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Theorem fourierdlem93 40930
Description: Integral by substitution (the domain is shifted by 𝑋) for a piecewise continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem93.1 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem93.2 𝐻 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋))
fourierdlem93.3 (𝜑𝑀 ∈ ℕ)
fourierdlem93.4 (𝜑𝑄 ∈ (𝑃𝑀))
fourierdlem93.5 (𝜑𝑋 ∈ ℝ)
fourierdlem93.6 (𝜑𝐹:(-π[,]π)⟶ℂ)
fourierdlem93.7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem93.8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
fourierdlem93.9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
Assertion
Ref Expression
fourierdlem93 (𝜑 → ∫(-π[,]π)(𝐹𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠)
Distinct variable groups:   𝑖,𝐹,𝑠,𝑡   𝑖,𝐻,𝑠,𝑡   𝑡,𝐿   𝑖,𝑀,𝑚,𝑝   𝑀,𝑠,𝑡   𝑄,𝑖,𝑝   𝑄,𝑠,𝑡   𝑡,𝑅   𝑖,𝑋,𝑠,𝑡   𝜑,𝑖,𝑠,𝑡
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝑃(𝑡,𝑖,𝑚,𝑠,𝑝)   𝑄(𝑚)   𝑅(𝑖,𝑚,𝑠,𝑝)   𝐹(𝑚,𝑝)   𝐻(𝑚,𝑝)   𝐿(𝑖,𝑚,𝑠,𝑝)   𝑋(𝑚,𝑝)

Proof of Theorem fourierdlem93
Dummy variables 𝑟 𝑥 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem93.4 . . . . . . . 8 (𝜑𝑄 ∈ (𝑃𝑀))
2 fourierdlem93.3 . . . . . . . . 9 (𝜑𝑀 ∈ ℕ)
3 fourierdlem93.1 . . . . . . . . . 10 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
43fourierdlem2 40840 . . . . . . . . 9 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
52, 4syl 17 . . . . . . . 8 (𝜑 → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
61, 5mpbid 222 . . . . . . 7 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
76simprd 483 . . . . . 6 (𝜑 → (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
87simplld 751 . . . . 5 (𝜑 → (𝑄‘0) = -π)
98eqcomd 2777 . . . 4 (𝜑 → -π = (𝑄‘0))
107simplrd 753 . . . . 5 (𝜑 → (𝑄𝑀) = π)
1110eqcomd 2777 . . . 4 (𝜑 → π = (𝑄𝑀))
129, 11oveq12d 6813 . . 3 (𝜑 → (-π[,]π) = ((𝑄‘0)[,](𝑄𝑀)))
1312itgeq1d 40687 . 2 (𝜑 → ∫(-π[,]π)(𝐹𝑡) d𝑡 = ∫((𝑄‘0)[,](𝑄𝑀))(𝐹𝑡) d𝑡)
14 0zd 11595 . . 3 (𝜑 → 0 ∈ ℤ)
15 nnuz 11929 . . . . 5 ℕ = (ℤ‘1)
162, 15syl6eleq 2860 . . . 4 (𝜑𝑀 ∈ (ℤ‘1))
17 1e0p1 11758 . . . . . 6 1 = (0 + 1)
1817a1i 11 . . . . 5 (𝜑 → 1 = (0 + 1))
1918fveq2d 6337 . . . 4 (𝜑 → (ℤ‘1) = (ℤ‘(0 + 1)))
2016, 19eleqtrd 2852 . . 3 (𝜑𝑀 ∈ (ℤ‘(0 + 1)))
213, 2, 1fourierdlem15 40853 . . . 4 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
22 pire 24430 . . . . . . 7 π ∈ ℝ
2322renegcli 10547 . . . . . 6 -π ∈ ℝ
24 iccssre 12459 . . . . . 6 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
2523, 22, 24mp2an 672 . . . . 5 (-π[,]π) ⊆ ℝ
2625a1i 11 . . . 4 (𝜑 → (-π[,]π) ⊆ ℝ)
2721, 26fssd 6198 . . 3 (𝜑𝑄:(0...𝑀)⟶ℝ)
287simprd 483 . . . 4 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2928r19.21bi 3081 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
30 fourierdlem93.6 . . . . 5 (𝜑𝐹:(-π[,]π)⟶ℂ)
3130adantr 466 . . . 4 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝐹:(-π[,]π)⟶ℂ)
32 simpr 471 . . . . 5 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
3312eqcomd 2777 . . . . . 6 (𝜑 → ((𝑄‘0)[,](𝑄𝑀)) = (-π[,]π))
3433adantr 466 . . . . 5 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → ((𝑄‘0)[,](𝑄𝑀)) = (-π[,]π))
3532, 34eleqtrd 2852 . . . 4 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝑡 ∈ (-π[,]π))
3631, 35ffvelrnd 6505 . . 3 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → (𝐹𝑡) ∈ ℂ)
3727adantr 466 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
38 elfzofz 12692 . . . . . 6 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
3938adantl 467 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
4037, 39ffvelrnd 6505 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
41 fzofzp1 12772 . . . . . 6 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
4241adantl 467 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
4337, 42ffvelrnd 6505 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
4430feqmptd 6393 . . . . . . . . . 10 (𝜑𝐹 = (𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)))
4544adantr 466 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹 = (𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)))
4645reseq1d 5532 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
47 ioossicc 12463 . . . . . . . . . . 11 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
4847a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
4923rexri 10302 . . . . . . . . . . . . . 14 -π ∈ ℝ*
5049a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → -π ∈ ℝ*)
5122rexri 10302 . . . . . . . . . . . . . 14 π ∈ ℝ*
5251a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → π ∈ ℝ*)
5321ad2antrr 705 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
