Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem70 Structured version   Visualization version   GIF version

Theorem fourierdlem70 40894
 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 8398 . . 3 {ran 𝑄, ran 𝐼} ∈ Fin
21a1i 11 . 2 (𝜑 → {ran 𝑄, ran 𝐼} ∈ Fin)
3 simpr 479 . . . . . . 7 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 {ran 𝑄, ran 𝐼})
4 fourierdlem70.q . . . . . . . . . . 11 (𝜑𝑄:(0...𝑀)⟶ℝ)
5 ovex 6839 . . . . . . . . . . 11 (0...𝑀) ∈ V
6 fex 6651 . . . . . . . . . . 11 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
74, 5, 6sylancl 697 . . . . . . . . . 10 (𝜑𝑄 ∈ V)
8 rnexg 7261 . . . . . . . . . 10 (𝑄 ∈ V → ran 𝑄 ∈ V)
97, 8syl 17 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ V)
10 fzofi 12965 . . . . . . . . . . . 12 (0..^𝑀) ∈ Fin
11 fourierdlem70.i . . . . . . . . . . . . 13 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
1211rnmptfi 39848 . . . . . . . . . . . 12 ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
1310, 12ax-mp 5 . . . . . . . . . . 11 ran 𝐼 ∈ Fin
1413elexi 3351 . . . . . . . . . 10 ran 𝐼 ∈ V
1514uniex 7116 . . . . . . . . 9 ran 𝐼 ∈ V
16 uniprg 4600 . . . . . . . . 9 ((ran 𝑄 ∈ V ∧ ran 𝐼 ∈ V) → {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
179, 15, 16sylancl 697 . . . . . . . 8 (𝜑 {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
1817adantr 472 . . . . . . 7 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
193, 18eleqtrd 2839 . . . . . 6 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 ∈ (ran 𝑄 ran 𝐼))
20 eqid 2758 . . . . . . . . . . 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 10217 . . . . . . . . . . . . . . 15 ℝ ∈ V
2322, 5elmap 8050 . . . . . . . . . . . . . 14 (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ)
244, 23sylibr 224 . . . . . . . . . . . . 13 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
25 fourierdlem70.q0 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) = 𝐴)
26 fourierdlem70.qm . . . . . . . . . . . . . 14 (𝜑 → (𝑄𝑀) = 𝐵)
2725, 26jca 555 . . . . . . . . . . . . 13 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
28 fourierdlem70.qlt . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2928ralrimiva 3102 . . . . . . . . . . . . 13 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
3024, 27, 29jca32 559 . . . . . . . . . . . 12 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
3120fourierdlem2 40827 . . . . . . . . . . . . 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 247 . . . . . . . . . . 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 40840 . . . . . . . . . 10 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
35 frn 6212 . . . . . . . . . 10 (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) → ran 𝑄 ⊆ (𝐴[,]𝐵))
3634, 35syl 17 . . . . . . . . 9 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
3736sselda 3742 . . . . . . . 8 ((𝜑𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
3837adantlr 753 . . . . . . 7 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
39 simpll 807 . . . . . . . 8 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝜑)
40 elunnel1 3895 . . . . . . . . 9 ((𝑠 ∈ (ran 𝑄 ran 𝐼) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ran 𝐼)
4140adantll 752 . . . . . . . 8 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ran 𝐼)
42 simpr 479 . . . . . . . . . 10 ((𝜑𝑠 ran 𝐼) → 𝑠 ran 𝐼)
4311funmpt2 6086 . . . . . . . . . . 11 Fun 𝐼
44 elunirn 6670 . . . . . . . . . . 11 (Fun 𝐼 → (𝑠 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼𝑖)))
4543, 44mp1i 13 . . . . . . . . . 10 ((𝜑𝑠 ran 𝐼) → (𝑠 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼𝑖)))
4642, 45mpbid 222 . . . . . . . . 9 ((𝜑𝑠 ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼𝑖))
47 id 22 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ dom 𝐼𝑖 ∈ dom 𝐼)
48 ovex 6839 . . . . . . . . . . . . . . . . . . 19 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
4948, 11dmmpti 6182 . . . . . . . . . . . . . . . . . 18 dom 𝐼 = (0..^𝑀)
5047, 49syl6eleq 2847 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ dom 𝐼𝑖 ∈ (0..^𝑀))
5111fvmpt2 6451 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5250, 48, 51sylancl 697 . . . . . . . . . . . . . . . 16 (𝑖 ∈ dom 𝐼 → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5352adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
54 ioossicc 12450 . . . . . . . . . . . . . . . 16 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
55 fourierdlem70.a . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ ℝ)
5655rexrd 10279 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ ℝ*)
