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Theorem fourierdlem80 44417
Description: The derivative of 𝑂 is bounded on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem80.f (𝜑𝐹:ℝ⟶ℝ)
fourierdlem80.xre (𝜑𝑋 ∈ ℝ)
fourierdlem80.a (𝜑𝐴 ∈ ℝ)
fourierdlem80.b (𝜑𝐵 ∈ ℝ)
fourierdlem80.ab (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
fourierdlem80.n0 (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
fourierdlem80.c (𝜑𝐶 ∈ ℝ)
fourierdlem80.o 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
fourierdlem80.i 𝐼 = ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))
fourierdlem80.fbdioo ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤)
fourierdlem80.fdvbdioo ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
fourierdlem80.sf (𝜑𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
fourierdlem80.slt ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) < (𝑆‘(𝑗 + 1)))
fourierdlem80.sjss ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
fourierdlem80.relioo (((𝜑𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
fdv ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹𝐼)):𝐼⟶ℝ)
fourierdlem80.y 𝑌 = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
fourierdlem80.ch (𝜒 ↔ (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
Assertion
Ref Expression
fourierdlem80 (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏)
Distinct variable groups:   𝐴,𝑏,𝑟,𝑠,𝑡   𝐵,𝑏,𝑟,𝑠,𝑡   𝐶,𝑏,𝑟,𝑠,𝑡   𝐹,𝑏,𝑟,𝑠,𝑡   𝑤,𝐹,𝑧,𝑠,𝑡   𝑤,𝐼,𝑧   𝑁,𝑏,𝑗,𝑟,𝑠   𝑘,𝑁,𝑗,𝑟   𝑤,𝑁,𝑧,𝑗   𝑂,𝑏,𝑗,𝑟   𝑤,𝑂,𝑧   𝑆,𝑏,𝑗,𝑟,𝑠,𝑡   𝑆,𝑘   𝑤,𝑆,𝑧   𝑋,𝑏,𝑟,𝑠,𝑡   𝑌,𝑠   𝜑,𝑏,𝑗,𝑟,𝑠   𝜒,𝑠,𝑡   𝜑,𝑤,𝑧
Allowed substitution hints:   𝜑(𝑡,𝑘)   𝜒(𝑧,𝑤,𝑗,𝑘,𝑟,𝑏)   𝐴(𝑧,𝑤,𝑗,𝑘)   𝐵(𝑧,𝑤,𝑗,𝑘)   𝐶(𝑧,𝑤,𝑗,𝑘)   𝐹(𝑗,𝑘)   𝐼(𝑡,𝑗,𝑘,𝑠,𝑟,𝑏)   𝑁(𝑡)   𝑂(𝑡,𝑘,𝑠)   𝑋(𝑧,𝑤,𝑗,𝑘)   𝑌(𝑧,𝑤,𝑡,𝑗,𝑘,𝑟,𝑏)

Proof of Theorem fourierdlem80
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fourierdlem80.o . . . . . . . . 9 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
2 oveq2 7365 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → (𝑋 + 𝑠) = (𝑋 + 𝑡))
32fveq2d 6846 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑡)))
43oveq1d 7372 . . . . . . . . . . 11 (𝑠 = 𝑡 → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) = ((𝐹‘(𝑋 + 𝑡)) − 𝐶))
5 oveq1 7364 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → (𝑠 / 2) = (𝑡 / 2))
65fveq2d 6846 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (sin‘(𝑠 / 2)) = (sin‘(𝑡 / 2)))
76oveq2d 7373 . . . . . . . . . . 11 (𝑠 = 𝑡 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝑡 / 2))))
84, 7oveq12d 7375 . . . . . . . . . 10 (𝑠 = 𝑡 → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) = (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))
98cbvmptv 5218 . . . . . . . . 9 (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))
101, 9eqtr2i 2765 . . . . . . . 8 (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2))))) = 𝑂
1110oveq2i 7368 . . . . . . 7 (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = (ℝ D 𝑂)
1211dmeqi 5860 . . . . . 6 dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = dom (ℝ D 𝑂)
1312ineq2i 4169 . . . . 5 (ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2))))))) = (ran 𝑆 ∩ dom (ℝ D 𝑂))
1413sneqi 4597 . . . 4 {(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} = {(ran 𝑆 ∩ dom (ℝ D 𝑂))}
1514uneq1i 4119 . . 3 ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
16 snfi 8988 . . . . 5 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∈ Fin
17 fzofi 13879 . . . . . 6 (0..^𝑁) ∈ Fin
18 eqid 2736 . . . . . . 7 (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
1918rnmptfi 43378 . . . . . 6 ((0..^𝑁) ∈ Fin → ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ Fin)
2017, 19ax-mp 5 . . . . 5 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ Fin
21 unfi 9116 . . . . 5 (({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∈ Fin ∧ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ Fin) → ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ∈ Fin)
2216, 20, 21mp2an 690 . . . 4 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ∈ Fin
2322a1i 11 . . 3 (𝜑 → ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ∈ Fin)
2415, 23eqeltrid 2842 . 2 (𝜑 → ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ∈ Fin)
25 id 22 . . . 4 (𝑠 ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑠 ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
2615unieqi 4878 . . . 4 ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
2725, 26eleqtrdi 2848 . . 3 (𝑠 ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
28 simpl 483 . . . . 5 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → 𝜑)
29 uniun 4891 . . . . . . . . 9 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ( {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
3029eleq2i 2829 . . . . . . . 8 (𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ 𝑠 ∈ ( {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