54 simplr 752 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
55 simpr 471 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
5650, 52, 53, 54, 55fourierdlem1 40839 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑡 ∈ (-π[,]π))
5756ralrimiva 3115 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))𝑡 ∈ (-π[,]π))
58 dfss3 3741 . . . . . . . . . . 11 (((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π) ↔ ∀𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))𝑡 ∈ (-π[,]π))
5957, 58sylibr 224 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
6048, 59sstrd 3762 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
6160resmptd 5592 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)))
6246, 61eqtrd 2805 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)))
6362eqcomd 2777 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
64 fourierdlem93.7 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
6563, 64eqeltrd 2850 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
66 fourierdlem93.9 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
6762oveq1d 6810 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄‘(𝑖 + 1))))
6866, 67eleqtrd 2852 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄‘(𝑖 + 1))))
69 fourierdlem93.8 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
7062oveq1d 6810 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄𝑖)))
7169, 70eleqtrd 2852 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄𝑖)))
7240, 43, 65, 68, 71iblcncfioo 40708 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) ∈ 𝐿1)
7330ad2antrr 705 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝐹:(-π[,]π)⟶ℂ)
7473, 56ffvelrnd 6505 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹𝑡) ∈ ℂ)
7540, 43, 72, 74ibliooicc 40701 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) ∈ 𝐿1)
7614, 20, 27, 29, 36, 75itgspltprt 40709 . 2 (𝜑 → ∫((𝑄‘0)[,](𝑄𝑀))(𝐹𝑡) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡)
77 fvres 6350 . . . . . . . 8 (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡) = (𝐹𝑡))
7877eqcomd 2777 . . . . . . 7 (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) → (𝐹𝑡) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡))
7978adantl 467 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹𝑡) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡))
8079itgeq2dv 23767 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡) d𝑡)
81 eqid 2771 . . . . . 6 (𝑥 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)))) = (𝑥 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥))))
8230adantr 466 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹:(-π[,]π)⟶ℂ)
8382, 59fssresd 6212 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))):((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
8448resabs1d 5568 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
8584, 64eqeltrd 2850 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
8684oveq1d 6810 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
8740, 43, 29, 83limcicciooub 40384 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
8886, 87eqtr3d 2807 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
8966, 88eleqtrd 2852 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
9084eqcomd 2777 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
9190oveq1d 6810 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = (((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
9240, 43, 29, 83limciccioolb 40368 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
9391, 92eqtrd 2805 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
9469, 93eleqtrd 2852 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
95 fourierdlem93.5 . . . . . . 7 (𝜑𝑋 ∈ ℝ)
9695adantr 466 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
9781, 40, 43, 29, 83, 85, 89, 94, 96fourierdlem82 40919 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑡)) d𝑡)
9840adantr 466 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑄𝑖) ∈ ℝ)
9943adantr 466 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
10095ad2antrr 705 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑋 ∈ ℝ)
10198, 100resubcld 10663 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
10299, 100resubcld 10663 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
103 simpr 471 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
104 eliccre 40246 . . . . . . . . . 10 ((((𝑄𝑖) − 𝑋) ∈ ℝ ∧ ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ ℝ)
105101, 102, 103, 104syl3anc 1476 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ ℝ)
