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Theorem fourierdlem70 39697
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 8179 . . 3 {ran 𝑄, ran 𝐼} ∈ Fin
21a1i 11 . 2 (𝜑 → {ran 𝑄, ran 𝐼} ∈ Fin)
3 simpr 477 . . . . . . 7 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 {ran 𝑄, ran 𝐼})
4 fourierdlem70.q . . . . . . . . . . 11 (𝜑𝑄:(0...𝑀)⟶ℝ)
5 ovex 6632 . . . . . . . . . . 11 (0...𝑀) ∈ V
6 fex 6444 . . . . . . . . . . 11 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
74, 5, 6sylancl 693 . . . . . . . . . 10 (𝜑𝑄 ∈ V)
8 rnexg 7045 . . . . . . . . . 10 (𝑄 ∈ V → ran 𝑄 ∈ V)
97, 8syl 17 . . . . . . . . 9 (𝜑 → ran 𝑄 ∈ V)
10 fzofi 12713 . . . . . . . . . . . 12 (0..^𝑀) ∈ Fin
11 fourierdlem70.i . . . . . . . . . . . . 13 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
1211rnmptfi 38822 . . . . . . . . . . . 12 ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
1310, 12ax-mp 5 . . . . . . . . . . 11 ran 𝐼 ∈ Fin
1413elexi 3199 . . . . . . . . . 10 ran 𝐼 ∈ V
1514uniex 6906 . . . . . . . . 9 ran 𝐼 ∈ V
16 uniprg 4416 . . . . . . . . 9 ((ran 𝑄 ∈ V ∧ ran 𝐼 ∈ V) → {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
179, 15, 16sylancl 693 . . . . . . . 8 (𝜑 {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
1817adantr 481 . . . . . . 7 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → {ran 𝑄, ran 𝐼} = (ran 𝑄 ran 𝐼))
193, 18eleqtrd 2700 . . . . . 6 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 ∈ (ran 𝑄 ran 𝐼))
20 eqid 2621 . . . . . . . . . . 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 9971 . . . . . . . . . . . . . . 15 ℝ ∈ V
2322, 5elmap 7830 . . . . . . . . . . . . . 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 554 . . . . . . . . . . . . 13 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
28 fourierdlem70.qlt . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2928ralrimiva 2960 . . . . . . . . . . . . 13 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
3024, 27, 29jca32 557 . . . . . . . . . . . 12 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
3120fourierdlem2 39630 . . . . . . . . . . . . 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 39643 . . . . . . . . . 10 (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
35 frn 6010 . . . . . . . . . 10 (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) → ran 𝑄 ⊆ (𝐴[,]𝐵))
3634, 35syl 17 . . . . . . . . 9 (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
3736sselda 3583 . . . . . . . 8 ((𝜑𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
3837adantlr 750 . . . . . . 7 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
39 simpll 789 . . . . . . . 8 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝜑)
40 elunnel1 3732 . . . . . . . . 9 ((𝑠 ∈ (ran 𝑄 ran 𝐼) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ran 𝐼)
4140adantll 749 . . . . . . . 8 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ran 𝐼)
42 simpr 477 . . . . . . . . . 10 ((𝜑𝑠 ran 𝐼) → 𝑠 ran 𝐼)
4311funmpt2 5885 . . . . . . . . . . 11 Fun 𝐼
44 elunirn 6463 . . . . . . . . . . 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 6632 . . . . . . . . . . . . . . . . . . 19 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
4948, 11dmmpti 5980 . . . . . . . . . . . . . . . . . 18 dom 𝐼 = (0..^𝑀)
5047, 49syl6eleq 2708 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ dom 𝐼𝑖 ∈ (0..^𝑀))
5111fvmpt2 6248 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5250, 48, 51sylancl 693 . . . . . . . . . . . . . . . 16 (𝑖 ∈ dom 𝐼 → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
5352adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
54 ioossicc 12201 . . . . . . . . . . . . . . . 16 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
55 fourierdlem70.a . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ ℝ)
5655rexrd 10033 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ ℝ*)
5756adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝐴 ∈ ℝ*)
58 fourierdlem70.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ)
5958rexrd 10033 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ ℝ*)
6059adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝐵 ∈ ℝ*)
6134adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
6250adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
6357, 60, 61, 62fourierdlem8 39636 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6454, 63syl5ss 3594 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ dom 𝐼) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
6553, 64eqsstrd 3618 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ dom 𝐼) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
66653adant3 1079 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → (𝐼𝑖) ⊆ (𝐴[,]𝐵))
67 simp3 1061 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐼𝑖))
