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Theorem fourierdlem80 41191
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 6918 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → (𝑋 + 𝑠) = (𝑋 + 𝑡))
32fveq2d 6441 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑡)))
43oveq1d 6925 . . . . . . . . . . 11 (𝑠 = 𝑡 → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) = ((𝐹‘(𝑋 + 𝑡)) − 𝐶))
5 oveq1 6917 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → (𝑠 / 2) = (𝑡 / 2))
65fveq2d 6441 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (sin‘(𝑠 / 2)) = (sin‘(𝑡 / 2)))
76oveq2d 6926 . . . . . . . . . . 11 (𝑠 = 𝑡 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝑡 / 2))))
84, 7oveq12d 6928 . . . . . . . . . 10 (𝑠 = 𝑡 → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) = (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))
98cbvmptv 4975 . . . . . . . . 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 2850 . . . . . . . 8 (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2))))) = 𝑂
1110oveq2i 6921 . . . . . . 7 (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = (ℝ D 𝑂)
1211dmeqi 5561 . . . . . 6 dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = dom (ℝ D 𝑂)
1312ineq2i 4040 . . . . 5 (ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2))))))) = (ran 𝑆 ∩ dom (ℝ D 𝑂))
1413sneqi 4410 . . . 4 {(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} = {(ran 𝑆 ∩ dom (ℝ D 𝑂))}
1514uneq1i 3992 . . 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 8313 . . . . 5 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∈ Fin
17 fzofi 13075 . . . . . 6 (0..^𝑁) ∈ Fin
18 eqid 2825 . . . . . . 7 (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
1918rnmptfi 40155 . . . . . 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 8502 . . . . 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 683 . . . 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, 23syl5eqel 2910 . 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 4669 . . . 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, 26syl6eleq 2916 . . 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 476 . . . . 5 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → 𝜑)
29 uniun 4681 . . . . . . . . 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 2898 . . . . . . . 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 3982 . . . . . . . 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 211 . . . . . . 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 475 . . . . . 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 6942 . . . . . . . . . . . . . 14 (0...𝑁) ∈ V
3635a1i 11 . . . . . . . . . . . . 13 (𝜑 → (0...𝑁) ∈ V)
37 fex 6750 . . . . . . . . . . . . 13 ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (0...𝑁) ∈ V) → 𝑆 ∈ V)
3834, 36, 37syl2anc 579 . . . . . . . . . . . 12 (𝜑𝑆 ∈ V)
39 rnexg 7364 . . . . . . . . . . . 12 (𝑆 ∈ V → ran 𝑆 ∈ V)
4038, 39syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑆 ∈ V)
41 inex1g 5028 . . . . . . . . . . 11 (ran 𝑆 ∈ V → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V)
4240, 41syl 17 . . . . . . . . . 10 (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V)
43 unisng 4675 . . . . . . . . . 10 ((ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V → {(ran 𝑆 ∩ dom (ℝ D 𝑂))} = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
4442, 43syl 17 . . . . . . . . 9 (𝜑 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
4544eleq2d 2892 . . . . . . . 8 (𝜑 → (𝑠 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ↔ 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))))
4645adantr 474 . . . . . . 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))))
4746orbi1d 945 . . . . . 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)))))))
4833, 47mpbid 224 . . . . 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))))))
49 dvf 24077 . . . . . . . . 9 (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ
5049a1i 11 . . . . . . . 8 (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ)
51 elinel2 4029 . . . . . . . 8 (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → 𝑠 ∈ dom (ℝ D 𝑂))
5250, 51ffvelrnd 6614 . . . . . . 7 (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
5352adantl 475 . . . . . 6 ((𝜑𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
54 ovex 6942 . . . . . . . . . . . 12 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ∈ V
5554dfiun3 5617 . . . . . . . . . . 11 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
5655eleq2i 2898 . . . . . . . . . 10 (𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
5756biimpri 220 . . . . . . . . 9 (𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
5857adantl 475 . . . . . . . 8 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
59 eliun 4746 . . . . . . . 8 (𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6058, 59sylib 210 . . . . . . 7 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
61 nfv 2013 . . . . . . . . 9 𝑗𝜑