31 elun 4108 . . . . . . . 8 (𝑠 ∈ ( {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ (𝑠 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
3230, 31sylbb 218 . . . . . . 7 (𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → (𝑠 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
3332adantl 482 . . . . . 6 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (𝑠 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
34 fourierdlem80.sf . . . . . . . . . . . . 13 (𝜑𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
35 ovex 7390 . . . . . . . . . . . . . 14 (0...𝑁) ∈ V
3635a1i 11 . . . . . . . . . . . . 13 (𝜑 → (0...𝑁) ∈ V)
3734, 36fexd 7177 . . . . . . . . . . . 12 (𝜑𝑆 ∈ V)
38 rnexg 7841 . . . . . . . . . . . 12 (𝑆 ∈ V → ran 𝑆 ∈ V)
3937, 38syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑆 ∈ V)
40 inex1g 5276 . . . . . . . . . . 11 (ran 𝑆 ∈ V → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V)
4139, 40syl 17 . . . . . . . . . 10 (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V)
42 unisng 4886 . . . . . . . . . 10 ((ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V → {(ran 𝑆 ∩ dom (ℝ D 𝑂))} = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
4341, 42syl 17 . . . . . . . . 9 (𝜑 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
4443eleq2d 2823 . . . . . . . 8 (𝜑 → (𝑠 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ↔ 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))))
4544adantr 481 . . . . . . 7 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (𝑠 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ↔ 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))))
4645orbi1d 915 . . . . . 6 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((𝑠 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∨ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))))
4733, 46mpbid 231 . . . . 5 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∨ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
48 dvf 25271 . . . . . . . . 9 (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ
4948a1i 11 . . . . . . . 8 (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ)
50 elinel2 4156 . . . . . . . 8 (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → 𝑠 ∈ dom (ℝ D 𝑂))
5149, 50ffvelcdmd 7036 . . . . . . 7 (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
5251adantl 482 . . . . . 6 ((𝜑𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
53 ovex 7390 . . . . . . . . . . . 12 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ∈ V
5453dfiun3 5921 . . . . . . . . . . 11 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
5554eleq2i 2829 . . . . . . . . . 10 (𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
5655biimpri 227 . . . . . . . . 9 (𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
5756adantl 482 . . . . . . . 8 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
58 eliun 4958 . . . . . . . 8 (𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
5957, 58sylib 217 . . . . . . 7 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
60 nfv 1917 . . . . . . . . 9 𝑗𝜑
61 nfmpt1 5213 . . . . . . . . . . . 12 𝑗(𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6261nfrn 5907 . . . . . . . . . . 11 𝑗ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6362nfuni 4872 . . . . . . . . . 10 𝑗 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6463nfcri 2894 . . . . . . . . 9 𝑗 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6560, 64nfan 1902 . . . . . . . 8 𝑗(𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
66 nfv 1917 . . . . . . . 8 𝑗((ℝ D 𝑂)‘𝑠) ∈ ℂ
6748a1i 11 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ)
68 fourierdlem80.y . . . . . . . . . . . . . . . . . . 19 𝑌 = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
691reseq1i 5933 . . . . . . . . . . . . . . . . . . . 20 (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
70 ioossicc 13350 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑆𝑗)[,](𝑆‘(𝑗 + 1)))
71 fourierdlem80.sjss . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
7270, 71sstrid 3955 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
7372resmptd 5994 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
7469, 73eqtrid 2788 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
7568, 74eqtr4id 2795 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑌 = (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
7675oveq2d 7373 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
77 ax-resscn 11108 . . . . . . . . . . . . . . . . . . . . 21 ℝ ⊆ ℂ
7877a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ℝ ⊆ ℂ)
79 fourierdlem80.f . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐹:ℝ⟶ℝ)
8079adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℝ)
