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Theorem fourierdlem70 46191
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 9363 . . 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 7464 . . . . . . . . . . 11 (0...𝑀) ∈ V
6 fex 7246 . . . . . . . . . . 11 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
74, 5, 6sylancl 586 . . . . . . . . . 10 (𝜑𝑄 ∈ V)
8 rnexg 7924 . . . . . . . . . 10 (𝑄 ∈ V → ran 𝑄 ∈ V)
97, 8syl 17 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ V)
10 fzofi 14015 . . . . . . . . . . . 12 (0..^𝑀) ∈ Fin
11 fourierdlem70.i . . . . . . . . . . . . 13 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
1211rnmptfi 45176 . . . . . . . . . . . 12 ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
1310, 12ax-mp 5 . . . . . . . . . . 11 ran 𝐼 ∈ Fin
1413elexi 3503 . . . . . . . . . 10 ran 𝐼 ∈ V
1514uniex 7761 . . . . . . . . 9 ran 𝐼 ∈ V
16 uniprg 4923 . . . . . . . . 9 ((ran 𝑄 ∈ V ∧ ran 𝐼 ∈ V) → {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
179, 15, 16sylancl 586 . . . . . . . 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 2843 . . . . . 6 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 ∈ (ran 𝑄 ran 𝐼))
20 eqid 2737 . . . . . . . . . . 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 11246 . . . . . . . . . . . . . . 15 ℝ ∈ V
2322, 5elmap 8911 . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . 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 46124 . . . . . . . . . . . . 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 46137 . . . . . . . . . 10 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
3534frnd 6744 . . . . . . . . 9 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
3635sselda 3983 . . . . . . . 8 ((𝜑𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
3736adantlr 715 . . . . . . 7 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
38 simpll 767 . . . . . . . 8 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝜑)
39 elunnel1 4154 . . . . . . . . 9 ((𝑠 ∈ (ran 𝑄 ran 𝐼) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ran 𝐼)
4039adantll 714 . . . . . . . 8 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ran 𝐼)
41 simpr 484 . . . . . . . . . 10 ((𝜑𝑠 ran 𝐼) → 𝑠 ran 𝐼)
4211funmpt2 6605 . . . . . . . . . . 11 Fun 𝐼
43 elunirn 7271 . . . . . . . . . . 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 7464 . . . . . . . . . . . . . . . . . . 19 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
4847, 11dmmpti 6712 . . . . . . . . . . . . . . . . . 18 dom 𝐼 = (0..^𝑀)
4946, 48eleqtrdi 2851 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ dom 𝐼𝑖 ∈ (0..^𝑀))
5011fvmpt2 7027 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5149, 47, 50sylancl 586 . . . . . . . . . . . . . . . 16 (𝑖 ∈ dom 𝐼 → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5251adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
53 ioossicc 13473 . . . . . . . . . . . . . . . 16 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
54 fourierdlem70.a . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ ℝ)
5554rexrd 11311 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ ℝ*)
5655adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝐴 ∈ ℝ*)
57 fourierdlem70.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
5857rexrd 11311 . . . . . . . . . . . . . . . . . 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 46130 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6353, 62sstrid 3995 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6452, 63eqsstrd 4018 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
65643adant3 1133 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
66 simp3 1139 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐼𝑖))
6765, 66sseldd 3984 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐴[,]𝐵))
68673exp 1120 . . . . . . . . . . 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 3153 . . . . . . . . 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 584 . . . . . . 7 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
7337, 72pm2.61dan 813 . . . . . 6 ((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) → 𝑠 ∈ (𝐴[,]𝐵))
7419, 73syldan 591 . . . . 5 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 ∈ (𝐴[,]𝐵))
75 fourierdlem70.f . . . . . 6 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
7675ffvelcdmda 7104 . . . . 5 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹𝑠) ∈ ℝ)
7774, 76syldan 591 . . . 4 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (𝐹𝑠) ∈ ℝ)
7877recnd 11289 . . 3 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (𝐹𝑠) ∈ ℂ)
7978abscld 15475 . 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 14014 . . . . . . 7 ((𝜑𝑤 = ran 𝑄) → (0...𝑀) ∈ Fin)
83 rnffi 45180 . . . . . . 7 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
8481, 82, 83syl2anc 584 . . . . . 6 ((𝜑𝑤 = ran 𝑄) → ran 𝑄 ∈ Fin)
8580, 84eqeltrd 2841 . . . . 5 ((𝜑𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
8685adantlr 715 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
8775ad2antrr 726 . . . . . . . . 9 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
88 simpll 767 . . . . . . . . . 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 2843 . . . . . . . . . . 11 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑠 ∈ ran 𝑄)
9291adantll 714 . . . . . . . . . 10 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝑠 ∈ ran 𝑄)
9388, 92, 36syl2anc 584 . . . . . . . . 9 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝑠 ∈ (𝐴[,]𝐵))
9487, 93ffvelcdmd 7105 . . . . . . . 8 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (𝐹𝑠) ∈ ℝ)
9594recnd 11289 . . . . . . 7 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (𝐹𝑠) ∈ ℂ)
9695abscld 15475 . . . . . 6 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (abs‘(𝐹𝑠)) ∈ ℝ)
9796ralrimiva 3146 . . . . 5 ((𝜑𝑤 = ran 𝑄) → ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ)
