Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem80 Structured version   Visualization version   GIF version

Theorem fourierdlem80 39079
 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 6557 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → (𝑋 + 𝑠) = (𝑋 + 𝑡))
32fveq2d 6107 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑡)))
43oveq1d 6564 . . . . . . . . . . 11 (𝑠 = 𝑡 → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) = ((𝐹‘(𝑋 + 𝑡)) − 𝐶))
5 oveq1 6556 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → (𝑠 / 2) = (𝑡 / 2))
65fveq2d 6107 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (sin‘(𝑠 / 2)) = (sin‘(𝑡 / 2)))
76oveq2d 6565 . . . . . . . . . . 11 (𝑠 = 𝑡 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝑡 / 2))))
84, 7oveq12d 6567 . . . . . . . . . 10 (𝑠 = 𝑡 → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) = (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))
98cbvmptv 4678 . . . . . . . . 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 2633 . . . . . . . 8 (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2))))) = 𝑂
1110oveq2i 6560 . . . . . . 7 (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = (ℝ D 𝑂)
1211dmeqi 5247 . . . . . 6 dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = dom (ℝ D 𝑂)
1312ineq2i 3773 . . . . 5 (ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2))))))) = (ran 𝑆 ∩ dom (ℝ D 𝑂))
1413sneqi 4136 . . . 4 {(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} = {(ran 𝑆 ∩ dom (ℝ D 𝑂))}
1514uneq1i 3725 . . 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 7923 . . . . 5 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∈ Fin
17 fzofi 12635 . . . . . 6 (0..^𝑁) ∈ Fin
18 eqid 2610 . . . . . . 7 (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
1918rnmptfi 38346 . . . . . 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 8112 . . . . 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 704 . . . 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 2692 . 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 4381 . . . 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 2698 . . 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 472 . . . . 5 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → 𝜑)
29 uniun 4392 . . . . . . . . 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 2680 . . . . . . . 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 3715 . . . . . . . 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 208 . . . . . . 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 481 . . . . . 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 6577 . . . . . . . . . . . . . 14 (0...𝑁) ∈ V
3635a1i 11 . . . . . . . . . . . . 13 (𝜑 → (0...𝑁) ∈ V)
37 fex 6394 . . . . . . . . . . . . 13 ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (0...𝑁) ∈ V) → 𝑆 ∈ V)
3834, 36, 37syl2anc 691 . . . . . . . . . . . 12 (𝜑𝑆 ∈ V)
39 rnexg 6990 . . . . . . . . . . . 12 (𝑆 ∈ V → ran 𝑆 ∈ V)
4038, 39syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑆 ∈ V)
41 inex1g 4729 . . . . . . . . . . 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 4388 . . . . . . . . . 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 2673 . . . . . . . 8 (𝜑 → (𝑠 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ↔ 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))))
4645adantr 480 . . . . . . 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 735 . . . . . 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 221 . . . . 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 23477 . . . . . . . . 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 3762 . . . . . . . 8 (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → 𝑠 ∈ dom (ℝ D 𝑂))
5250, 51ffvelrnd 6268 . . . . . . 7 (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
5352adantl 481 . . . . . 6 ((𝜑𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
54 ovex 6577 . . . . . . . . . . . 12 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ∈ V
5554dfiun3 5301 . . . . . . . . . . 11 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
5655eleq2i 2680 . . . . . . . . . 10 (𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
5756biimpri 217 . . . . . . . . 9 (𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
5857adantl 481 . . . . . . . 8 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
59 eliun 4460 . . . . . . . 8 (𝑠 𝑗 ∈ (0..^𝑁)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6058, 59sylib 207 . . . . . . 7 ((𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
61 nfv 1830 . . . . . . . . 9 𝑗𝜑
62 nfmpt1 4675 . . . . . . . . . . . 12 𝑗(𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6362nfrn 5289 . . . . . . . . . . 11 𝑗ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6463nfuni 4378 . . . . . . . . . 10 𝑗 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6564nfcri 2745 . . . . . . . . 9 𝑗 𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
6661, 65nfan 1816 . . . . . . . 8 𝑗(𝜑𝑠 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
67 nfv 1830 . . . . . . . 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 5313 . . . . . . . . . . . . . . . . . . . 20 (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
70 ioossicc 12130 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑆𝑗)[,](𝑆‘(𝑗 + 1)))
71 fourierdlem80.sjss . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
7270, 71syl5ss 3579 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
7372resmptd 5371 . . . . . . . . . . . . . . . . . . . 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 2656 . . . . . . . . . . . . . . . . . . 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 2663 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑌 = (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