81 fourierdlem80.xre . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑋 ∈ ℝ)
8281adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
83 fourierdlem80.a . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐴 ∈ ℝ)
84 fourierdlem80.b . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐵 ∈ ℝ)
8583, 84iccssred 13351 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8685sselda 3944 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
8782, 86readdcld 11184 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
8880, 87ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
8988recnd 11183 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
90 fourierdlem80.c . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐶 ∈ ℝ)
9190recnd 11183 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐶 ∈ ℂ)
9291adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)
9389, 92subcld 11512 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
94 2cnd 12231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 2 ∈ ℂ)
9585, 78sstrd 3954 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
9695sselda 3944 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
9796halfcld 12398 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝑠 / 2) ∈ ℂ)
9897sincld 16012 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
9994, 98mulcld 11175 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
100 2ne0 12257 . . . . . . . . . . . . . . . . . . . . . . . 24 2 ≠ 0
101100a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 2 ≠ 0)
102 fourierdlem80.ab . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
103102sselda 3944 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ (-π[,]π))
104 eqcom 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑠 = 0 ↔ 0 = 𝑠)
105104biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = 0 → 0 = 𝑠)
106105adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 = 𝑠)
107 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 𝑠 ∈ (𝐴[,]𝐵))
108106, 107eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
109108adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
110 fourierdlem80.n0 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
111110ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → ¬ 0 ∈ (𝐴[,]𝐵))
112109, 111pm2.65da 815 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → ¬ 𝑠 = 0)
113112neqned 2950 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ≠ 0)
114 fourierdlem44 44382 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
115103, 113, 114syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
11694, 98, 101, 115mulne0d 11807 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
11793, 99, 116divcld 11931 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
118117, 1fmptd 7062 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑂:(𝐴[,]𝐵)⟶ℂ)
119 ioossre 13325 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ
120119a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ)
121 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
122121tgioo2 24166 . . . . . . . . . . . . . . . . . . . . 21 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
123121, 122dvres 25275 . . . . . . . . . . . . . . . . . . . 20 (((ℝ ⊆ ℂ ∧ 𝑂:(𝐴[,]𝐵)⟶ℂ) ∧ ((𝐴[,]𝐵) ⊆ ℝ ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ)) → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
12478, 118, 85, 120, 123syl22anc 837 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
125 ioontr 43739 . . . . . . . . . . . . . . . . . . . 20 ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))
126125reseq2i 5934 . . . . . . . . . . . . . . . . . . 19 ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
127124, 126eqtrdi 2792 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
128127adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
12976, 128eqtr2d 2777 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0..^𝑁)) → ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (ℝ D 𝑌))
130129dmeqd 5861 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑁)) → dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = dom (ℝ D 𝑌))
13179adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ)
13281adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑋 ∈ ℝ)
13385adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ ℝ)
13434adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
135 elfzofz 13588 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ (0...𝑁))
136135adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0...𝑁))
137134, 136ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) ∈ (𝐴[,]𝐵))
138133, 137sseldd 3945 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) ∈ ℝ)
139 fzofzp1 13669 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0..^𝑁) → (𝑗 + 1) ∈ (0...𝑁))
