . . . . . . . . . 11 (𝑤 = (𝐸𝑥) → (𝐹𝑤) = (𝐹‘(𝐸𝑥)))
269268fveq2d 6356 . . . . . . . . . 10 (𝑤 = (𝐸𝑥) → (abs‘(𝐹𝑤)) = (abs‘(𝐹‘(𝐸𝑥))))
270269breq1d 4814 . . . . . . . . 9 (𝑤 = (𝐸𝑥) → ((abs‘(𝐹𝑤)) ≤ 𝑦 ↔ (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦))
271270rspccva 3448 . . . . . . . 8 ((∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑤)) ≤ 𝑦 ∧ (𝐸𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦)
272245, 267, 271syl2anc 696 . . . . . . 7 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘(𝐸𝑥))) ≤ 𝑦)
273239, 272eqbrtrd 4826 . . . . . 6 (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹𝑥)) ≤ 𝑦)
274273ex 449 . . . . 5 ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) → (𝑥 ∈ dom 𝐹 → (abs‘(𝐹𝑥)) ≤ 𝑦))
275200, 274ralrimi 3095 . . . 4 ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦) → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦)
276275ex 449 . . 3 (𝜑 → (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦))
277276reximdv 3154 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹𝑥)) ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦))
278197, 277mpd 15 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cun 3713  cin 3714  wss 3715  {cpr 4323   cuni 4588   class class class wbr 4804  cmpt 4881  dom cdm 5266  ran crn 5267  cres 5268  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  Fincfn 8121  supcsup 8511  cc 10126  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   · cmul 10133   < clt 10266  cle 10267  cmin 10458   / cdiv 10876  cn 11212  cz 11569  (,)cioo 12368  (,]cioc 12369  [,]cicc 12371  ...cfz 12519  ..^cfzo 12659  cfl 12785  abscabs 14173  cnccncf 22880   lim climc 23825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-fi 8482  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ioo 12372  df-ioc 12373  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-starv 16158  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-unif 16167  df-hom 16168  df-cco 16169  df-rest 16285  df-topn 16286  df-0g 16304  df-gsum 16305  df-topgen 16306  df-pt 16307  df-prds 16310  df-xrs 16364  df-qtop 16369  df-imas 16370  df-xps 16372  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-cnfld 19949  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027  df-cn 21233  df-cnp 21234  df-cmp 21392  df-tx 21567  df-hmeo 21760  df-xms 22326  df-ms 22327  df-tms 22328  df-cncf 22882  df-limc 23829
This theorem is referenced by:  fourierdlem94  40920  fourierdlem113  40939
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