7776oveq2d 6565 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
78 ax-resscn 9872 . . . . . . . . . . . . . . . . . . . . 21 ℝ ⊆ ℂ
7978a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ℝ ⊆ ℂ)
80 fourierdlem80.f . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐹:ℝ⟶ℝ)
8180adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℝ)
82 fourierdlem80.xre . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑋 ∈ ℝ)
8382adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
84 fourierdlem80.a . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐴 ∈ ℝ)
85 fourierdlem80.b . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐵 ∈ ℝ)
8684, 85iccssred 38574 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8786sselda 3568 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
8883, 87readdcld 9948 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
8981, 88ffvelrnd 6268 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
9089recnd 9947 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
91 fourierdlem80.c . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐶 ∈ ℝ)
9291recnd 9947 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐶 ∈ ℂ)
9392adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)
9490, 93subcld 10271 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
95 2cnd 10970 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 2 ∈ ℂ)
9686, 79sstrd 3578 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
9796sselda 3568 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
9897halfcld 11154 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (𝑠 / 2) ∈ ℂ)
9998sincld 14699 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
10095, 99mulcld 9939 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
101 2ne0 10990 . . . . . . . . . . . . . . . . . . . . . . . 24 2 ≠ 0
102101a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 2 ≠ 0)
103 fourierdlem80.ab . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
104103sselda 3568 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ (-π[,]π))
105 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑠 = 0 ↔ 0 = 𝑠)
106105biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = 0 → 0 = 𝑠)
107106adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 = 𝑠)
108 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 𝑠 ∈ (𝐴[,]𝐵))
109107, 108eqeltrd 2688 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
110109adantll 746 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
111 fourierdlem80.n0 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
112111ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → ¬ 0 ∈ (𝐴[,]𝐵))
113110, 112pm2.65da 598 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → ¬ 𝑠 = 0)
114113neqned 2789 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ≠ 0)
115 fourierdlem44 39044 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
116104, 114, 115syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
11795, 99, 102, 116mulne0d 10558 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
11894, 100, 117divcld 10680 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ (𝐴[,]𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
119118, 1fmptd 6292 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑂:(𝐴[,]𝐵)⟶ℂ)
120 ioossre 12106 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ
121120a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ)
122 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
123122tgioo2 22414 . . . . . . . . . . . . . . . . . . . . 21 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
124122, 123dvres 23481 . . . . . . . . . . . . . . . . . . . 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 1319 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
126 ioontr 38583 . . . . . . . . . . . . . . . . . . . 20 ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))
127126reseq2i 5314 . . . . . . . . . . . . . . . . . . 19 ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
128125, 127syl6eq 2660 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
129128adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
13077, 129eqtr2d 2645 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0..^𝑁)) → ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (ℝ D 𝑌))
131130dmeqd 5248 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑁)) → dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = dom (ℝ D 𝑌))
13280adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ)
13382adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑋 ∈ ℝ)
13486adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ ℝ)
13534adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
136 elfzofz 12354 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ (0...𝑁))
137136adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0...𝑁))
138135, 137ffvelrnd 6268 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) ∈ (𝐴[,]𝐵))
139134, 138sseldd 3569 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) ∈ ℝ)
140 fzofzp1 12431 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0..^𝑁) → (𝑗 + 1) ∈ (0...𝑁))
141140adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑗 + 1) ∈ (0...𝑁))
142135, 141ffvelrnd 6268 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ (𝐴[,]𝐵))
143134, 142sseldd 3569 . . . . . . . . . . . . . . . . . . 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 5950 . . . . . . . . . . . . . . . . . . . . 21 ((ℝ D (𝐹𝐼)):𝐼⟶ℝ ↔ (ℝ D (𝐹𝐼)):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
147144, 146sylib 207 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹𝐼)):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