140139adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑗 + 1) ∈ (0...𝑁))
141134, 140ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ (𝐴[,]𝐵))
142133, 141sseldd 3945 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ ℝ)
143 fdv . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹𝐼)):𝐼⟶ℝ)
144 fourierdlem80.i . . . . . . . . . . . . . . . . . . . . . 22 𝐼 = ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))
145144feq2i 6660 . . . . . . . . . . . . . . . . . . . . 21 ((ℝ D (𝐹𝐼)):𝐼⟶ℝ ↔ (ℝ D (𝐹𝐼)):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
146143, 145sylib 217 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹𝐼)):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
147144reseq2i 5934 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹𝐼) = (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))
148147oveq2i 7368 . . . . . . . . . . . . . . . . . . . . 21 (ℝ D (𝐹𝐼)) = (ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))
149148feq1i 6659 . . . . . . . . . . . . . . . . . . . 20 ((ℝ D (𝐹𝐼)):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ ↔ (ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
150146, 149sylib 217 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
151102adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ (-π[,]π))
15272, 151sstrd 3954 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
153110adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ (𝐴[,]𝐵))
15472, 153ssneldd 3947 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
15590adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐶 ∈ ℝ)
156131, 132, 138, 142, 150, 152, 154, 155, 68fourierdlem57 44394 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) → ((ℝ D 𝑌):((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ ∧ (ℝ D 𝑌) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘(𝑋 + 𝑠)) · (2 · (sin‘(𝑠 / 2)))) − ((cos‘(𝑠 / 2)) · ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) / ((2 · (sin‘(𝑠 / 2)))↑2))))) ∧ (ℝ D (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (2 · (sin‘(𝑠 / 2))))) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (cos‘(𝑠 / 2))))
157156simpli 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0..^𝑁)) → ((ℝ D 𝑌):((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ ∧ (ℝ D 𝑌) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘(𝑋 + 𝑠)) · (2 · (sin‘(𝑠 / 2)))) − ((cos‘(𝑠 / 2)) · ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) / ((2 · (sin‘(𝑠 / 2)))↑2)))))
158157simpld 495 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌):((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ)
159 fdm 6677 . . . . . . . . . . . . . . . 16 ((ℝ D 𝑌):((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ → dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
160158, 159syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑁)) → dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
161130, 160eqtr2d 2777 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
162 resss 5962 . . . . . . . . . . . . . . 15 ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ (ℝ D 𝑂)
163 dmss 5858 . . . . . . . . . . . . . . 15 (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ (ℝ D 𝑂) → dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ dom (ℝ D 𝑂))
164162, 163mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑁)) → dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ dom (ℝ D 𝑂))
165161, 164eqsstrd 3982 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
1661653adant3 1132 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
167 simp3 1138 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
168166, 167sseldd 3945 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ dom (ℝ D 𝑂))
16967, 168ffvelcdmd 7036 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
1701693exp 1119 . . . . . . . . 9 (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)))
171170adantr 481 . . . . . . . 8 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → (𝑗 ∈ (0..^𝑁) → (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)))
17265, 66, 171rexlimd 3249 . . . . . . 7 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → (∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ))
17359, 172mpd 15 . . . . . 6 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
17452, 173jaodan 956 . . . . 5 ((𝜑 ∧ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∨ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
17528, 47, 174syl2anc 584 . . . 4 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
176175abscld 15321 . . 3 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
17727, 176sylan2 593 . 2 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
178 id 22 . . . 4 (𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
179178, 15eleqtrdi 2848 . . 3 (𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