6866, 67sseldd 3584 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ dom 𝐼𝑠 ∈ (𝐼𝑖)) → 𝑠 ∈ (𝐴[,]𝐵))
69683exp 1261 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
7069adantr 481 . . . . . . . . . 10 ((𝜑𝑠 ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
7170rexlimdv 3023 . . . . . . . . 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 692 . . . . . . 7 (((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
7438, 73pm2.61dan 831 . . . . . 6 ((𝜑𝑠 ∈ (ran 𝑄 ran 𝐼)) → 𝑠 ∈ (𝐴[,]𝐵))
7519, 74syldan 487 . . . . 5 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → 𝑠 ∈ (𝐴[,]𝐵))
76 fourierdlem70.f . . . . . 6 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
7776ffvelrnda 6315 . . . . 5 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹𝑠) ∈ ℝ)
7875, 77syldan 487 . . . 4 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (𝐹𝑠) ∈ ℝ)
7978recnd 10012 . . 3 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (𝐹𝑠) ∈ ℂ)
8079abscld 14109 . 2 ((𝜑𝑠 {ran 𝑄, ran 𝐼}) → (abs‘(𝐹𝑠)) ∈ ℝ)
81 simpr 477 . . . . . 6 ((𝜑𝑤 = ran 𝑄) → 𝑤 = ran 𝑄)
824adantr 481 . . . . . . 7 ((𝜑𝑤 = ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
83 fzfid 12712 . . . . . . 7 ((𝜑𝑤 = ran 𝑄) → (0...𝑀) ∈ Fin)
84 rnffi 38827 . . . . . . 7 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
8582, 83, 84syl2anc 692 . . . . . 6 ((𝜑𝑤 = ran 𝑄) → ran 𝑄 ∈ Fin)
8681, 85eqeltrd 2698 . . . . 5 ((𝜑𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
8786adantlr 750 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
8876ad2antrr 761 . . . . . . . . 9 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
89 simpll 789 . . . . . . . . . 10 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝜑)
90 simpr 477 . . . . . . . . . . . 12 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑠𝑤)
91 simpl 473 . . . . . . . . . . . 12 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑤 = ran 𝑄)
9290, 91eleqtrd 2700 . . . . . . . . . . 11 ((𝑤 = ran 𝑄𝑠𝑤) → 𝑠 ∈ ran 𝑄)
9392adantll 749 . . . . . . . . . 10 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝑠 ∈ ran 𝑄)
9489, 93, 37syl2anc 692 . . . . . . . . 9 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → 𝑠 ∈ (𝐴[,]𝐵))
9588, 94ffvelrnd 6316 . . . . . . . 8 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (𝐹𝑠) ∈ ℝ)
9695recnd 10012 . . . . . . 7 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (𝐹𝑠) ∈ ℂ)
9796abscld 14109 . . . . . 6 (((𝜑𝑤 = ran 𝑄) ∧ 𝑠𝑤) → (abs‘(𝐹𝑠)) ∈ ℝ)
9897ralrimiva 2960 . . . . 5 ((𝜑𝑤 = ran 𝑄) → ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ)
9998adantlr 750 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ)
100 fimaxre3 10914 . . . 4 ((𝑤 ∈ Fin ∧ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
10187, 99, 100syl2anc 692 . . 3 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
102 simpll 789 . . . 4 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝜑)
103 neqne 2798 . . . . . 6 𝑤 = ran 𝑄𝑤 ≠ ran 𝑄)
104 elprn1 39266 . . . . . 6 ((𝑤 ∈ {ran 𝑄, ran 𝐼} ∧ 𝑤 ≠ ran 𝑄) → 𝑤 = ran 𝐼)
105103, 104sylan2 491 . . . . 5 ((𝑤 ∈ {ran 𝑄, ran 𝐼} ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ran 𝐼)
106105adantll 749 . . . 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 9937 . . . . . . . . . 10 ℝ ⊆ ℂ
109108a1i 11 . . . . . . . . 9 (𝜑 → ℝ ⊆ ℂ)
11076, 109fssd 6014 . . . . . . . 8 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
111110ad2antrr 761 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
11272adantlr 750 . . . . . . 7 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
113111, 112ffvelrnd 6316 . . . . . 6 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → (𝐹𝑠) ∈ ℂ)
114113abscld 14109 . . . . 5 (((𝜑𝑤 = ran 𝐼) ∧ 𝑠 ran 𝐼) → (abs‘(𝐹𝑠)) ∈ ℝ)
11548, 11fnmpti 5979 . . . . . . . . . 10 𝐼 Fn (0..^𝑀)
116 fvelrnb 6200 . . . . . . . . . 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 482 . . . . . . 7 ((𝜑𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼𝑖) = 𝑡)
1204adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
121 elfzofz 12426 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
122121adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
123120, 122ffvelrnd 6316 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
124 fzofzp1 12506 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
125124adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
126120, 125ffvelrnd 6316 . . . . . . . . . . . . . 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 39411 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏)
131 fvres 6164 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠) = (𝐹𝑠))
132131fveq2d 6152 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) = (abs‘(𝐹𝑠)))
133132breq1d 4623 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹𝑠)) ≤ 𝑏))
134133adantl 482 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹𝑠)) ≤ 𝑏))