106100, 105readdcld 10274 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
107 elicc2 12442 . . . . . . . . . . . 12 ((((𝑄𝑖) − 𝑋) ∈ ℝ ∧ ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ) → (𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ ((𝑄𝑖) − 𝑋) ≤ 𝑡𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋))))
108101, 102, 107syl2anc 573 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ ((𝑄𝑖) − 𝑋) ≤ 𝑡𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋))))
109103, 108mpbid 222 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑡 ∈ ℝ ∧ ((𝑄𝑖) − 𝑋) ≤ 𝑡𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋)))
110109simp2d 1137 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄𝑖) − 𝑋) ≤ 𝑡)
11198, 100, 105lesubadd2d 10831 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (((𝑄𝑖) − 𝑋) ≤ 𝑡 ↔ (𝑄𝑖) ≤ (𝑋 + 𝑡)))
112110, 111mpbid 222 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑄𝑖) ≤ (𝑋 + 𝑡))
113109simp3d 1138 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋))
114100, 105, 99leaddsub2d 10834 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑋 + 𝑡) ≤ (𝑄‘(𝑖 + 1)) ↔ 𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋)))
115113, 114mpbird 247 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ≤ (𝑄‘(𝑖 + 1)))
11698, 99, 106, 112, 115eliccd 40244 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
117 fvres 6350 . . . . . . 7 ((𝑋 + 𝑡) ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑡)) = (𝐹‘(𝑋 + 𝑡)))
118116, 117syl 17 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑡)) = (𝐹‘(𝑋 + 𝑡)))
119118itgeq2dv 23767 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑡)) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
12080, 97, 1193eqtrd 2809 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
121120sumeq2dv 14640 . . 3 (𝜑 → Σ𝑖 ∈ (0..^𝑀)∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
122 oveq2 6803 . . . . . . 7 (𝑠 = 𝑡 → (𝑋 + 𝑠) = (𝑋 + 𝑡))
123122fveq2d 6337 . . . . . 6 (𝑠 = 𝑡 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑡)))
124123cbvitgv 23762 . . . . 5 ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡
125124a1i 11 . . . 4 (𝜑 → ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
126 fourierdlem93.2 . . . . . . . . 9 𝐻 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋))
127126a1i 11 . . . . . . . 8 (𝜑𝐻 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋)))
128 fveq2 6333 . . . . . . . . . 10 (𝑖 = 0 → (𝑄𝑖) = (𝑄‘0))
129128oveq1d 6810 . . . . . . . . 9 (𝑖 = 0 → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
130129adantl 467 . . . . . . . 8 ((𝜑𝑖 = 0) → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
1312nnzd 11687 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
13214, 131, 143jca 1122 . . . . . . . . 9 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ))
133 0le0 11315 . . . . . . . . . . 11 0 ≤ 0
134133a1i 11 . . . . . . . . . 10 (𝜑 → 0 ≤ 0)
135 0red 10246 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
1362nnred 11240 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℝ)
1372nngt0d 11269 . . . . . . . . . . 11 (𝜑 → 0 < 𝑀)
138135, 136, 137ltled 10390 . . . . . . . . . 10 (𝜑 → 0 ≤ 𝑀)
139134, 138jca 501 . . . . . . . . 9 (𝜑 → (0 ≤ 0 ∧ 0 ≤ 𝑀))
140 elfz2 12539 . . . . . . . . 9 (0 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
141132, 139, 140sylanbrc 572 . . . . . . . 8 (𝜑 → 0 ∈ (0...𝑀))
1428, 23syl6eqel 2858 . . . . . . . . 9 (𝜑 → (𝑄‘0) ∈ ℝ)
143142, 95resubcld 10663 . . . . . . . 8 (𝜑 → ((𝑄‘0) − 𝑋) ∈ ℝ)
144127, 130, 141, 143fvmptd 6432 . . . . . . 7 (𝜑 → (𝐻‘0) = ((𝑄‘0) − 𝑋))
1458oveq1d 6810 . . . . . . 7 (𝜑 → ((𝑄‘0) − 𝑋) = (-π − 𝑋))
146144, 145eqtr2d 2806 . . . . . 6 (𝜑 → (-π − 𝑋) = (𝐻‘0))
147 fveq2 6333 . . . . . . . . . 10 (𝑖 = 𝑀 → (𝑄𝑖) = (𝑄𝑀))
148147oveq1d 6810 . . . . . . . . 9 (𝑖 = 𝑀 → ((𝑄𝑖) − 𝑋) = ((𝑄𝑀) − 𝑋))
149148adantl 467 . . . . . . . 8 ((𝜑𝑖 = 𝑀) → ((𝑄𝑖) − 𝑋) = ((𝑄𝑀) − 𝑋))
15014, 131, 1313jca 1122 . . . . . . . . 9 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
151136leidd 10799 . . . . . . . . . 10 (𝜑𝑀𝑀)
152138, 151jca 501 . . . . . . . . 9 (𝜑 → (0 ≤ 𝑀𝑀𝑀))
153 elfz2 12539 . . . . . . . . 9 (𝑀 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
154150, 152, 153sylanbrc 572 . . . . . . . 8 (𝜑𝑀 ∈ (0...𝑀))
15510, 22syl6eqel 2858 . . . . . . . . 9 (𝜑 → (𝑄𝑀) ∈ ℝ)
156155, 95resubcld 10663 . . . . . . . 8 (𝜑 → ((𝑄𝑀) − 𝑋) ∈ ℝ)
157127, 149, 154, 156fvmptd 6432 . . . . . . 7 (𝜑 → (𝐻𝑀) = ((𝑄𝑀) − 𝑋))
15810oveq1d 6810 . . . . . . 7 (𝜑 → ((𝑄𝑀) − 𝑋) = (π − 𝑋))
159157, 158eqtr2d 2806 . . . . . 6 (𝜑 → (π − 𝑋) = (𝐻𝑀))
160146, 159oveq12d 6813 . . . . 5 (𝜑 → ((-π − 𝑋)[,](π − 𝑋)) = ((𝐻‘0)[,](𝐻𝑀)))