62 nfmpt1 4972 . . . . . . . . . . . 12 𝑗(𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6362nfrn 5605 . . . . . . . . . . 11 𝑗ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6463nfuni 4666 . . . . . . . . . 10 𝑗 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6564nfcri 2963 . . . . . . . . 9 𝑗 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6661, 65nfan 2002 . . . . . . . 8 𝑗(𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
67 nfv 2013 . . . . . . . 8 𝑗((ℝ D 𝑂)‘𝑠) ∈ ℂ
6849a1i 11 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ)
691reseq1i 5629 . . . . . . . . . . . . . . . . . . . 20 (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
70 ioossicc 12554 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑆𝑗)[,](𝑆‘(𝑗 + 1)))
71 fourierdlem80.sjss . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
7270, 71syl5ss 3838 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
7372resmptd 5693 . . . . . . . . . . . . . . . . . . . 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, 73syl5eq 2873 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
75 fourierdlem80.y . . . . . . . . . . . . . . . . . . 19 𝑌 = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
7674, 75syl6reqr 2880 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑌 = (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
7776oveq2d 6926 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
78 ax-resscn 10316 . . . . . . . . . . . . . . . . . . . . 21 ℝ ⊆ ℂ
7978a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ℝ ⊆ ℂ)
80 fourierdlem80.f . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐹:ℝ⟶ℝ)
8180adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℝ)
82 fourierdlem80.xre . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑋 ∈ ℝ)
8382adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
84 fourierdlem80.a . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐴 ∈ ℝ)
85 fourierdlem80.b . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐵 ∈ ℝ)
8684, 85iccssred 40520 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8786sselda 3827 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
8883, 87readdcld 10393 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
8981, 88ffvelrnd 6614 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
9089recnd 10392 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
91 fourierdlem80.c . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐶 ∈ ℝ)
9291recnd 10392 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐶 ∈ ℂ)
9392adantr 474 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)
9490, 93subcld 10720 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
95 2cnd 11436 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 2 ∈ ℂ)
9686, 79sstrd 3837 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
9796sselda 3827 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
9897halfcld 11610 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝑠 / 2) ∈ ℂ)
9998sincld 15239 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
10095, 99mulcld 10384 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
101 2ne0 11469 . . . . . . . . . . . . . . . . . . . . . . . 24 2 ≠ 0
102101a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 2 ≠ 0)
103 fourierdlem80.ab . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
104103sselda 3827 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ (-π[,]π))
105 eqcom 2832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑠 = 0 ↔ 0 = 𝑠)
106105biimpi 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = 0 → 0 = 𝑠)
107106adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 = 𝑠)
108 simpl 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 𝑠 ∈ (𝐴[,]𝐵))
109107, 108eqeltrd 2906 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
110109adantll 705 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
111 fourierdlem80.n0 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
112111ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → ¬ 0 ∈ (𝐴[,]𝐵))
113110, 112pm2.65da 851 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → ¬ 𝑠 = 0)
114113neqned 3006 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ≠ 0)
115 fourierdlem44 41156 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
116104, 114, 115syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
11795, 99, 102, 116mulne0d 11011 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
11894, 100, 117divcld 11134 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
119118, 1fmptd 6638 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑂:(𝐴[,]𝐵)⟶ℂ)
120 ioossre 12530 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ
121120a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ)
122 eqid 2825 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
123122tgioo2 22983 . . . . . . . . . . . . . . . . . . . . 21 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
124122, 123dvres 24081 . . . . . . . . . . . . . . . . . . . 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))))))
12579, 119, 86, 121, 124syl22anc 872 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
126 ioontr 40527 . . . . . . . . . . . . . . . . . . . 20 ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))
127126reseq2i 5630 . . . . . . . . . . . . . . . . . . 19 ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
128125, 127syl6eq 2877 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