135134ralbidva 2979 . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . 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 1079 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏)
13948, 51mpan2 706 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → (𝐼𝑖) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
140139eqcomd 2627 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑀) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
141140adantr 481 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼𝑖))
142 simpr 477 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (𝐼𝑖) = 𝑡)
143141, 142eqtrd 2655 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
144143raleqdv 3133 . . . . . . . . . . . . 13 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
145144rexbidv 3045 . . . . . . . . . . . 12 ((𝑖 ∈ (0..^𝑀) ∧ (𝐼𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏))
1461453adant1 1077 . . . . . . . . . . 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 1261 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)))
149148adantr 481 . . . . . . . 8 ((𝜑𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)))
150149rexlimdv 3023 . . . . . . 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 750 . . . . 5 (((𝜑𝑤 = ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠𝑡 (abs‘(𝐹𝑠)) ≤ 𝑏)
153 eqimss 3636 . . . . . 6 (𝑤 = ran 𝐼𝑤 ran 𝐼)
154153adantl 482 . . . . 5 ((𝜑𝑤 = ran 𝐼) → 𝑤 ran 𝐼)
155107, 114, 152, 154ssfiunibd 38984 . . . 4 ((𝜑𝑤 = ran 𝐼) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
156102, 106, 155syl2anc 692 . . 3 (((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
157101, 156pm2.61dan 831 . 2 ((𝜑𝑤 ∈ {ran 𝑄, ran 𝐼}) → ∃𝑧 ∈ ℝ ∀𝑠𝑤 (abs‘(𝐹𝑠)) ≤ 𝑧)
15821ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
1594ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
160 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
16125eqcomd 2627 . . . . . . . . . . . . . . . 16 (𝜑𝐴 = (𝑄‘0))
16226eqcomd 2627 . . . . . . . . . . . . . . . 16 (𝜑𝐵 = (𝑄𝑀))
163161, 162oveq12d 6622 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
164163adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄𝑀)))
165160, 164eleqtrd 2700 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
166165adantr 481 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
167 simpr 477 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ¬ 𝑡 ∈ ran 𝑄)
168 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑄𝑘) = (𝑄𝑗))
169168breq1d 4623 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((𝑄𝑘) < 𝑡 ↔ (𝑄𝑗) < 𝑡))
170169cbvrabv 3185 . . . . . . . . . . . . 13 {𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}
171170supeq1i 8297 . . . . . . . . . . . 12 sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝑡}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) < 𝑡}, ℝ, < )
172158, 159, 166, 167, 171fourierdlem25 39653 . . . . . . . . . . 11 (((𝜑𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
173139eleq2d 2684 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑀) → (𝑡 ∈ (𝐼𝑖) ↔ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
174173rexbiia 3033 . . . . . . . . . . 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 2630 . . . . . . . . . . 11 (0..^𝑀) = dom 𝐼
177176rexeqi 3132 . . . . . . . . . 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 6463 . . . . . . . . . 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 450 . . . . . . 7 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (¬ 𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
183182orrd 393 . . . . . 6 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
184 elun 3731 . . . . . 6 (𝑡 ∈ (ran 𝑄 ran 𝐼) ↔ (𝑡 ∈ ran 𝑄𝑡 ran 𝐼))
185183, 184sylibr 224 . . . . 5 ((𝜑𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (ran 𝑄 ran 𝐼))
186185ralrimiva 2960 . . . 4 (𝜑 → ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ran 𝐼))
187 dfss3 3573 . . . 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 3621 . 2 (𝜑 → (𝐴[,]𝐵) ⊆ {ran 𝑄, ran 𝐼})
1902, 80, 157, 189ssfiunibd 38984 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cun 3553  wss 3555  {cpr 4150   cuni 4402   class class class wbr 4613  cmpt 4673  dom cdm 5074  ran crn 5075  cres 5076  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  Fincfn 7899  supcsup 8290  cc 9878  cr 9879  0cc0 9880  1c1 9881   + caddc 9883  *cxr 10017   < clt 10018  cle 10019  cn 10964  (,)cioo 12117  [,]cicc 12120  ...cfz 12268  ..^cfzo 12406  abscabs 13908  cnccncf 22587   lim climc 23532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ioc 12122  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-cn 20941  df-cnp 20942  df-cmp 21100  df-tx 21275  df-hmeo 21468  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589  df-limc 23536
This theorem is referenced by:  fourierdlem103  39730  fourierdlem104  39731
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