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Theorem fourierdlem70 46097
Description: A piecewise continuous function is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem70.a (𝜑𝐴 ∈ ℝ)
fourierdlem70.2 (𝜑𝐵 ∈ ℝ)
fourierdlem70.aleb (𝜑𝐴𝐵)
fourierdlem70.f (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
fourierdlem70.m (𝜑𝑀 ∈ ℕ)
fourierdlem70.q (𝜑𝑄:(0...𝑀)⟶ℝ)
fourierdlem70.q0 (𝜑 → (𝑄‘0) = 𝐴)
fourierdlem70.qm (𝜑 → (𝑄𝑀) = 𝐵)
fourierdlem70.qlt ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
fourierdlem70.fcn ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem70.r ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
fourierdlem70.l ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
fourierdlem70.i 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
Assertion
Ref Expression
fourierdlem70 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑠)) ≤ 𝑥)
Distinct variable groups:   𝐴,𝑖   𝐵,𝑖   𝑖,𝐹,𝑠   𝑥,𝐹,𝑠   𝑖,𝐼,𝑠   𝑥,𝐼   𝐿,𝑠   𝑖,𝑀,𝑠   𝑄,𝑖,𝑠   𝑥,𝑄   𝑅,𝑠   𝜑,𝑖,𝑠   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑠)   𝐵(𝑥,𝑠)   𝑅(𝑥,𝑖)   𝐿(𝑥,𝑖)   𝑀(𝑥)

Proof of Theorem fourierdlem70
Dummy variables 𝑡 𝑣 𝑦 𝑤 𝑏 𝑧 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfi 9391 . . 3 {ran 𝑄, ran 𝐼} ∈ Fin
21a1i 11 . 2 (𝜑 → {ran 𝑄, ran 𝐼} ∈ Fin)
3 simpr 484 . . . . . . 7 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 {ran 𝑄, ran 𝐼})
4 fourierdlem70.q . . . . . . . . . . 11 (𝜑𝑄:(0...𝑀)⟶ℝ)
5 ovex 7481 . . . . . . . . . . 11 (0...𝑀) ∈ V
6 fex 7263 . . . . . . . . . . 11 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
74, 5, 6sylancl 585 . . . . . . . . . 10 (𝜑𝑄 ∈ V)
8 rnexg 7942 . . . . . . . . . 10 (𝑄 ∈ V → ran 𝑄 ∈ V)
97, 8syl 17 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ V)
10 fzofi 14025 . . . . . . . . . . . 12 (0..^𝑀) ∈ Fin
11 fourierdlem70.i . . . . . . . . . . . . 13 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
1211rnmptfi 45078 . . . . . . . . . . . 12 ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
1310, 12ax-mp 5 . . . . . . . . . . 11 ran 𝐼 ∈ Fin
1413elexi 3511 . . . . . . . . . 10 ran 𝐼 ∈ V
1514uniex 7776 . . . . . . . . 9 ran 𝐼 ∈ V
16 uniprg 4947 . . . . . . . . 9 ((ran 𝑄 ∈ V ∧ ran 𝐼 ∈ V) → {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
179, 15, 16sylancl 585 . . . . . . . 8 (𝜑 {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
1817adantr 480 . . . . . . 7 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
193, 18eleqtrd 2846 . . . . . 6 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 ∈ (ran 𝑄 ran 𝐼))
20 eqid 2740 . . . . . . . . . . 11 (𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣𝑖) < (𝑣‘(𝑖 + 1)))}) = (𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣𝑖) < (𝑣‘(𝑖 + 1)))})
21 fourierdlem70.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℕ)
22 reex 11275 . . . . . . . . . . . . . . 15 ℝ ∈ V
2322, 5elmap 8929 . . . . . . . . . . . . . 14 (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ)
244, 23sylibr 234 . . . . . . . . . . . . 13 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
25 fourierdlem70.q0 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) = 𝐴)
26 fourierdlem70.qm . . . . . . . . . . . . . 14 (𝜑 → (𝑄𝑀) = 𝐵)
2725, 26jca 511 . . . . . . . . . . . . 13 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
28 fourierdlem70.qlt . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2928ralrimiva 3152 . . . . . . . . . . . . 13 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
3024, 27, 29jca32 515 . . . . . . . . . . . 12 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
3120fourierdlem2 46030 . . . . . . . . . . . . 13 (𝑀 ∈ ℕ → (𝑄 ∈ ((𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣𝑖) < (𝑣‘(𝑖 + 1)))})‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
3221, 31syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑄 ∈ ((𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣𝑖) < (𝑣‘(𝑖 + 1)))})‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
3330, 32mpbird 257 . . . . . . . . . . 11 (𝜑𝑄 ∈ ((𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣𝑖) < (𝑣‘(𝑖 + 1)))})‘𝑀))
3420, 21, 33fourierdlem15 46043 . . . . . . . . . 10 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
3534frnd 6755 . . . . . . . . 9 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
3635sselda 4008 . . . . . . . 8 ((𝜑𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
3736adantlr 714 . . . . . . 7 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
38 simpll 766 . . . . . . . 8 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝜑)
39 elunnel1 4177 . . . . . . . . 9 ((𝑠 ∈ (ran 𝑄 ran 𝐼) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ran 𝐼)