129128adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
13077, 129eqtr2d 2862 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0..^𝑁)) → ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (ℝ D 𝑌))
131130dmeqd 5562 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑁)) → dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = dom (ℝ D 𝑌))
13280adantr 474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ)
13382adantr 474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑋 ∈ ℝ)
13486adantr 474 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ ℝ)
13534adantr 474 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
136 elfzofz 12787 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ (0...𝑁))
137136adantl 475 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0...𝑁))
138135, 137ffvelrnd 6614 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) ∈ (𝐴[,]𝐵))
139134, 138sseldd 3828 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) ∈ ℝ)
140 fzofzp1 12867 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0..^𝑁) → (𝑗 + 1) ∈ (0...𝑁))
141140adantl 475 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑗 + 1) ∈ (0...𝑁))
142135, 141ffvelrnd 6614 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ (𝐴[,]𝐵))
143134, 142sseldd 3828 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ ℝ)
144 fdv . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹𝐼)):𝐼⟶ℝ)
145 fourierdlem80.i . . . . . . . . . . . . . . . . . . . . . 22 𝐼 = ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))
146145feq2i 6274 . . . . . . . . . . . . . . . . . . . . 21 ((ℝ D (𝐹𝐼)):𝐼⟶ℝ ↔ (ℝ D (𝐹𝐼)):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
147144, 146sylib 210 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹𝐼)):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
148145reseq2i 5630 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹𝐼) = (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))
149148oveq2i 6921 . . . . . . . . . . . . . . . . . . . . 21 (ℝ D (𝐹𝐼)) = (ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))
150149feq1i 6273 . . . . . . . . . . . . . . . . . . . 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)
151147, 150sylib 210 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
152103adantr 474 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ (-π[,]π))
15372, 152sstrd 3837 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
154111adantr 474 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ (𝐴[,]𝐵))
15572, 154ssneldd 3830 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
15691adantr 474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐶 ∈ ℝ)
157132, 133, 139, 143, 151, 153, 155, 156, 75fourierdlem57 41168 . . . . . . . . . . . . . . . . . 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))))
158157simpli 478 . . . . . . . . . . . . . . . . 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)))))
159158simpld 490 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌):((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ)
160 fdm 6290 . . . . . . . . . . . . . . . 16 ((ℝ D 𝑌):((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ → dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
161159, 160syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑁)) → dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
162131, 161eqtr2d 2862 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
163 resss 5662 . . . . . . . . . . . . . . 15 ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ (ℝ D 𝑂)
164 dmss 5559 . . . . . . . . . . . . . . 15 (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ (ℝ D 𝑂) → dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ dom (ℝ D 𝑂))
165163, 164mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑁)) → dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ dom (ℝ D 𝑂))
166162, 165eqsstrd 3864 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
1671663adant3 1166 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
168 simp3 1172 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
169167, 168sseldd 3828 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ dom (ℝ D 𝑂))
17068, 169ffvelrnd 6614 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
1711703exp 1152 . . . . . . . . 9 (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)))
172171adantr 474 . . . . . . . 8 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → (𝑗 ∈ (0..^𝑁) → (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)))
17366, 67, 172rexlimd 3235 . . . . . . 7 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → (∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ))
17460, 173mpd 15 . . . . . 6 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
17553, 174jaodan 985 . . . . 5 ((𝜑 ∧ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∨ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
17628, 48, 175syl2anc 579 . . . 4 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
177176abscld 14559 . . 3 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
17827, 177sylan2 586 . 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)
179 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))))))