180 elsni 4603 . . . . . 6 (𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
181 simpr 485 . . . . . . . 8 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
182 fzfid 13878 . . . . . . . . . . 11 (𝜑 → (0...𝑁) ∈ Fin)
183 rnffi 43382 . . . . . . . . . . 11 ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (0...𝑁) ∈ Fin) → ran 𝑆 ∈ Fin)
18434, 182, 183syl2anc 584 . . . . . . . . . 10 (𝜑 → ran 𝑆 ∈ Fin)
185 infi 9212 . . . . . . . . . 10 (ran 𝑆 ∈ Fin → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
186184, 185syl 17 . . . . . . . . 9 (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
187186adantr 481 . . . . . . . 8 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
188181, 187eqeltrd 2838 . . . . . . 7 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 ∈ Fin)
189 nfv 1917 . . . . . . . . 9 𝑠𝜑
190 nfcv 2907 . . . . . . . . . . 11 𝑠ran 𝑆
191 nfcv 2907 . . . . . . . . . . . . 13 𝑠
192 nfcv 2907 . . . . . . . . . . . . 13 𝑠 D
193 nfmpt1 5213 . . . . . . . . . . . . . 14 𝑠(𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
1941, 193nfcxfr 2905 . . . . . . . . . . . . 13 𝑠𝑂
195191, 192, 194nfov 7387 . . . . . . . . . . . 12 𝑠(ℝ D 𝑂)
196195nfdm 5906 . . . . . . . . . . 11 𝑠dom (ℝ D 𝑂)
197190, 196nfin 4176 . . . . . . . . . 10 𝑠(ran 𝑆 ∩ dom (ℝ D 𝑂))
198197nfeq2 2924 . . . . . . . . 9 𝑠 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))
199189, 198nfan 1902 . . . . . . . 8 𝑠(𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
200 simpr 485 . . . . . . . . . . . . 13 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑠𝑟)
201 simpl 483 . . . . . . . . . . . . 13 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
202200, 201eleqtrd 2840 . . . . . . . . . . . 12 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
203202, 50syl 17 . . . . . . . . . . 11 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑠 ∈ dom (ℝ D 𝑂))
204203adantll 712 . . . . . . . . . 10 (((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) ∧ 𝑠𝑟) → 𝑠 ∈ dom (ℝ D 𝑂))
20548ffvelcdmi 7034 . . . . . . . . . . 11 (𝑠 ∈ dom (ℝ D 𝑂) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
206205abscld 15321 . . . . . . . . . 10 (𝑠 ∈ dom (ℝ D 𝑂) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
207204, 206syl 17 . . . . . . . . 9 (((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) ∧ 𝑠𝑟) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
208207ex 413 . . . . . . . 8 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (𝑠𝑟 → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ))
209199, 208ralrimi 3240 . . . . . . 7 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
210 fimaxre3 12101 . . . . . . 7 ((𝑟 ∈ Fin ∧ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
211188, 209, 210syl2anc 584 . . . . . 6 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
212180, 211sylan2 593 . . . . 5 ((𝜑𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
213212adantlr 713 . . . 4 (((𝜑𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) ∧ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
214 simpll 765 . . . . 5 (((𝜑𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) ∧ ¬ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → 𝜑)
215 elunnel1 4109 . . . . . 6 ((𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ∧ ¬ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
216215adantll 712 . . . . 5 (((𝜑𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) ∧ ¬ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
217 vex 3449 . . . . . . . . 9 𝑟 ∈ V
21818elrnmpt 5911 . . . . . . . . 9 (𝑟 ∈ V → (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
219217, 218ax-mp 5 . . . . . . . 8 (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
220219biimpi 215 . . . . . . 7 (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
221220adantl 482 . . . . . 6 ((𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
22262nfcri 2894 . . . . . . . 8 𝑗 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
22360, 222nfan 1902 . . . . . . 7 𝑗(𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
224 nfv 1917 . . . . . . 7 𝑗𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦
225 fourierdlem80.fbdioo . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤)
226 fourierdlem80.fdvbdioo . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
227 reeanv 3217 . . . . . . . . . . . . 13 (∃𝑤 ∈ ℝ ∃𝑧 ∈ ℝ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) ↔ (∃𝑤 ∈ ℝ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∃𝑧 ∈ ℝ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
228225, 226, 227sylanbrc 583 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∃𝑧 ∈ ℝ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
229 simp1 1136 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → (𝜑𝑗 ∈ (0..^𝑁)))