4039adantll 713 . . . . . . . 8 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ran 𝐼)
41 simpr 484 . . . . . . . . . 10 ((𝜑𝑠 ran 𝐼) → 𝑠 ran 𝐼)
4211funmpt2 6617 . . . . . . . . . . 11 Fun 𝐼
43 elunirn 7288 . . . . . . . . . . 11 (Fun 𝐼 → (𝑠 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼𝑖)))
4442, 43mp1i 13 . . . . . . . . . 10 ((𝜑𝑠 ran 𝐼) → (𝑠 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼𝑖)))
4541, 44mpbid 232 . . . . . . . . 9 ((𝜑𝑠 ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼𝑖))
46 id 22 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ dom 𝐼𝑖 ∈ dom 𝐼)
47 ovex 7481 . . . . . . . . . . . . . . . . . . 19 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
4847, 11dmmpti 6724 . . . . . . . . . . . . . . . . . 18 dom 𝐼 = (0..^𝑀)
4946, 48eleqtrdi 2854 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ dom 𝐼𝑖 ∈ (0..^𝑀))
5011fvmpt2 7040 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5149, 47, 50sylancl 585 . . . . . . . . . . . . . . . 16 (𝑖 ∈ dom 𝐼 → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5251adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
53 ioossicc 13493 . . . . . . . . . . . . . . . 16 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
54 fourierdlem70.a . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ ℝ)
5554rexrd 11340 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ ℝ*)
5655adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝐴 ∈ ℝ*)
57 fourierdlem70.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
5857rexrd 11340 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ ℝ*)
5958adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝐵 ∈ ℝ*)
6034adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
6149adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
6256, 59, 60, 61fourierdlem8 46036 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6353, 62sstrid 4020 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6452, 63eqsstrd 4047 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
65643adant3 1132 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
66 simp3 1138 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐼𝑖))
6765, 66sseldd 4009 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐴[,]𝐵))
68673exp 1119 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
6968adantr 480 . . . . . . . . . 10 ((𝜑𝑠 ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
7069rexlimdv 3159 . . . . . . . . 9 ((𝜑𝑠 ran 𝐼) → (∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼𝑖) → 𝑠 ∈ (𝐴[,]𝐵)))
7145, 70mpd 15 . . . . . . . 8 ((𝜑𝑠 ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
7238, 40, 71syl2anc 583 . . . . . . 7 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
7337, 72pm2.61dan 812 . . . . . 6 ((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) → 𝑠 ∈ (𝐴[,]𝐵))
7419, 73syldan 590 . . . . 5 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 ∈ (𝐴[,]𝐵))
75 fourierdlem70.f . . . . . 6 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
7675ffvelcdmda 7118 . . . . 5 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹𝑠) ∈ ℝ)
7774, 76syldan 590 . . . 4 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (𝐹𝑠) ∈ ℝ)
7877recnd 11318 . . 3 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (𝐹𝑠) ∈ ℂ)
7978abscld 15485 . 2 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (abs‘(𝐹𝑠)) ∈ ℝ)
80 simpr 484 . . . . . 6 ((𝜑𝑤 = ran 𝑄) → 𝑤 = ran 𝑄)
814adantr 480 . . . . . . 7 ((𝜑𝑤 = ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
82 fzfid 14024 . . . . . . 7 ((𝜑𝑤 = ran 𝑄) → (0...𝑀) ∈ Fin)
83 rnffi 45082 . . . . . . 7 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
8481, 82, 83syl2anc 583 . . . . . 6 ((𝜑𝑤 = ran 𝑄) → ran 𝑄 ∈ Fin)
8580, 84eqeltrd 2844 . . . . 5 ((𝜑𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
8685adantlr 714 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
8775ad2antrr 725 . . . . . . . . 9 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
88 simpll 766 . . . . . . . . . 10 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝜑)
89 simpr 484 . . . . . . . . . . . 12 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑠𝑤)
90 simpl 482 . . . . . . . . . . . 12 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑤 = ran 𝑄)
9189, 90eleqtrd 2846 . . . . . . . . . . 11 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑠 ∈ ran 𝑄)
9291adantll 713 . . . . . . . . . 10 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝑠 ∈ ran 𝑄)
9388, 92, 36syl2anc 583 . . . . . . . . 9 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝑠 ∈ (𝐴[,]𝐵))