180179, 15syl6eleq 2916 . . 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))))))
181 elsni 4416 . . . . . 6 (𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
182 simpr 479 . . . . . . . 8 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
183 fzfid 13074 . . . . . . . . . . 11 (𝜑 → (0...𝑁) ∈ Fin)
184 rnffi 40160 . . . . . . . . . . 11 ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (0...𝑁) ∈ Fin) → ran 𝑆 ∈ Fin)
18534, 183, 184syl2anc 579 . . . . . . . . . 10 (𝜑 → ran 𝑆 ∈ Fin)
186 infi 8459 . . . . . . . . . 10 (ran 𝑆 ∈ Fin → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
187185, 186syl 17 . . . . . . . . 9 (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
188187adantr 474 . . . . . . . 8 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
189182, 188eqeltrd 2906 . . . . . . 7 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 ∈ Fin)
190 nfv 2013 . . . . . . . . 9 𝑠𝜑
191 nfcv 2969 . . . . . . . . . . 11 𝑠ran 𝑆
192 nfcv 2969 . . . . . . . . . . . . 13 𝑠
193 nfcv 2969 . . . . . . . . . . . . 13 𝑠 D
194 nfmpt1 4972 . . . . . . . . . . . . . 14 𝑠(𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
1951, 194nfcxfr 2967 . . . . . . . . . . . . 13 𝑠𝑂
196192, 193, 195nfov 6940 . . . . . . . . . . . 12 𝑠(ℝ D 𝑂)
197196nfdm 5604 . . . . . . . . . . 11 𝑠dom (ℝ D 𝑂)
198191, 197nfin 4047 . . . . . . . . . 10 𝑠(ran 𝑆 ∩ dom (ℝ D 𝑂))
199198nfeq2 2985 . . . . . . . . 9 𝑠 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))
200190, 199nfan 2002 . . . . . . . 8 𝑠(𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
201 simpr 479 . . . . . . . . . . . . 13 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑠𝑟)
202 simpl 476 . . . . . . . . . . . . 13 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
203201, 202eleqtrd 2908 . . . . . . . . . . . 12 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
204203, 51syl 17 . . . . . . . . . . 11 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑠 ∈ dom (ℝ D 𝑂))
205204adantll 705 . . . . . . . . . 10 (((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) ∧ 𝑠𝑟) → 𝑠 ∈ dom (ℝ D 𝑂))
20649ffvelrni 6612 . . . . . . . . . . 11 (𝑠 ∈ dom (ℝ D 𝑂) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
207206abscld 14559 . . . . . . . . . 10 (𝑠 ∈ dom (ℝ D 𝑂) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
208205, 207syl 17 . . . . . . . . 9 (((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) ∧ 𝑠𝑟) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
209208ex 403 . . . . . . . 8 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (𝑠𝑟 → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ))
210200, 209ralrimi 3166 . . . . . . 7 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
211 fimaxre3 11307 . . . . . . 7 ((𝑟 ∈ Fin ∧ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
212189, 210, 211syl2anc 579 . . . . . 6 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
213181, 212sylan2 586 . . . . 5 ((𝜑𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
214213adantlr 706 . . . 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)
215 simpll 783 . . . . 5 (((𝜑𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) ∧ ¬ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → 𝜑)
216 elunnel1 3983 . . . . . 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)))))
217216adantll 705 . . . . 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)))))
218 vex 3417 . . . . . . . . 9 𝑟 ∈ V
21918elrnmpt 5609 . . . . . . . . 9 (𝑟 ∈ V → (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
220218, 219ax-mp 5 . . . . . . . 8 (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
221220biimpi 208 . . . . . . 7 (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
222221adantl 475 . . . . . 6 ((𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
22363nfcri 2963 . . . . . . . 8 𝑗 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
22461, 223nfan 2002 . . . . . . 7 𝑗(𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
225 nfv 2013 . . . . . . 7 𝑗𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦
226 fourierdlem80.fbdioo . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤)
227 fourierdlem80.fdvbdioo . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
228 reeanv 3317 . . . . . . . . . . . . 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))
229226, 227, 228sylanbrc 578 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∃𝑧 ∈ ℝ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
230 simp1 1170 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → (𝜑𝑗 ∈ (0..^𝑁)))
231 simp2l 1260 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → 𝑤 ∈ ℝ)
232 simp2r 1261 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → 𝑧 ∈ ℝ)
233230, 231, 232jca31 510 . . . . . . . . . . . . . . . . 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))
234 simp3l 1262 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤)
235 simp3r 1263 . . . . . . . . . . . . . . . . 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)
236233, 234, 235jca31 510 . . . . . . . . . . . . . . . 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))
237 fourierdlem80.ch . . . . . . . . . . . . . . . 16 (𝜒 ↔ (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