161160itgeq1d 40687 . . . 4 (𝜑 → ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐻‘0)[,](𝐻𝑀))(𝐹‘(𝑋 + 𝑡)) d𝑡)
16227ffvelrnda 6504 . . . . . . . 8 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
16395adantr 466 . . . . . . . 8 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
164162, 163resubcld 10663 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
165164, 126fmptd 6529 . . . . . 6 (𝜑𝐻:(0...𝑀)⟶ℝ)
16640, 43, 96, 29ltsub1dd 10844 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) − 𝑋) < ((𝑄‘(𝑖 + 1)) − 𝑋))
16739, 164syldan 579 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
168126fvmpt2 6435 . . . . . . . 8 ((𝑖 ∈ (0...𝑀) ∧ ((𝑄𝑖) − 𝑋) ∈ ℝ) → (𝐻𝑖) = ((𝑄𝑖) − 𝑋))
16939, 167, 168syl2anc 573 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) = ((𝑄𝑖) − 𝑋))
170 fveq2 6333 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
171170oveq1d 6810 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑄𝑖) − 𝑋) = ((𝑄𝑗) − 𝑋))
172171cbvmptv 4885 . . . . . . . . . 10 (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋))
173126, 172eqtri 2793 . . . . . . . . 9 𝐻 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋))
174173a1i 11 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐻 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋)))
175 fveq2 6333 . . . . . . . . . 10 (𝑗 = (𝑖 + 1) → (𝑄𝑗) = (𝑄‘(𝑖 + 1)))
176175oveq1d 6810 . . . . . . . . 9 (𝑗 = (𝑖 + 1) → ((𝑄𝑗) − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
177176adantl 467 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑄𝑗) − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
17843, 96resubcld 10663 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
179174, 177, 42, 178fvmptd 6432 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) − 𝑋))
180166, 169, 1793brtr4d 4819 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) < (𝐻‘(𝑖 + 1)))
181 frn 6190 . . . . . . . . 9 (𝐹:(-π[,]π)⟶ℂ → ran 𝐹 ⊆ ℂ)
18230, 181syl 17 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℂ)
183182adantr 466 . . . . . . 7 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → ran 𝐹 ⊆ ℂ)
184 ffun 6187 . . . . . . . . . 10 (𝐹:(-π[,]π)⟶ℂ → Fun 𝐹)
18530, 184syl 17 . . . . . . . . 9 (𝜑 → Fun 𝐹)
186185adantr 466 . . . . . . . 8 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → Fun 𝐹)
18723a1i 11 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → -π ∈ ℝ)
18822a1i 11 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → π ∈ ℝ)
18995adantr 466 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑋 ∈ ℝ)
190144, 143eqeltrd 2850 . . . . . . . . . . . . 13 (𝜑 → (𝐻‘0) ∈ ℝ)
191190adantr 466 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻‘0) ∈ ℝ)
192157, 156eqeltrd 2850 . . . . . . . . . . . . 13 (𝜑 → (𝐻𝑀) ∈ ℝ)
193192adantr 466 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻𝑀) ∈ ℝ)
194 simpr 471 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)))
195 eliccre 40246 . . . . . . . . . . . 12 (((𝐻‘0) ∈ ℝ ∧ (𝐻𝑀) ∈ ℝ ∧ 𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ℝ)
196191, 193, 194, 195syl3anc 1476 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ℝ)
197189, 196readdcld 10274 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ ℝ)
198128adantl 467 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 = 0) → (𝑄𝑖) = (𝑄‘0))
199198oveq1d 6810 . . . . . . . . . . . . . . 15 ((𝜑𝑖 = 0) → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
200127, 199, 141, 143fvmptd 6432 . . . . . . . . . . . . . 14 (𝜑 → (𝐻‘0) = ((𝑄‘0) − 𝑋))
201200oveq2d 6811 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + (𝐻‘0)) = (𝑋 + ((𝑄‘0) − 𝑋)))
20295recnd 10273 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ ℂ)
203142recnd 10273 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) ∈ ℂ)
204202, 203pncan3d 10600 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + ((𝑄‘0) − 𝑋)) = (𝑄‘0))
205201, 204, 83eqtrrd 2810 . . . . . . . . . . . 12 (𝜑 → -π = (𝑋 + (𝐻‘0)))
206205adantr 466 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → -π = (𝑋 + (𝐻‘0)))
207 elicc2 12442 . . . . . . . . . . . . . . 15 (((𝐻‘0) ∈ ℝ ∧ (𝐻𝑀) ∈ ℝ) → (𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)) ↔ (𝑡 ∈ ℝ ∧ (𝐻‘0) ≤ 𝑡𝑡 ≤ (𝐻𝑀))))
208191, 193, 207syl2anc 573 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)) ↔ (𝑡 ∈ ℝ ∧ (𝐻‘0) ≤ 𝑡𝑡 ≤ (𝐻𝑀))))
209194, 208mpbid 222 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑡 ∈ ℝ ∧ (𝐻‘0) ≤ 𝑡𝑡 ≤ (𝐻𝑀)))
210209simp2d 1137 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻‘0) ≤ 𝑡)
211191, 196, 189, 210leadd2dd 10847 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + (𝐻‘0)) ≤ (𝑋 + 𝑡))