5756adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝐴 ∈ ℝ*)
58 fourierdlem70.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
5958rexrd 10279 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ ℝ*)
6059adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝐵 ∈ ℝ*)
6134adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
6250adantl 473 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
6357, 60, 61, 62fourierdlem8 40833 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6454, 63syl5ss 3753 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6553, 64eqsstrd 3778 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
66653adant3 1127 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
67 simp3 1133 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐼𝑖))
6866, 67sseldd 3743 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐴[,]𝐵))
69683exp 1113 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
7069adantr 472 . . . . . . . . . 10 ((𝜑𝑠 ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
7170rexlimdv 3166 . . . . . . . . 9 ((𝜑𝑠 ran 𝐼) → (∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼𝑖) → 𝑠 ∈ (𝐴[,]𝐵)))
7246, 71mpd 15 . . . . . . . 8 ((𝜑𝑠 ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
7339, 41, 72syl2anc 696 . . . . . . 7 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
7438, 73pm2.61dan 867 . . . . . 6 ((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) → 𝑠 ∈ (𝐴[,]𝐵))
7519, 74syldan 488 . . . . 5 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 ∈ (𝐴[,]𝐵))
76 fourierdlem70.f . . . . . 6 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
7776ffvelrnda 6520 . . . . 5 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹𝑠) ∈ ℝ)
7875, 77syldan 488 . . . 4 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (𝐹𝑠) ∈ ℝ)
7978recnd 10258 . . 3 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (𝐹𝑠) ∈ ℂ)
8079abscld 14372 . 2 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (abs‘(𝐹𝑠)) ∈ ℝ)
81 simpr 479 . . . . . 6 ((𝜑𝑤 = ran 𝑄) → 𝑤 = ran 𝑄)
824adantr 472 . . . . . . 7 ((𝜑𝑤 = ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
83 fzfid 12964 . . . . . . 7 ((𝜑𝑤 = ran 𝑄) → (0...𝑀) ∈ Fin)
84 rnffi 39853 . . . . . . 7 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
8582, 83, 84syl2anc 696 . . . . . 6 ((𝜑𝑤 = ran 𝑄) → ran 𝑄 ∈ Fin)
8681, 85eqeltrd 2837 . . . . 5 ((𝜑𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
8786adantlr 753 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
8876ad2antrr 764 . . . . . . . . 9 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
89 simpll 807 . . . . . . . . . 10 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝜑)
90 simpr 479 . . . . . . . . . . . 12 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑠𝑤)
91 simpl 474 . . . . . . . . . . . 12 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑤 = ran 𝑄)
9290, 91eleqtrd 2839 . . . . . . . . . . 11 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑠 ∈ ran 𝑄)
9392adantll 752 . . . . . . . . . 10 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝑠 ∈ ran 𝑄)
9489, 93, 37syl2anc 696 . . . . . . . . 9 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝑠 ∈ (𝐴[,]𝐵))
9588, 94ffvelrnd 6521 . . . . . . . 8 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (𝐹𝑠) ∈ ℝ)
9695recnd 10258 . . . . . . 7 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (𝐹𝑠) ∈ ℂ)
9796abscld 14372 . . . . . 6 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (abs‘(𝐹𝑠)) ∈ ℝ)
9897ralrimiva 3102 . . . . 5 ((𝜑𝑤 = ran 𝑄) → ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ)
9998adantlr 753 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ)
100 fimaxre3 11160 . . . 4 ((𝑤 ∈ Fin ∧ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
10187, 99, 100syl2anc 696 . . 3 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
102 simpll 807 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝜑)
103 neqne 2938 . . . . . 6 𝑤 = ran 𝑄𝑤 ≠ ran 𝑄)
104 elprn1 40366 . . . . . 6 ((𝑤 ∈ {ran 𝑄, ran 𝐼} ∧ 𝑤 ≠ ran 𝑄) → 𝑤 = ran 𝐼)
105103, 104sylan2 492 . . . . 5 ((𝑤 ∈ {ran 𝑄, ran 𝐼} ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ran 𝐼)
106105adantll 752 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ran 𝐼)
10710, 12mp1i 13 . . . . 5 ((𝜑𝑤 = ran 𝐼) → ran 𝐼 ∈ Fin)
108 ax-resscn 10183 . . . . . . . . . 10 ℝ ⊆ ℂ
109108a1i 11 . . . . . . . . 9 (𝜑 → ℝ ⊆ ℂ)
11076, 109fssd 6216 . . . . . . . 8 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
111110ad2antrr 764 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