230 simp2l 1199 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → 𝑤 ∈ ℝ)
231 simp2r 1200 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → 𝑧 ∈ ℝ)
232229, 230, 231jca31 515 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → (((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ))
233 simp3l 1201 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤)
234 simp3r 1202 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
235232, 233, 234jca31 515 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
236 fourierdlem80.ch . . . . . . . . . . . . . . . 16 (𝜒 ↔ (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
237235, 236sylibr 233 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → 𝜒)
238236biimpi 215 . . . . . . . . . . . . . . . . . . . . 21 (𝜒 → (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
239 simp-5l 783 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → 𝜑)
240238, 239syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜒𝜑)
241240, 79syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝐹:ℝ⟶ℝ)
242240, 81syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝑋 ∈ ℝ)
243 simp-4l 781 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → (𝜑𝑗 ∈ (0..^𝑁)))
244238, 243syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜒 → (𝜑𝑗 ∈ (0..^𝑁)))
245244, 138syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → (𝑆𝑗) ∈ ℝ)
246244, 142syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → (𝑆‘(𝑗 + 1)) ∈ ℝ)
247 fourierdlem80.slt . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) < (𝑆‘(𝑗 + 1)))
248244, 247syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → (𝑆𝑗) < (𝑆‘(𝑗 + 1)))
24971, 151sstrd 3954 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
250244, 249syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
25171, 153ssneldd 3947 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))))
252244, 251syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → ¬ 0 ∈ ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))))
253244, 150syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → (ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
254 simp-4r 782 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → 𝑤 ∈ ℝ)
255238, 254syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝑤 ∈ ℝ)
256238simplrd 768 . . . . . . . . . . . . . . . . . . . 20 (𝜒 → ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤)
257 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) → 𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))
258257, 144eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) → 𝑡𝐼)
259 rspa 3231 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤𝑡𝐼) → (abs‘(𝐹𝑡)) ≤ 𝑤)
260256, 258, 259syl2an 596 . . . . . . . . . . . . . . . . . . 19 ((𝜒𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))) → (abs‘(𝐹𝑡)) ≤ 𝑤)
261 simpllr 774 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → 𝑧 ∈ ℝ)
262238, 261syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝑧 ∈ ℝ)
263148fveq1i 6843 . . . . . . . . . . . . . . . . . . . . . 22 ((ℝ D (𝐹𝐼))‘𝑡) = ((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)
264263fveq2i 6845 . . . . . . . . . . . . . . . . . . . . 21 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) = (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡))
265238simprd 496 . . . . . . . . . . . . . . . . . . . . . 22 (𝜒 → ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
266265r19.21bi 3234 . . . . . . . . . . . . . . . . . . . . 21 ((𝜒𝑡𝐼) → (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
267264, 266eqbrtrrid 5141 . . . . . . . . . . . . . . . . . . . 20 ((𝜒𝑡𝐼) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)) ≤ 𝑧)
268258, 267sylan2 593 . . . . . . . . . . . . . . . . . . 19 ((𝜒𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)) ≤ 𝑧)
269240, 90syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝐶 ∈ ℝ)
270241, 242, 245, 246, 248, 250, 252, 253, 255, 260, 262, 268, 269, 68fourierdlem68 44405 . . . . . . . . . . . . . . . . . 18 (𝜒 → (dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ∧ ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
271270simprd 496 . . . . . . . . . . . . . . . . 17 (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
272270simpld 495 . . . . . . . . . . . . . . . . . . 19 (𝜒 → dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
273272raleqdv 3313 . . . . . . . . . . . . . . . . . 18 (𝜒 → (∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
274273rexbidv 3175 . . . . . . . . . . . . . . . . 17 (𝜒 → (∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
275271, 274mpbid 231 . . . . . . . . . . . . . . . 16 (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
276125eqcomi 2745 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
277276reseq2i 5934 . . . . . . . . . . . . . . . . . . . . . 22 ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