238236, 237sylibr 226 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → 𝜒)
239237biimpi 208 . . . . . . . . . . . . . . . . . . . . 21 (𝜒 → (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
240 simp-5l 805 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → 𝜑)
241239, 240syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜒𝜑)
242241, 80syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝐹:ℝ⟶ℝ)
243241, 82syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝑋 ∈ ℝ)
244 simp-4l 801 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → (𝜑𝑗 ∈ (0..^𝑁)))
245239, 244syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜒 → (𝜑𝑗 ∈ (0..^𝑁)))
246245, 139syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → (𝑆𝑗) ∈ ℝ)
247245, 143syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → (𝑆‘(𝑗 + 1)) ∈ ℝ)
248 fourierdlem80.slt . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) < (𝑆‘(𝑗 + 1)))
249245, 248syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → (𝑆𝑗) < (𝑆‘(𝑗 + 1)))
25071, 152sstrd 3837 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
251245, 250syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
25271, 154ssneldd 3830 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))))
253245, 252syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → ¬ 0 ∈ ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))))
254245, 151syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → (ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
255 simp-4r 803 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → 𝑤 ∈ ℝ)
256239, 255syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝑤 ∈ ℝ)
257239simplrd 786 . . . . . . . . . . . . . . . . . . . 20 (𝜒 → ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤)
258 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) → 𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))
259258, 145syl6eleqr 2917 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) → 𝑡𝐼)
260 rspa 3139 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤𝑡𝐼) → (abs‘(𝐹𝑡)) ≤ 𝑤)
261257, 259, 260syl2an 589 . . . . . . . . . . . . . . . . . . 19 ((𝜒𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))) → (abs‘(𝐹𝑡)) ≤ 𝑤)
262 simpllr 793 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧) → 𝑧 ∈ ℝ)
263239, 262syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝑧 ∈ ℝ)
264149fveq1i 6438 . . . . . . . . . . . . . . . . . . . . . 22 ((ℝ D (𝐹𝐼))‘𝑡) = ((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)
265264fveq2i 6440 . . . . . . . . . . . . . . . . . . . . 21 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) = (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡))
266239simprd 491 . . . . . . . . . . . . . . . . . . . . . 22 (𝜒 → ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
267266r19.21bi 3141 . . . . . . . . . . . . . . . . . . . . 21 ((𝜒𝑡𝐼) → (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
268265, 267syl5eqbrr 4911 . . . . . . . . . . . . . . . . . . . 20 ((𝜒𝑡𝐼) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)) ≤ 𝑧)
269259, 268sylan2 586 . . . . . . . . . . . . . . . . . . 19 ((𝜒𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)) ≤ 𝑧)
270241, 91syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒𝐶 ∈ ℝ)
271242, 243, 246, 247, 249, 251, 253, 254, 256, 261, 263, 269, 270, 75fourierdlem68 41179 . . . . . . . . . . . . . . . . . 18 (𝜒 → (dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ∧ ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
272271simprd 491 . . . . . . . . . . . . . . . . 17 (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
273271simpld 490 . . . . . . . . . . . . . . . . . . 19 (𝜒 → dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
274273raleqdv 3356 . . . . . . . . . . . . . . . . . 18 (𝜒 → (∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
275274rexbidv 3262 . . . . . . . . . . . . . . . . 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))
276272, 275mpbid 224 . . . . . . . . . . . . . . . 16 (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
277126eqcomi 2834 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
278277reseq2i 5630 . . . . . . . . . . . . . . . . . . . . . 22 ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
279278fveq1i 6438 . . . . . . . . . . . . . . . . . . . . 21 (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠)
280 fvres 6456 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = ((ℝ D 𝑂)‘𝑠))
281280adantl 475 . . . . . . . . . . . . . . . . . . . . 21 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = ((ℝ D 𝑂)‘𝑠))
282245, 72syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜒 → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
283282resmptd 5693 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
28469, 283syl5eq 2873 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜒 → (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
285284, 75syl6reqr 2880 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜒𝑌 = (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
286285oveq2d 6926 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜒 → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