148145reseq2i 5314 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹𝐼) = (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))
149148oveq2i 6560 . . . . . . . . . . . . . . . . . . . . 21 (ℝ D (𝐹𝐼)) = (ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))
150149feq1i 5949 . . . . . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
152103adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ (-π[,]π))
15372, 152sstrd 3578 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
154111adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ (𝐴[,]𝐵))
15572, 154ssneldd 3571 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
15691adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐶 ∈ ℝ)
157132, 133, 139, 143, 151, 153, 155, 156, 75fourierdlem57 39056 . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌):((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ)
160 fdm 5964 . . . . . . . . . . . . . . . 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 2645 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = dom ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
163 resss 5342 . . . . . . . . . . . . . . 15 ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ (ℝ D 𝑂)
164 dmss 5245 . . . . . . . . . . . . . . 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 3602 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
1671663adant3 1074 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
168 simp3 1056 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
169167, 168sseldd 3569 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ dom (ℝ D 𝑂))
17068, 169ffvelrnd 6268 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
1711703exp 1256 . . . . . . . . 9 (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)))
172171adantr 480 . . . . . . . 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 3008 . . . . . . 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 822 . . . . 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 691 . . . 4 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
177176abscld 14023 . . 3 ((𝜑𝑠 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
17827, 177sylan2 490 . 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 2698 . . 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 4142 . . . . . 6 (𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
182 simpr 476 . . . . . . . 8 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
183 fzfid 12634 . . . . . . . . . . 11 (𝜑 → (0...𝑁) ∈ Fin)
184 rnffi 38351 . . . . . . . . . . 11 ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (0...𝑁) ∈ Fin) → ran 𝑆 ∈ Fin)
18534, 183, 184syl2anc 691 . . . . . . . . . 10 (𝜑 → ran 𝑆 ∈ Fin)
186 infi 8069 . . . . . . . . . 10 (ran 𝑆 ∈ Fin → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
187185, 186syl 17 . . . . . . . . 9 (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
188187adantr 480 . . . . . . . 8 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
189182, 188eqeltrd 2688 . . . . . . 7 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 ∈ Fin)
190 nfv 1830 . . . . . . . . 9 𝑠𝜑
191 nfcv 2751 . . . . . . . . . . 11 𝑠ran 𝑆
192 nfcv 2751 . . . . . . . . . . . . 13 𝑠
193 nfcv 2751 . . . . . . . . . . . . 13 𝑠 D
194 nfmpt1 4675 . . . . . . . . . . . . . 14 𝑠(𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
1951, 194nfcxfr 2749 . . . . . . . . . . . . 13 𝑠𝑂
196192, 193, 195nfov 6575 . . . . . . . . . . . 12 𝑠(ℝ D 𝑂)
197196nfdm 5288 . . . . . . . . . . 11 𝑠dom (ℝ D 𝑂)
198191, 197nfin 3782 . . . . . . . . . 10 𝑠(ran 𝑆 ∩ dom (ℝ D 𝑂))
199198nfeq2 2766 . . . . . . . . 9 𝑠 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))
200190, 199nfan 1816 . . . . . . . 8 𝑠(𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
201 simpr 476 . . . . . . . . . . . . 13 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑠𝑟)
202 simpl 472 . . . . . . . . . . . . 13 ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠𝑟) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
203201, 202eleqtrd 2690 . . . . . . . . . . . 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 746 . . . . . . . . . 10 (((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) ∧ 𝑠𝑟) → 𝑠 ∈ dom (ℝ D 𝑂))
20649ffvelrni 6266 . . . . . . . . . . 11 (𝑠 ∈ dom (ℝ D 𝑂) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
207206abscld 14023 . . . . . . . . . 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 449 . . . . . . . 8 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (𝑠𝑟 → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ))
210200, 209ralrimi 2940 . . . . . . 7 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
211 fimaxre3 10849 . . . . . . 7 ((𝑟 ∈ Fin ∧ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
212189, 210, 211syl2anc 691 . . . . . 6 ((𝜑𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
213181, 212sylan2 490 . . . . 5 ((𝜑𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
214213adantlr 747 . . . 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 786 . . . . 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 3716 . . . . . 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 746 . . . . 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 3176 . . . . . . . . 9 𝑟 ∈ V
21918elrnmpt 5293 . . . . . . . . 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 205 . . . . . . 7 (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
222221adantl 481 . . . . . 6 ((𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
22363nfcri 2745 . . . . . . . 8 𝑗 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
22461, 223nfan 1816 . . . . . . 7 𝑗(𝜑𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