212206, 211eqbrtrd 4809 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → -π ≤ (𝑋 + 𝑡))
213209simp3d 1138 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ≤ (𝐻𝑀))
214196, 193, 189, 213leadd2dd 10847 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ≤ (𝑋 + (𝐻𝑀)))
215157oveq2d 6811 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + (𝐻𝑀)) = (𝑋 + ((𝑄𝑀) − 𝑋)))
216155recnd 10273 . . . . . . . . . . . . . 14 (𝜑 → (𝑄𝑀) ∈ ℂ)
217202, 216pncan3d 10600 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + ((𝑄𝑀) − 𝑋)) = (𝑄𝑀))
218215, 217, 103eqtrrd 2810 . . . . . . . . . . . 12 (𝜑 → π = (𝑋 + (𝐻𝑀)))
219218adantr 466 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → π = (𝑋 + (𝐻𝑀)))
220214, 219breqtrrd 4815 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ≤ π)
221187, 188, 197, 212, 220eliccd 40244 . . . . . . . . 9 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ (-π[,]π))
222 fdm 6192 . . . . . . . . . . . 12 (𝐹:(-π[,]π)⟶ℂ → dom 𝐹 = (-π[,]π))
22330, 222syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = (-π[,]π))
224223eqcomd 2777 . . . . . . . . . 10 (𝜑 → (-π[,]π) = dom 𝐹)
225224adantr 466 . . . . . . . . 9 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (-π[,]π) = dom 𝐹)
226221, 225eleqtrd 2852 . . . . . . . 8 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ dom 𝐹)
227 fvelrn 6497 . . . . . . . 8 ((Fun 𝐹 ∧ (𝑋 + 𝑡) ∈ dom 𝐹) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
228186, 226, 227syl2anc 573 . . . . . . 7 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
229183, 228sseldd 3753 . . . . . 6 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
230169, 167eqeltrd 2850 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℝ)
231179, 178eqeltrd 2850 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
23282, 60fssresd 6212 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
23340rexrd 10294 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ*)
234233adantr 466 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) ∈ ℝ*)
23543rexrd 10294 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
236235adantr 466 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
23795ad2antrr 705 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
238 elioore 12409 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
239238adantl 467 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ℝ)
240237, 239readdcld 10274 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ℝ)
241169oveq2d 6811 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻𝑖)) = (𝑋 + ((𝑄𝑖) − 𝑋)))
242202adantr 466 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
24340recnd 10273 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℂ)
244242, 243pncan3d 10600 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑄𝑖) − 𝑋)) = (𝑄𝑖))
245 eqidd 2772 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = (𝑄𝑖))
246241, 244, 2453eqtrrd 2810 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = (𝑋 + (𝐻𝑖)))
247246adantr 466 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) = (𝑋 + (𝐻𝑖)))
248230adantr 466 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) ∈ ℝ)
249 simpr 471 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
250248rexrd 10294 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) ∈ ℝ*)
251231rexrd 10294 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℝ*)
252251adantr 466 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻‘(𝑖 + 1)) ∈ ℝ*)
253 elioo2 12420 . . . . . . . . . . . . . . . . . 18 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ*) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↔ (𝑡 ∈ ℝ ∧ (𝐻𝑖) < 𝑡𝑡 < (𝐻‘(𝑖 + 1)))))
254250, 252, 253syl2anc 573 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↔ (𝑡 ∈ ℝ ∧ (𝐻𝑖) < 𝑡𝑡 < (𝐻‘(𝑖 + 1)))))
255249, 254mpbid 222 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑡 ∈ ℝ ∧ (𝐻𝑖) < 𝑡𝑡 < (𝐻‘(𝑖 + 1))))
256255simp2d 1137 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) < 𝑡)
257248, 239, 237, 256ltadd2dd 10401 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + (𝐻𝑖)) < (𝑋 + 𝑡))
258247, 257eqbrtrd 4809 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) < (𝑋 + 𝑡))
259231adantr 466 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
260255simp3d 1138 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 < (𝐻‘(𝑖 + 1)))
261239, 259, 237, 260ltadd2dd 10401 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) < (𝑋 + (𝐻‘(𝑖 + 1))))
262179oveq2d 6811 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑋 + ((𝑄‘(𝑖 + 1)) − 𝑋)))
26343recnd 10273 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