287286fveq1d 6439 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜒 → ((ℝ D 𝑌)‘𝑠) = ((ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
288125fveq1d 6439 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
289241, 288syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜒 → ((ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
290287, 289eqtr2d 2862 . . . . . . . . . . . . . . . . . . . . . 22 (𝜒 → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
291290adantr 474 . . . . . . . . . . . . . . . . . . . . 21 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
292279, 281, 2913eqtr3a 2885 . . . . . . . . . . . . . . . . . . . 20 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) = ((ℝ D 𝑌)‘𝑠))
293292fveq2d 6441 . . . . . . . . . . . . . . . . . . 19 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (abs‘((ℝ D 𝑂)‘𝑠)) = (abs‘((ℝ D 𝑌)‘𝑠)))
294293breq1d 4885 . . . . . . . . . . . . . . . . . 18 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ (abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
295294ralbidva 3194 . . . . . . . . . . . . . . . . 17 (𝜒 → (∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
296295rexbidv 3262 . . . . . . . . . . . . . . . 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))
297276, 296mpbird 249 . . . . . . . . . . . . . . 15 (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
298238, 297syl 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)
2992983exp 1152 . . . . . . . . . . . . 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)))
300299rexlimdvv 3247 . . . . . . . . . . . 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))
301229, 300mpd 15 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
3023013adant3 1166 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
303 raleq 3350 . . . . . . . . . . . 12 (𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → (∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
3043033ad2ant3 1169 . . . . . . . . . . 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))
305304rexbidv 3262 . . . . . . . . . 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))
306302, 305mpbird 249 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
3073063exp 1152 . . . . . . . 8 (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)))
308307adantr 474 . . . . . . 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)))
309224, 225, 308rexlimd 3235 . . . . . 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))
310222, 309mpd 15 . . . . 5 ((𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
311215, 217, 310syl2anc 579 . . . 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)
312214, 311pm2.61dan 847 . . 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)
313180, 312sylan2 586 . 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)
314 pm3.22 453 . . . . . . . . . . . 12 ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → (𝑟 ∈ ran 𝑆𝑟 ∈ dom (ℝ D 𝑂)))
315 elin 4025 . . . . . . . . . . . 12 (𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ↔ (𝑟 ∈ ran 𝑆𝑟 ∈ dom (ℝ D 𝑂)))
316314, 315sylibr 226 . . . . . . . . . . 11 ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
317316adantll 705 . . . . . . . . . 10 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
31844eqcomd 2831 . . . . . . . . . . 11 (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
319318ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
320317, 319eleqtrd 2908 . . . . . . . . 9 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
321320orcd 904 . . . . . . . 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))))))
322 simpll 783 . . . . . . . . . . 11 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝜑)
32378a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → ℝ ⊆ ℂ)
324119adantr 474 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑂:(𝐴[,]𝐵)⟶ℂ)
32584adantr 474 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝐴 ∈ ℝ)
32685adantr 474 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝐵 ∈ ℝ)
327325, 326iccssred 40520 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → (𝐴[,]𝐵) ⊆ ℝ)
328323, 324, 327dvbss 24071 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → dom (ℝ D 𝑂) ⊆ (𝐴[,]𝐵))
329 simpr 479 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ dom (ℝ D 𝑂))
330328, 329sseldd 3828 . . . . . . . . . . . 12 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ (𝐴[,]𝐵))
331330adantr 474 . . . . . . . . . . 11 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (𝐴[,]𝐵))
332 simpr 479 . . . . . . . . . . 11 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ¬ 𝑟 ∈ ran 𝑆)
333 fourierdlem80.relioo . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
334 fveq2 6437 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑘 → (𝑆𝑗) = (𝑆𝑘))
335 oveq1 6917 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
336335fveq2d 6441 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑘 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝑘 + 1)))
337334, 336oveq12d 6928 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
338 ovex 6942 . . . . . . . . . . . . . . . 16 ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))) ∈ V