9487, 93ffvelcdmd 7119 . . . . . . . 8 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (𝐹𝑠) ∈ ℝ)
9594recnd 11318 . . . . . . 7 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (𝐹𝑠) ∈ ℂ)
9695abscld 15485 . . . . . 6 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (abs‘(𝐹𝑠)) ∈ ℝ)
9796ralrimiva 3152 . . . . 5 ((𝜑𝑤 = ran 𝑄) → ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ)
9897adantlr 714 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ)
99 fimaxre3 12241 . . . 4 ((𝑤 ∈ Fin ∧ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
10086, 98, 99syl2anc 583 . . 3 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
101 simpll 766 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝜑)
102 neqne 2954 . . . . . 6 𝑤 = ran 𝑄𝑤 ≠ ran 𝑄)
103 elprn1 45554 . . . . . 6 ((𝑤 ∈ {ran 𝑄, ran 𝐼} ∧ 𝑤 ≠ ran 𝑄) → 𝑤 = ran 𝐼)
104102, 103sylan2 592 . . . . 5 ((𝑤 ∈ {ran 𝑄, ran 𝐼} ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ran 𝐼)
105104adantll 713 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ran 𝐼)
10610, 12mp1i 13 . . . . 5 ((𝜑𝑤 = ran 𝐼) → ran 𝐼 ∈ Fin)
107 ax-resscn 11241 . . . . . . . . . 10 ℝ ⊆ ℂ
108107a1i 11 . . . . . . . . 9 (𝜑 → ℝ ⊆ ℂ)
10975, 108fssd 6764 . . . . . . . 8 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
110109ad2antrr 725 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
11171adantlr 714 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
112110, 111ffvelcdmd 7119 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → (𝐹𝑠) ∈ ℂ)
113112abscld 15485 . . . . 5 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → (abs‘(𝐹𝑠)) ∈ ℝ)
11447, 11fnmpti 6723 . . . . . . . . . 10 𝐼 Fn (0..^𝑀)
115 fvelrnb 6982 . . . . . . . . . 10 (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡))
116114, 115ax-mp 5 . . . . . . . . 9 (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
117116biimpi 216 . . . . . . . 8 (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
118117adantl 481 . . . . . . 7 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
1194adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
120 elfzofz 13732 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
121120adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
122119, 121ffvelcdmd 7119 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
123 fzofzp1 13814 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
124123adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
125119, 124ffvelcdmd 7119 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
126 fourierdlem70.fcn . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
127 fourierdlem70.l . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
128 fourierdlem70.r . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
129122, 125, 126, 127, 128cncfioobd 45818 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏)
130 fvres 6939 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠) = (𝐹𝑠))
131130fveq2d 6924 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) = (abs‘(𝐹𝑠)))
132131breq1d 5176 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹𝑠)) ≤ 𝑏))
133132adantl 481 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹𝑠)) ≤ 𝑏))
134133ralbidva 3182 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏))
135134rexbidv 3185 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏))
136129, 135mpbid 232 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏)
1371363adant3 1132 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏)
13847, 50mpan2 690 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
139138eqcomd 2746 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
140139adantr 480 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
141 simpr 484 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (𝐼𝑖) = 𝑡)
142140, 141eqtrd 2780 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
143142raleqdv 3334 . . . . . . . . . . . . 13 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
144143rexbidv 3185 . . . . . . . . . . . 12 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
1451443adant1 1130 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
146137, 145mpbid 232 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)
1471463exp 1119 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)))
148147adantr 480 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)))
149148rexlimdv 3159 . . . . . . 7 ((𝜑𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
150118, 149mpd 15 . . . . . 6 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)
151150adantlr 714 . . . . 5 (((𝜑𝑤 = ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)
152 eqimss 4067 . . . . . 6 (𝑤 = ran 𝐼𝑤 ran 𝐼)