225 nfv 1830 . . . . . . 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 3086 . . . . . . . . . . . . 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 695 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∃𝑧 ∈ ℝ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
230 simp1 1054 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → (𝜑𝑗 ∈ (0..^𝑁)))
231 simp2l 1080 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → 𝑤 ∈ ℝ)
232 simp2r 1081 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → 𝑧 ∈ ℝ)
233230, 231, 232jca31 555 . . . . . . . . . . . . . . . . 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 1082 . . . . . . . . . . . . . . . . 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 1083 . . . . . . . . . . . . . . . . 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 555 . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤 ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)) → 𝜒)
239237biimpi 205 . . . . . . . . . . . . . . . . . . . . 21 (𝜒 → (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))
240 simp-5l 804 . . . . . . . . . . . . . . . . . . . . 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 802 . . . . . . . . . . . . . . . . . . . . 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 3578 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
251245, 250syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜒 → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
25271, 154ssneldd 3571 . . . . . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . . . . 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 2699 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) → 𝑡𝐼)
260 rspa 2914 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤𝑡𝐼) → (abs‘(𝐹𝑡)) ≤ 𝑤)
261257, 259, 260syl2an 493 . . . . . . . . . . . . . . . . . . 19 ((𝜒𝑡 ∈ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))) → (abs‘(𝐹𝑡)) ≤ 𝑤)
262 simpllr 795 . . . . . . . . . . . . . . . . . . . 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 6104 . . . . . . . . . . . . . . . . . . . . . 22 ((ℝ D (𝐹𝐼))‘𝑡) = ((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)
265264fveq2i 6106 . . . . . . . . . . . . . . . . . . . . 21 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) = (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡))
266239simprd 478 . . . . . . . . . . . . . . . . . . . . . 22 (𝜒 → ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
267266r19.21bi 2916 . . . . . . . . . . . . . . . . . . . . 21 ((𝜒𝑡𝐼) → (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)
268265, 267syl5eqbrr 4619 . . . . . . . . . . . . . . . . . . . 20 ((𝜒𝑡𝐼) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)) ≤ 𝑧)
269259, 268sylan2 490 . . . . . . . . . . . . . . . . . . 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 39067 . . . . . . . . . . . . . . . . . 18 (𝜒 → (dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ∧ ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
272271simprd 478 . . . . . . . . . . . . . . . . 17 (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
273271simpld 474 . . . . . . . . . . . . . . . . . . 19 (𝜒 → dom (ℝ D 𝑌) = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
274273raleqdv 3121 . . . . . . . . . . . . . . . . . 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 3034 . . . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . . . 16 (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
277126eqcomi 2619 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
278277reseq2i 5314 . . . . . . . . . . . . . . . . . . . . . 22 ((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
279278fveq1i 6104 . . . . . . . . . . . . . . . . . . . . 21 (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠)
280 fvres 6117 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → (((ℝ D 𝑂) ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = ((ℝ D 𝑂)‘𝑠))
281280adantl 481 . . . . . . . . . . . . . . . . . . . . 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 5371 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2656 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜒 → (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
285284, 75syl6reqr 2663 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜒𝑌 = (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
286285oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜒 → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
287286fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜒 → ((ℝ D 𝑌)‘𝑠) = ((ℝ D (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
288125fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . . 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 2645 . . . . . . . . . . . . . . . . . . . . . 22 (𝜒 → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
291290adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
292279, 281, 2913eqtr3a 2668 . . . . . . . . . . . . . . . . . . . 20 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) = ((ℝ D 𝑌)‘𝑠))
293292fveq2d 6107 . . . . . . . . . . . . . . . . . . 19 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → (abs‘((ℝ D 𝑂)‘𝑠)) = (abs‘((ℝ D 𝑌)‘𝑠)))
294293breq1d 4593 . . . . . . . . . . . . . . . . . 18 ((𝜒𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ (abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
295294ralbidva 2968 . . . . . . . . . . . . . . . . 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 3034 . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . 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 1256 . . . . . . . . . . . . 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 3019 . . . . . . . . . . . 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 1074 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