339337, 18, 338fvmpt 6533 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0..^𝑁) → ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) = ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
340339eleq2d 2892 . . . . . . . . . . . . . 14 (𝑘 ∈ (0..^𝑁) → (𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ 𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1)))))
341340rexbiia 3250 . . . . . . . . . . . . 13 (∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
342333, 341sylibr 226 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
34354, 18dmmpti 6260 . . . . . . . . . . . . 13 dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (0..^𝑁)
344343rexeqi 3355 . . . . . . . . . . . 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))
345342, 344sylibr 226 . . . . . . . . . . 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))
346322, 331, 332, 345syl21anc 871 . . . . . . . . . 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))
347 funmpt 6165 . . . . . . . . . . 11 Fun (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
348 elunirn 6769 . . . . . . . . . . 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)))
349347, 348mp1i 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)))
350346, 349mpbird 249 . . . . . . . . 9 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
351350olcd 905 . . . . . . . 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))))))
352321, 351pm2.61dan 847 . . . . . . 7 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → (𝑟 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
353 elun 3982 . . . . . . 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))))))
354352, 353sylibr 226 . . . . . 6 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ ( {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
355354, 29syl6eleqr 2917 . . . . 5 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
356355ralrimiva 3175 . . . 4 (𝜑 → ∀𝑟 ∈ dom (ℝ D 𝑂)𝑟 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
357 dfss3 3816 . . . 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))))))
358356, 357sylibr 226 . . 3 (𝜑 → dom (ℝ D 𝑂) ⊆ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
359358, 26syl6sseqr 3877 . 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))))))
36024, 178, 313, 359ssfiunibd 40315 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 198  wa 386  wo 878  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wral 3117  wrex 3118  Vcvv 3414  cun 3796  cin 3797  wss 3798  {csn 4399   cuni 4660   ciun 4742   class class class wbr 4875  cmpt 4954  dom cdm 5346  ran crn 5347  cres 5348  Fun wfun 6121  wf 6123  cfv 6127  (class class class)co 6910  Fincfn 8228  cc 10257  cr 10258  0cc0 10259  1c1 10260   + caddc 10262   · cmul 10264   < clt 10398  cle 10399  cmin 10592  -cneg 10593   / cdiv 11016  2c2 11413  (,)cioo 12470  [,]cicc 12473  ...cfz 12626  ..^cfzo 12767  cexp 13161  abscabs 14358  sincsin 15173  cosccos 15174  πcpi 15176  TopOpenctopn 16442  topGenctg 16458  fldccnfld 20113  intcnt 21199   D cdv 24033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337  ax-addf 10338  ax-mulf 10339
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-of 7162  df-om 7332  df-1st 7433  df-2nd 7434  df-supp 7565  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-ixp 8182  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-fsupp 8551  df-fi 8592  df-sup 8623  df-inf 8624  df-oi 8691  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-5 11424  df-6 11425  df-7 11426  df-8 11427  df-9 11428  df-n0 11626  df-z 11712  df-dec 11829  df-uz 11976  df-q 12079  df-rp 12120  df-xneg 12239  df-xadd 12240  df-xmul 12241  df-ioo 12474  df-ioc 12475  df-ico 12476  df-icc 12477  df-fz 12627  df-fzo 12768  df-fl 12895  df-mod 12971  df-seq 13103  df-exp 13162  df-fac 13361  df-bc 13390  df-hash 13418  df-shft 14191  df-cj 14223  df-re 14224  df-im 14225  df-sqrt 14359  df-abs 14360  df-limsup 14586  df-clim 14603  df-rlim 14604  df-sum 14801  df-ef 15177  df-sin 15179  df-cos 15180  df-pi 15182  df-struct 16231  df-ndx 16232  df-slot 16233  df-base 16235  df-sets 16236  df-ress 16237  df-plusg 16325  df-mulr 16326  df-starv 16327  df-sca 16328  df-vsca 16329  df-ip 16330  df-tset 16331  df-ple 16332  df-ds 16334  df-unif 16335  df-hom 16336  df-cco 16337  df-rest 16443  df-topn 16444  df-0g 16462  df-gsum 16463  df-topgen 16464  df-pt 16465  df-prds 16468  df-xrs 16522  df-qtop 16527  df-imas 16528  df-xps 16530  df-mre 16606  df-mrc 16607  df-acs 16609  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-submnd 17696  df-mulg 17902  df-cntz 18107  df-cmn 18555  df-psmet 20105  df-xmet 20106  df-met 20107  df-bl 20108  df-mopn 20109  df-fbas 20110  df-fg 20111  df-cnfld 20114  df-top 21076  df-topon 21093  df-topsp 21115  df-bases 21128  df-cld 21201  df-ntr 21202  df-cls 21203  df-nei 21280  df-lp 21318  df-perf 21319  df-cn 21409  df-cnp 21410  df-t1 21496  df-haus 21497  df-cmp 21568  df-tx 21743  df-hmeo 21936  df-fil 22027  df-fm 22119  df-flim 22120  df-flf 22121  df-xms 22502  df-ms 22503  df-tms 22504  df-cncf 23058  df-limc 24036  df-dv 24037
This theorem is referenced by:  fourierdlem103  41214  fourierdlem104  41215
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