264242, 263pncan3d 10600 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑄‘(𝑖 + 1)) − 𝑋)) = (𝑄‘(𝑖 + 1)))
265262, 264eqtrd 2805 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
266265adantr 466 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
267261, 266breqtrd 4813 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) < (𝑄‘(𝑖 + 1)))
268234, 236, 240, 258, 267eliood 40238 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
269 eqid 2771 . . . . . . . . . . . 12 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
270268, 269fmptd 6529 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)):((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))⟶((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
271 fcompt 6545 . . . . . . . . . . 11 (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ ∧ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)):((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))⟶((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) = (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))))
272232, 270, 271syl2anc 573 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) = (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))))
273 oveq2 6803 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑟 → (𝑋 + 𝑡) = (𝑋 + 𝑟))
274273cbvmptv 4885 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))
275274fveq1i 6334 . . . . . . . . . . . . . 14 ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠) = ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠)
276275fveq2i 6336 . . . . . . . . . . . . 13 ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠))
277276mpteq2i 4876 . . . . . . . . . . . 12 (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) = (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠)))
278277a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) = (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠))))
279 fveq2 6333 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠) = ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡))
280279fveq2d 6337 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠)) = ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡)))
281280cbvmptv 4885 . . . . . . . . . . . 12 (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠))) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡)))
282281a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠))) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡))))
283 eqidd 2772 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟)) = (𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟)))
284 oveq2 6803 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑡 → (𝑋 + 𝑟) = (𝑋 + 𝑡))
285284adantl 467 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) ∧ 𝑟 = 𝑡) → (𝑋 + 𝑟) = (𝑋 + 𝑡))
286283, 285, 249, 240fvmptd 6432 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡) = (𝑋 + 𝑡))
287286fveq2d 6337 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡)) = ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑡)))
288 fvres 6350 . . . . . . . . . . . . . 14 ((𝑋 + 𝑡) ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑡)) = (𝐹‘(𝑋 + 𝑡)))
289268, 288syl 17 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑡)) = (𝐹‘(𝑋 + 𝑡)))
290287, 289eqtrd 2805 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡)) = (𝐹‘(𝑋 + 𝑡)))
291290mpteq2dva 4879 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡))) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))))
292278, 282, 2913eqtrd 2809 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))))
293272, 292eqtr2d 2806 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) = ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))))
294 eqid 2771 . . . . . . . . . . 11 (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡))
295 ssid 3773 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
296295a1i 11 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → ℂ ⊆ ℂ)
297 id 22 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → 𝑋 ∈ ℂ)
298296, 297, 296constcncfg 40599 . . . . . . . . . . . . 13 (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
299 cncfmptid 22934 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
300295, 295, 299mp2an 672 . . . . . . . . . . . . . 14 (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ)
301300a1i 11 . . . . . . . . . . . . 13 (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
302298, 301addcncf 40601 . . . . . . . . . . . 12 (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡)) ∈ (ℂ–cn→ℂ))
303242, 302syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡)) ∈ (ℂ–cn→ℂ))
