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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  cc 9813  cr 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  fldccnfld 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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