153152adantl 481 . . . . 5 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
154106, 113, 151, 153ssfiunibd 45224 . . . 4 ((𝜑𝑤 = ran 𝐼) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
155101, 105, 154syl2anc 583 . . 3 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
156100, 155pm2.61dan 812 . 2 ((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
15721ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
1584ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
159 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
16025eqcomd 2746 . . . . . . . . . . . . . . . 16 (𝜑𝐴 = (𝑄‘0))
16126eqcomd 2746 . . . . . . . . . . . . . . . 16 (𝜑𝐵 = (𝑄𝑀))
162160, 161oveq12d 7466 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
163162adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
164159, 163eleqtrd 2846 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
165164adantr 480 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
166 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ¬ 𝑡 ∈ ran 𝑄)
167 fveq2 6920 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
168167breq1d 5176 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑡 ↔ (𝑄𝑗) < 𝑡))
169168cbvrabv 3454 . . . . . . . . . . . . 13 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}
170169supeq1i 9516 . . . . . . . . . . . 12 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}, ℝ, < )
171157, 158, 165, 166, 170fourierdlem25 46053 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
172138eleq2d 2830 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑀) → (𝑡 ∈ (𝐼𝑖) ↔ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
173172rexbiia 3098 . . . . . . . . . . 11 (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
174171, 173sylibr 234 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼𝑖))
17548eqcomi 2749 . . . . . . . . . . 11 (0..^𝑀) = dom 𝐼
176175rexeqi 3333 . . . . . . . . . 10 (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼𝑖))
177174, 176sylib 218 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼𝑖))
178 elunirn 7288 . . . . . . . . . 10 (Fun 𝐼 → (𝑡 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼𝑖)))
17942, 178mp1i 13 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → (𝑡 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼𝑖)))
180177, 179mpbird 257 . . . . . . . 8 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ran 𝐼)
181180ex 412 . . . . . . 7 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (¬ 𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
182181orrd 862 . . . . . 6 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
183 elun 4176 . . . . . 6 (𝑡 ∈ (ran 𝑄 ran 𝐼) ↔ (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
184182, 183sylibr 234 . . . . 5 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (ran 𝑄 ran 𝐼))
185184ralrimiva 3152 . . . 4 (𝜑 → ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ran 𝐼))
186 dfss3 3997 . . . 4 ((𝐴[,]𝐵) ⊆ (ran 𝑄 ran 𝐼) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ran 𝐼))
187185, 186sylibr 234 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ (ran 𝑄 ran 𝐼))
188187, 17sseqtrrd 4050 . 2 (𝜑 → (𝐴[,]𝐵) ⊆ {ran 𝑄, ran 𝐼})
1892, 79, 156, 188ssfiunibd 45224 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑠)) ≤ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 846  w3a 1087   = wceq 1537  wcel 2108  wne 2946  wral 3067  wrex 3076  {crab 3443  Vcvv 3488  cun 3974  wss 3976  {cpr 4650   cuni 4931   class class class wbr 5166  cmpt 5249  dom cdm 5700  ran crn 5701  cres 5702  Fun wfun 6567   Fn wfn 6568  wf 6569  cfv 6573  (class class class)co 7448  m cmap 8884  Fincfn 9003  supcsup 9509  cc 11182  cr 11183  0cc0 11184  1c1 11185   + caddc 11187  *cxr 11323   < clt 11324  cle 11325  cn 12293  (,)cioo 13407  [,]cicc 13410  ...cfz 13567  ..^cfzo 13711  abscabs 15283  cnccncf 24921   lim climc 25917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ioc 13412  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-rest 17482  df-topn 17483  df-0g 17501  df-gsum 17502  df-topgen 17503  df-pt 17504  df-prds 17507  df-xrs 17562  df-qtop 17567  df-imas 17568  df-xps 17570  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-mulg 19108  df-cntz 19357  df-cmn 19824  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-cnfld 21388  df-top 22921  df-topon 22938  df-topsp 22960  df-bases 22974  df-cld 23048  df-ntr 23049  df-cls 23050  df-cn 23256  df-cnp 23257  df-cmp 23416  df-tx 23591  df-hmeo 23784  df-xms 24351  df-ms 24352  df-tms 24353  df-cncf 24923  df-limc 25921
This theorem is referenced by:  fourierdlem103  46130  fourierdlem104  46131
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