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Theorem fourierdlem70 46132
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 9361 . . 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 7925 . . . . . . . . . 10 (𝑄 ∈ V → ran 𝑄 ∈ V)
97, 8syl 17 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ V)
10 fzofi 14012 . . . . . . . . . . . 12 (0..^𝑀) ∈ Fin
11 fourierdlem70.i . . . . . . . . . . . . 13 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
1211rnmptfi 45114 . . . . . . . . . . . 12 ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
1310, 12ax-mp 5 . . . . . . . . . . 11 ran 𝐼 ∈ Fin
1413elexi 3501 . . . . . . . . . 10 ran 𝐼 ∈ V
1514uniex 7760 . . . . . . . . 9 ran 𝐼 ∈ V
16 uniprg 4928 . . . . . . . . 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 2841 . . . . . 6 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 ∈ (ran 𝑄 ran 𝐼))
20 eqid 2735 . . . . . . . . . . 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 11244 . . . . . . . . . . . . . . 15 ℝ ∈ V
2322, 5elmap 8910 . . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . 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 46065 . . . . . . . . . . . . 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 46078 . . . . . . . . . 10 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
3534frnd 6745 . . . . . . . . 9 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
3635sselda 3995 . . . . . . . 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 4164 . . . . . . . . 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 6607 . . . . . . . . . . 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 6713 . . . . . . . . . . . . . . . . . 18 dom 𝐼 = (0..^𝑀)
4946, 48eleqtrdi 2849 . . . . . . . . . . . . . . . . 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 13470 . . . . . . . . . . . . . . . 16 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
54 fourierdlem70.a . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ ℝ)
5554rexrd 11309 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ ℝ*)
5655adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝐴 ∈ ℝ*)
57 fourierdlem70.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
5857rexrd 11309 . . . . . . . . . . . . . . . . . 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 46071 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6353, 62sstrid 4007 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6452, 63eqsstrd 4034 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
65643adant3 1131 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
66 simp3 1137 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐼𝑖))
6765, 66sseldd 3996 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐴[,]𝐵))
68673exp 1118 . . . . . . . . . . 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 3151 . . . . . . . . 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 11287 . . 3 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (𝐹𝑠) ∈ ℂ)
7978abscld 15472 . 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 14011 . . . . . . 7 ((𝜑𝑤 = ran 𝑄) → (0...𝑀) ∈ Fin)
83 rnffi 45118 . . . . . . 7 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
8481, 82, 83syl2anc 584 . . . . . 6 ((𝜑𝑤 = ran 𝑄) → ran 𝑄 ∈ Fin)
8580, 84eqeltrd 2839 . . . . 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 2841 . . . . . . . . . . 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 11287 . . . . . . 7 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (𝐹𝑠) ∈ ℂ)
9695abscld 15472 . . . . . 6 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (abs‘(𝐹𝑠)) ∈ ℝ)
9796ralrimiva 3144 . . . . 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 12212 . . . 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 2946 . . . . . 6 𝑤 = ran 𝑄𝑤 ≠ ran 𝑄)
103 elprn1 45589 . . . . . 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 11210 . . . . . . . . . 10 ℝ ⊆ ℂ
108107a1i 11 . . . . . . . . 9 (𝜑 → ℝ ⊆ ℂ)
10975, 108fssd 6754 . . . . . . . 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 15472 . . . . 5 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → (abs‘(𝐹𝑠)) ∈ ℝ)
11447, 11fnmpti 6712 . . . . . . . . . 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 13712 . . . . . . . . . . . . . . . 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 13800 . . . . . . . . . . . . . . . 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 45853 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏)
130 fvres 6926 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠) = (𝐹𝑠))
131130fveq2d 6911 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) = (abs‘(𝐹𝑠)))
132131breq1d 5158 . . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . . . 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 3177 . . . . . . . . . . . . 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 1131 . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
143142raleqdv 3324 . . . . . . . . . . . . 13 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
144143rexbidv 3177 . . . . . . . . . . . 12 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
1451443adant1 1129 . . . . . . . . . . 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 1118 . . . . . . . . 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 3151 . . . . . . 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 4054 . . . . . 6 (𝑤 = ran 𝐼𝑤 ran 𝐼)
153152adantl 481 . . . . 5 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
154106, 113, 151, 153ssfiunibd 45260 . . . 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 2741 . . . . . . . . . . . . . . . 16 (𝜑𝐴 = (𝑄‘0))
16126eqcomd 2741 . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
168167breq1d 5158 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑡 ↔ (𝑄𝑗) < 𝑡))
169168cbvrabv 3444 . . . . . . . . . . . . 13 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}
170169supeq1i 9485 . . . . . . . . . . . 12 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}, ℝ, < )
171157, 158, 165, 166, 170fourierdlem25 46088 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
172138eleq2d 2825 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑀) → (𝑡 ∈ (𝐼𝑖) ↔ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
173172rexbiia 3090 . . . . . . . . . . 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 2744 . . . . . . . . . . 11 (0..^𝑀) = dom 𝐼
176175rexeqi 3323 . . . . . . . . . 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 863 . . . . . 6 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
183 elun 4163 . . . . . 6 (𝑡 ∈ (ran 𝑄 ran 𝐼) ↔ (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
184182, 183sylibr 234 . . . . 5 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (ran 𝑄 ran 𝐼))
185184ralrimiva 3144 . . . 4 (𝜑 → ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ran 𝐼))
186 dfss3 3984 . . . 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 4037 . 2 (𝜑 → (𝐴[,]𝐵) ⊆ {ran 𝑄, ran 𝐼})
1892, 79, 156, 188ssfiunibd 45260 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 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cun 3961  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  m cmap 8865  Fincfn 8984  supcsup 9478  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156  *cxr 11292   < clt 11293  cle 11294  cn 12264  (,)cioo 13384  [,]cicc 13387  ...cfz 13544  ..^cfzo 13691  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-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:  fourierdlem103  46165  fourierdlem104  46166
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