11272adantlr 753 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
113111, 112ffvelrnd 6521 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → (𝐹𝑠) ∈ ℂ)
114113abscld 14372 . . . . 5 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → (abs‘(𝐹𝑠)) ∈ ℝ)
11548, 11fnmpti 6181 . . . . . . . . . 10 𝐼 Fn (0..^𝑀)
116 fvelrnb 6403 . . . . . . . . . 10 (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡))
117115, 116ax-mp 5 . . . . . . . . 9 (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
118117biimpi 206 . . . . . . . 8 (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
119118adantl 473 . . . . . . 7 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
1204adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
121 elfzofz 12677 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
122121adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
123120, 122ffvelrnd 6521 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
124 fzofzp1 12757 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
125124adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
126120, 125ffvelrnd 6521 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
127 fourierdlem70.fcn . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
128 fourierdlem70.l . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
129 fourierdlem70.r . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
130123, 126, 127, 128, 129cncfioobd 40611 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏)
131 fvres 6366 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠) = (𝐹𝑠))
132131fveq2d 6354 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) = (abs‘(𝐹𝑠)))
133132breq1d 4812 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹𝑠)) ≤ 𝑏))
134133adantl 473 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹𝑠)) ≤ 𝑏))
135134ralbidva 3121 . . . . . . . . . . . . . 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))
136135rexbidv 3188 . . . . . . . . . . . . 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))
137130, 136mpbid 222 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏)
1381373adant3 1127 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏)
13948, 51mpan2 709 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
140139eqcomd 2764 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
141140adantr 472 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
142 simpr 479 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (𝐼𝑖) = 𝑡)
143141, 142eqtrd 2792 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
144143raleqdv 3281 . . . . . . . . . . . . 13 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
145144rexbidv 3188 . . . . . . . . . . . 12 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
1461453adant1 1125 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
147138, 146mpbid 222 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)
1481473exp 1113 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)))
149148adantr 472 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)))
150149rexlimdv 3166 . . . . . . 7 ((𝜑𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
151119, 150mpd 15 . . . . . 6 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)
152151adantlr 753 . . . . 5 (((𝜑𝑤 = ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)
153 eqimss 3796 . . . . . 6 (𝑤 = ran 𝐼𝑤 ran 𝐼)
154153adantl 473 . . . . 5 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
155107, 114, 152, 154ssfiunibd 40020 . . . 4 ((𝜑𝑤 = ran 𝐼) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
156102, 106, 155syl2anc 696 . . 3 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
157101, 156pm2.61dan 867 . 2 ((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
15821ad2antrr 764 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
1594ad2antrr 764 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
160 simpr 479 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
16125eqcomd 2764 . . . . . . . . . . . . . . . 16 (𝜑𝐴 = (𝑄‘0))
16226eqcomd 2764 . . . . . . . . . . . . . . . 16 (𝜑𝐵 = (𝑄𝑀))
163161, 162oveq12d 6829 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
164163adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
165160, 164eleqtrd 2839 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
166165adantr 472 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
167 simpr 479 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ¬ 𝑡 ∈ ran 𝑄)
168 fveq2 6350 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
169168breq1d 4812 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑡 ↔ (𝑄𝑗) < 𝑡))