278277fveq1i 6843 . . . . . . . . . . . . . . . . . . . . 21 (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠)
279 fvres 6861 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = ((ℝ D 𝑂)‘𝑠))
280279adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = ((ℝ D 𝑂)‘𝑠))
281244, 72syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜒 → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
282281resmptd 5994 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜒 → ((𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
28369, 282eqtrid 2788 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜒 → (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
28468, 283eqtr4id 2795 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜒𝑌 = (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
285284oveq2d 7373 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜒 → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
286285fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜒 → ((ℝ D 𝑌)‘𝑠) = ((ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
287124fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
288240, 287syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜒 → ((ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
289286, 288eqtr2d 2777 . . . . . . . . . . . . . . . . . . . . . 22 (𝜒 → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
290289adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
291278, 280, 2903eqtr3a 2800 . . . . . . . . . . . . . . . . . . . 20 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) = ((ℝ D 𝑌)‘𝑠))
292291fveq2d 6846 . . . . . . . . . . . . . . . . . . 19 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (abs‘((ℝ D 𝑂)‘𝑠)) = (abs‘((ℝ D 𝑌)‘𝑠)))
293292breq1d 5115 . . . . . . . . . . . . . . . . . 18 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ (abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
294293ralbidva 3172 . . . . . . . . . . . . . . . . 17 (𝜒 → (∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
295294rexbidv 3175 . . . . . . . . . . . . . . . 16 (𝜒 → (∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
296275, 295mpbird 256 . . . . . . . . . . . . . . 15 (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
297237, 296syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
2982973exp 1119 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)))
299298rexlimdvv 3204 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁)) → (∃𝑤 ∈ ℝ ∃𝑧 ∈ ℝ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
300228, 299mpd 15 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
3013003adant3 1132 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
302 raleq 3309 . . . . . . . . . . . 12 (𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → (∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
3033023ad2ant3 1135 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
304303rexbidv 3175 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
305301, 304mpbird 256 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
3063053exp 1119 . . . . . . . 8 (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)))
307306adantr 481 . . . . . . 7 ((𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → (𝑗 ∈ (0..^𝑁) → (𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)))
308223, 224, 307rexlimd 3249 . . . . . 6 ((𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → (∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
309221, 308mpd 15 . . . . 5 ((𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
310214, 216, 309syl2anc 584 . . . 4 (((𝜑𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) ∧ ¬ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
311213, 310pm2.61dan 811 . . 3 ((𝜑𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
312179, 311sylan2 593 . 2 ((𝜑𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
313 pm3.22 460 . . . . . . . . . . . 12 ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → (𝑟 ∈ ran 𝑆𝑟 ∈ dom (ℝ D 𝑂)))
314 elin 3926 . . . . . . . . . . . 12 (𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ↔ (𝑟 ∈ ran 𝑆𝑟 ∈ dom (ℝ D 𝑂)))
315313, 314sylibr 233 . . . . . . . . . . 11 ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
316315adantll 712 . . . . . . . . . 10 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
31743eqcomd 2742 . . . . . . . . . . 11 (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
318317ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
319316, 318eleqtrd 2840 . . . . . . . . 9 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
320319orcd 871 . . . . . . . 8 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → (𝑟 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