303 raleq 3115 . . . . . . . . . . . 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 1077 . . . . . . . . . . 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 3034 . . . . . . . . . 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 246 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
3073063exp 1256 . . . . . . . 8 (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑟 = ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) → ∃𝑦 ∈ ℝ ∀𝑠𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)))
308307adantr 480 . . . . . . 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 3008 . . . . . 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 691 . . . 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 828 . . 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 490 . 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 pm3.22 464 . . . . . . . . . . . 12 ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → (𝑟 ∈ ran 𝑆𝑟 ∈ dom (ℝ D 𝑂)))
315 elin 3758 . . . . . . . . . . . 12 (𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ↔ (𝑟 ∈ ran 𝑆𝑟 ∈ dom (ℝ D 𝑂)))
316314, 315sylibr 223 . . . . . . . . . . 11 ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
317316adantll 746 . . . . . . . . . 10 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
31844eqcomd 2616 . . . . . . . . . . 11 (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
319318ad2antrr 758 . . . . . . . . . 10 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
320317, 319eleqtrd 2690 . . . . . . . . 9 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
321320orcd 406 . . . . . . . 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 786 . . . . . . . . . . 11 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝜑)
32378a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → ℝ ⊆ ℂ)
324119adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑂:(𝐴[,]𝐵)⟶ℂ)
32584adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝐴 ∈ ℝ)
32685adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝐵 ∈ ℝ)
327325, 326iccssred 38574 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → (𝐴[,]𝐵) ⊆ ℝ)
328323, 324, 327dvbss 23471 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → dom (ℝ D 𝑂) ⊆ (𝐴[,]𝐵))
329 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ dom (ℝ D 𝑂))
330328, 329sseldd 3569 . . . . . . . . . . . 12 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ (𝐴[,]𝐵))
331330adantr 480 . . . . . . . . . . 11 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (𝐴[,]𝐵))
332 simpr 476 . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑘 → (𝑆𝑗) = (𝑆𝑘))
335 oveq1 6556 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
336335fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑘 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝑘 + 1)))
337334, 336oveq12d 6567 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) = ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
338 ovex 6577 . . . . . . . . . . . . . . . 16 ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))) ∈ V
339337, 18, 338fvmpt 6191 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0..^𝑁) → ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) = ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
340339eleq2d 2673 . . . . . . . . . . . . . 14 (𝑘 ∈ (0..^𝑁) → (𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ 𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1)))))
341340rexbiia 3022 . . . . . . . . . . . . 13 (∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))
342333, 341sylibr 223 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
34354, 18dmmpti 5936 . . . . . . . . . . . . 13 dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) = (0..^𝑁)
344343rexeqi 3120 . . . . . . . . . . . 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 223 . . . . . . . . . . 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 1317 . . . . . . . . . 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 5840 . . . . . . . . . . 11 Fun (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))
348 elunirn 6413 . . . . . . . . . . 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 246 . . . . . . . . 9 (((𝜑𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))))
351350olcd 407 . . . . . . . 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 828 . . . . . . 7 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → (𝑟 {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
353 elun 3715 . . . . . . 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 223 . . . . . 6 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ ( {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
355354, 29syl6eleqr 2699 . . . . 5 ((𝜑𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
356355ralrimiva 2949 . . . 4 (𝜑 → ∀𝑟 ∈ dom (ℝ D 𝑂)𝑟 ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
357 dfss3 3558 . . . 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 223 . . 3 (𝜑 → dom (ℝ D 𝑂) ⊆ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))))))
359358, 26syl6sseqr 3615 . 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 38464 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145  -cneg 10146   / cdiv 10563  2c2 10947  (,)cioo 12046  [,]cicc 12049  ...cfz 12197  ..^cfzo 12334  ↑cexp 12722  abscabs 13822  sincsin 14633  cosccos 14634  πcpi 14636  TopOpenctopn 15905  topGenctg 15921  ℂfldccnfld 19567  intcnt 20631   D cdv 23433 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-t1 20928  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437 This theorem is referenced by:  fourierdlem103  39102  fourierdlem104  39103
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