170169cbvrabv 3337 . . . . . . . . . . . . 13 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}
171170supeq1i 8516 . . . . . . . . . . . 12 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}, ℝ, < )
172158, 159, 166, 167, 171fourierdlem25 40850 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
173139eleq2d 2823 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑀) → (𝑡 ∈ (𝐼𝑖) ↔ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
174173rexbiia 3176 . . . . . . . . . . 11 (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
175172, 174sylibr 224 . . . . . . . . . 10 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼𝑖))
17649eqcomi 2767 . . . . . . . . . . 11 (0..^𝑀) = dom 𝐼
177176rexeqi 3280 . . . . . . . . . 10 (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼𝑖) ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼𝑖))
178175, 177sylib 208 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼𝑖))
179 elunirn 6670 . . . . . . . . . 10 (Fun 𝐼 → (𝑡 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼𝑖)))
18043, 179mp1i 13 . . . . . . . . 9 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → (𝑡 ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼𝑖)))
181178, 180mpbird 247 . . . . . . . 8 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ran 𝐼)
182181ex 449 . . . . . . 7 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (¬ 𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
183182orrd 392 . . . . . 6 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
184 elun 3894 . . . . . 6 (𝑡 ∈ (ran 𝑄 ran 𝐼) ↔ (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
185183, 184sylibr 224 . . . . 5 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (ran 𝑄 ran 𝐼))
186185ralrimiva 3102 . . . 4 (𝜑 → ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ran 𝐼))
187 dfss3 3731 . . . 4 ((𝐴[,]𝐵) ⊆ (ran 𝑄 ran 𝐼) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ran 𝐼))
188186, 187sylibr 224 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ (ran 𝑄 ran 𝐼))
189188, 17sseqtr4d 3781 . 2 (𝜑 → (𝐴[,]𝐵) ⊆ {ran 𝑄, ran 𝐼})
1902, 80, 157, 189ssfiunibd 40020 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑠)) ≤ 𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1630   ∈ wcel 2137   ≠ wne 2930  ∀wral 3048  ∃wrex 3049  {crab 3052  Vcvv 3338   ∪ cun 3711   ⊆ wss 3713  {cpr 4321  ∪ cuni 4586   class class class wbr 4802   ↦ cmpt 4879  dom cdm 5264  ran crn 5265   ↾ cres 5266  Fun wfun 6041   Fn wfn 6042  ⟶wf 6043  ‘cfv 6047  (class class class)co 6811   ↑𝑚 cmap 8021  Fincfn 8119  supcsup 8509  ℂcc 10124  ℝcr 10125  0cc0 10126  1c1 10127   + caddc 10129  ℝ*cxr 10263   < clt 10264   ≤ cle 10265  ℕcn 11210  (,)cioo 12366  [,]cicc 12369  ...cfz 12517  ..^cfzo 12657  abscabs 14171  –cn→ccncf 22878   limℂ climc 23823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203  ax-pre-sup 10204  ax-mulf 10206 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-of 7060  df-om 7229  df-1st 7331  df-2nd 7332  df-supp 7462  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-map 8023  df-pm 8024  df-ixp 8073  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fsupp 8439  df-fi 8480  df-sup 8511  df-inf 8512  df-oi 8578  df-card 8953  df-cda 9180  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-z 11568  df-dec 11684  df-uz 11878  df-q 11980  df-rp 12024  df-xneg 12137  df-xadd 12138  df-xmul 12139  df-ioo 12370  df-ioc 12371  df-ico 12372  df-icc 12373  df-fz 12518  df-fzo 12658  df-seq 12994  df-exp 13053  df-hash 13310  df-cj 14036  df-re 14037  df-im 14038  df-sqrt 14172  df-abs 14173  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16154  df-mulr 16155  df-starv 16156  df-sca 16157  df-vsca 16158  df-ip 16159  df-tset 16160  df-ple 16161  df-ds 16164  df-unif 16165  df-hom 16166  df-cco 16167  df-rest 16283  df-topn 16284  df-0g 16302  df-gsum 16303  df-topgen 16304  df-pt 16305  df-prds 16308  df-xrs 16362  df-qtop 16367  df-imas 16368  df-xps 16370  df-mre 16446  df-mrc 16447  df-acs 16449  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-submnd 17535  df-mulg 17740  df-cntz 17948  df-cmn 18393  df-psmet 19938  df-xmet 19939  df-met 19940  df-bl 19941  df-mopn 19942  df-cnfld 19947  df-top 20899  df-topon 20916  df-topsp 20937  df-bases 20950  df-cld 21023  df-ntr 21024  df-cls 21025  df-cn 21231  df-cnp 21232  df-cmp 21390  df-tx 21565  df-hmeo 21758  df-xms 22324  df-ms 22325  df-tms 22326  df-cncf 22880  df-limc 23827 This theorem is referenced by:  fourierdlem103  40927  fourierdlem104  40928
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