304 ioosscn 40234 . . . . . . . . . . . 12 ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ⊆ ℂ
305304a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ⊆ ℂ)
306 ioosscn 40234 . . . . . . . . . . . 12 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
307306a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
308294, 303, 305, 307, 268cncfmptssg 40598 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
309308, 64cncfco 22929 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
310293, 309eqeltrd 2850 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
311233adantr 466 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄𝑖) ∈ ℝ*)
312235adantr 466 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
313 simpr 471 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
314 vex 3354 . . . . . . . . . . . . . . . . . 18 𝑟 ∈ V
315269elrnmpt 5509 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ V → (𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ↔ ∃𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))𝑟 = (𝑋 + 𝑡)))
316314, 315ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ↔ ∃𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))𝑟 = (𝑋 + 𝑡))
317313, 316sylib 208 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ∃𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))𝑟 = (𝑋 + 𝑡))
318 nfv 1995 . . . . . . . . . . . . . . . . . 18 𝑡(𝜑𝑖 ∈ (0..^𝑀))
319 nfmpt1 4882 . . . . . . . . . . . . . . . . . . . 20 𝑡(𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
320319nfrn 5505 . . . . . . . . . . . . . . . . . . 19 𝑡ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
321320nfcri 2907 . . . . . . . . . . . . . . . . . 18 𝑡 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
322318, 321nfan 1980 . . . . . . . . . . . . . . . . 17 𝑡((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
323 nfv 1995 . . . . . . . . . . . . . . . . 17 𝑡 𝑟 ∈ ℝ
324 simp3 1132 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 = (𝑋 + 𝑡))
325953ad2ant1 1127 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑋 ∈ ℝ)
3262383ad2ant2 1128 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑡 ∈ ℝ)
327325, 326readdcld 10274 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑋 + 𝑡) ∈ ℝ)
328324, 327eqeltrd 2850 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 ∈ ℝ)
3293283exp 1112 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → (𝑟 = (𝑋 + 𝑡) → 𝑟 ∈ ℝ)))
330329ad2antrr 705 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → (𝑟 = (𝑋 + 𝑡) → 𝑟 ∈ ℝ)))
331322, 323, 330rexlimd 3174 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (∃𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))𝑟 = (𝑋 + 𝑡) → 𝑟 ∈ ℝ))
332317, 331mpd 15 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ ℝ)
333 nfv 1995 . . . . . . . . . . . . . . . . 17 𝑡(𝑄𝑖) < 𝑟
3342583adant3 1126 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑄𝑖) < (𝑋 + 𝑡))
335 simp3 1132 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 = (𝑋 + 𝑡))
336334, 335breqtrrd 4815 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑄𝑖) < 𝑟)
3373363exp 1112 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → (𝑟 = (𝑋 + 𝑡) → (𝑄𝑖) < 𝑟)))
338337adantr 466 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → (𝑟 = (𝑋 + 𝑡) → (𝑄𝑖) < 𝑟)))
339322, 333, 338rexlimd 3174 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (∃𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))𝑟 = (𝑋 + 𝑡) → (𝑄𝑖) < 𝑟))
340317, 339mpd 15 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄𝑖) < 𝑟)
341 nfv 1995 . . . . . . . . . . . . . . . . 17 𝑡 𝑟 < (𝑄‘(𝑖 + 1))
3422673adant3 1126 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑋 + 𝑡) < (𝑄‘(𝑖 + 1)))
343335, 342eqbrtrd 4809 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 < (𝑄‘(𝑖 + 1)))
3443433exp 1112 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → (𝑟 = (𝑋 + 𝑡) → 𝑟 < (𝑄‘(𝑖 + 1)))))
345344adantr 466 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → (𝑟 = (𝑋 + 𝑡) → 𝑟 < (𝑄‘(𝑖 + 1)))))
346322, 341, 345rexlimd 3174 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (∃𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))𝑟 = (𝑋 + 𝑡) → 𝑟 < (𝑄‘(𝑖 + 1))))
347317, 346mpd 15 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 < (𝑄‘(𝑖 + 1)))
348311, 312, 332, 340, 347eliood 40238 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
349223ineq2d 3965 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ dom 𝐹) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
350349adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ dom 𝐹) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