321 simpll 765 . . . . . . . . . . 11 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝜑)
32277a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → ℝ ⊆ ℂ)
323118adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑂:(𝐴[,]𝐵)⟶ℂ)
32483adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝐴 ∈ ℝ)
32584adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝐵 ∈ ℝ)
326324, 325iccssred 13351 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → (𝐴[,]𝐵) ⊆ ℝ)
327322, 323, 326dvbss 25265 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → dom (ℝ D 𝑂) ⊆ (𝐴[,]𝐵))
328 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ dom (ℝ D 𝑂))
329327, 328sseldd 3945 . . . . . . . . . . . 12 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ (𝐴[,]𝐵))
330329adantr 481 . . . . . . . . . . 11 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (𝐴[,]𝐵))
331 simpr 485 . . . . . . . . . . 11 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ¬ 𝑟 ∈ ran 𝑆)
332 fourierdlem80.relioo . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
333 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑘 → (𝑆𝑗) = (𝑆𝑘))
334 oveq1 7364 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
335334fveq2d 6846 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑘 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝑘 + 1)))
336333, 335oveq12d 7375 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
337 ovex 7390 . . . . . . . . . . . . . . . 16 ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))) ∈ V
338336, 18, 337fvmpt 6948 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0..^𝑁) → ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) = ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
339338eleq2d 2823 . . . . . . . . . . . . . 14 (𝑘 ∈ (0..^𝑁) → (𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ 𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1)))))
340339rexbiia 3095 . . . . . . . . . . . . 13 (∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
341332, 340sylibr 233 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
34253, 18dmmpti 6645 . . . . . . . . . . . . 13 dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (0..^𝑁)
343342rexeqi 3312 . . . . . . . . . . . 12 (∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
344341, 343sylibr 233 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
345321, 330, 331, 344syl21anc 836 . . . . . . . . . 10 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
346 funmpt 6539 . . . . . . . . . . 11 Fun (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
347 elunirn 7198 . . . . . . . . . . 11 (Fun (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘)))
348346, 347mp1i 13 . . . . . . . . . 10 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → (𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘)))
349345, 348mpbird 256 . . . . . . . . 9 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
350349olcd 872 . . . . . . . 8 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → (𝑟 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
351320, 350pm2.61dan 811 . . . . . . 7 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → (𝑟 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
352 elun 4108 . . . . . . 7 (𝑟 ∈ ( {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ (𝑟 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
353351, 352sylibr 233 . . . . . 6 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ ( {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
354353, 29eleqtrrdi 2849 . . . . 5 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
355354ralrimiva 3143 . . . 4 (𝜑 → ∀𝑟 ∈ dom (ℝ D 𝑂)𝑟 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
356 dfss3 3932 . . . 4 (dom (ℝ D 𝑂) ⊆ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ ∀𝑟 ∈ dom (ℝ D 𝑂)𝑟 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
357355, 356sylibr 233 . . 3 (𝜑 → dom (ℝ D 𝑂) ⊆ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
358357, 26sseqtrrdi 3995 . 2 (𝜑 → dom (ℝ D 𝑂) ⊆ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
35924, 177, 312, 358ssfiunibd 43533 1 (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cun 3908  cin 3909  wss 3910  {csn 4586   cuni 4865   ciun 4954   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634  cres 5635  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385  -cneg 11386   / cdiv 11812  2c2 12208  (,)cioo 13264  [,]cicc 13267  ...cfz 13424  ..^cfzo 13567  cexp 13967  abscabs 15119  sincsin 15946  cosccos 15947  πcpi 15949  TopOpenctopn 17303  topGenctg 17319  fldccnfld 20796  intcnt 22368   D cdv 25227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-t1 22665  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231
This theorem is referenced by:  fourierdlem103  44440  fourierdlem104  44441
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