9897adantlr 715 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ)
99 fimaxre3 12214 . . . 4 ((𝑤 ∈ Fin ∧ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
10086, 98, 99syl2anc 584 . . 3 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
101 simpll 767 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝜑)
102 neqne 2948 . . . . . 6 𝑤 = ran 𝑄𝑤 ≠ ran 𝑄)
103 elprn1 45648 . . . . . 6 ((𝑤 ∈ {ran 𝑄, ran 𝐼} ∧ 𝑤 ≠ ran 𝑄) → 𝑤 = ran 𝐼)
104102, 103sylan2 593 . . . . 5 ((𝑤 ∈ {ran 𝑄, ran 𝐼} ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ran 𝐼)
105104adantll 714 . . . 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 11212 . . . . . . . . . 10 ℝ ⊆ ℂ
108107a1i 11 . . . . . . . . 9 (𝜑 → ℝ ⊆ ℂ)
10975, 108fssd 6753 . . . . . . . 8 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
110109ad2antrr 726 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
11171adantlr 715 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
112110, 111ffvelcdmd 7105 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → (𝐹𝑠) ∈ ℂ)
113112abscld 15475 . . . . 5 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → (abs‘(𝐹𝑠)) ∈ ℝ)
11447, 11fnmpti 6711 . . . . . . . . . 10 𝐼 Fn (0..^𝑀)
115 fvelrnb 6969 . . . . . . . . . 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 13715 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
121120adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
122119, 121ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
123 fzofzp1 13803 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
124123adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
125119, 124ffvelcdmd 7105 . . . . . . . . . . . . . 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 45912 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏)
130 fvres 6925 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠) = (𝐹𝑠))
131130fveq2d 6910 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) = (abs‘(𝐹𝑠)))
132131breq1d 5153 . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . 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 3179 . . . . . . . . . . . . 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 1133 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏)
13847, 50mpan2 691 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
139138eqcomd 2743 . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
143142raleqdv 3326 . . . . . . . . . . . . 13 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
144143rexbidv 3179 . . . . . . . . . . . 12 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
1451443adant1 1131 . . . . . . . . . . 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 1120 . . . . . . . . 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 3153 . . . . . . 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 715 . . . . 5 (((𝜑𝑤 = ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)
152 eqimss 4042 . . . . . 6 (𝑤 = ran 𝐼𝑤 ran 𝐼)
153152adantl 481 . . . . 5 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
154106, 113, 151, 153ssfiunibd 45321 . . . 4 ((𝜑𝑤 = ran 𝐼) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
155101, 105, 154syl2anc 584 . . 3 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
156100, 155pm2.61dan 813 . 2 ((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
15721ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
1584ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
159 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
16025eqcomd 2743 . . . . . . . . . . . . . . . 16 (𝜑𝐴 = (𝑄‘0))
16126eqcomd 2743 . . . . . . . . . . . . . . . 16 (𝜑𝐵 = (𝑄𝑀))
162160, 161oveq12d 7449 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
163162adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
164159, 163eleqtrd 2843 . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
168167breq1d 5153 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑡 ↔ (𝑄𝑗) < 𝑡))
169168cbvrabv 3447 . . . . . . . . . . . . 13 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}
170169supeq1i 9487 . . . . . . . . . . . 12 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}, ℝ, < )
171157, 158, 165, 166, 170fourierdlem25 46147 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
172138eleq2d 2827 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑀) → (𝑡 ∈ (𝐼𝑖) ↔ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
173172rexbiia 3092 . . . . . . . . . . 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 2746 . . . . . . . . . . 11 (0..^𝑀) = dom 𝐼
176175rexeqi 3325 . . . . . . . . . 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 7271 . . . . . . . . . 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 864 . . . . . 6 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
183 elun 4153 . . . . . 6 (𝑡 ∈ (ran 𝑄 ran 𝐼) ↔ (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
184182, 183sylibr 234 . . . . 5 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (ran 𝑄 ran 𝐼))
185184ralrimiva 3146 . . . 4 (𝜑 → ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ran 𝐼))
186 dfss3 3972 . . . 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 4021 . 2 (𝜑 → (𝐴[,]𝐵) ⊆ {ran 𝑄, ran 𝐼})
1892, 79, 156, 188ssfiunibd 45321 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 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cun 3949  wss 3951  {cpr 4628   cuni 4907   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Fincfn 8985  supcsup 9480  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  *cxr 11294   < clt 11295  cle 11296  cn 12266  (,)cioo 13387  [,]cicc 13390  ...cfz 13547  ..^cfzo 13694  abscabs 15273  cnccncf 24902   lim climc 25897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-cn 23235  df-cnp 23236  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901
This theorem is referenced by:  fourierdlem103  46224  fourierdlem104  46225
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