351 dmres 5559 . . . . . . . . . . . . . . . . 17 dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ dom 𝐹)
352351a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ dom 𝐹))
353 dfss 3738 . . . . . . . . . . . . . . . . 17 (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π) ↔ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
35460, 353sylib 208 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
355350, 352, 3543eqtr4d 2815 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
356355adantr 466 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
357348, 356eleqtrrd 2853 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
358332, 347ltned 10378 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ≠ (𝑄‘(𝑖 + 1)))
359358neneqd 2948 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 = (𝑄‘(𝑖 + 1)))
360 velsn 4333 . . . . . . . . . . . . . 14 (𝑟 ∈ {(𝑄‘(𝑖 + 1))} ↔ 𝑟 = (𝑄‘(𝑖 + 1)))
361359, 360sylnibr 318 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 ∈ {(𝑄‘(𝑖 + 1))})
362357, 361eldifd 3734 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
363362ralrimiva 3115 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
364 dfss3 3741 . . . . . . . . . . 11 (ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ⊆ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}) ↔ ∀𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
365363, 364sylibr 224 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ⊆ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
366 eqid 2771 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠))
367202adantr 466 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
368 simpr 471 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
369367, 368addcomd 10443 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑠 ∈ ℂ) → (𝑋 + 𝑠) = (𝑠 + 𝑋))
370369mpteq2dva 4879 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)))
371 eqid 2771 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋))
372371addccncf 22938 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
373202, 372syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
374370, 373eqeltrd 2850 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
375374adantr 466 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
376230rexrd 10294 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℝ*)
377 iocssre 12457 . . . . . . . . . . . . . . . . . . 19 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℝ)
378376, 231, 377syl2anc 573 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℝ)
379 ax-resscn 10198 . . . . . . . . . . . . . . . . . 18 ℝ ⊆ ℂ
380378, 379syl6ss 3764 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℂ)
381295a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
382202ad2antrr 705 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℂ)
383380sselda 3752 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
384382, 383addcld 10264 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℂ)
385366, 375, 380, 381, 384cncfmptssg 40598 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))–cn→ℂ))
386 eqid 2771 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
387 eqid 2771 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
388386cnfldtop 22806 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘ℂfld) ∈ Top
389 unicntop 22808 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (TopOpen‘ℂfld)
390389restid 16301 . . . . . . . . . . . . . . . . . . . 20 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
391388, 390ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
392391eqcomi 2780 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
393386, 387, 392cncfcn 22931 . . . . . . . . . . . . . . . . 17 ((((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
394380, 381, 393syl2anc 573 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
395385, 394eleqtrd 2852 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
396386cnfldtopon 22805 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
397396a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
398 resttopon 21185 . . . . . . . . . . . . . . . . 17 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))))
399397, 380, 398syl2anc 573 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))))
400 cncnp 21304 . . . . . . . . . . . . . . . 16 ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → ((𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)) ↔ ((𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)):((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))⟶ℂ ∧ ∀𝑡 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡))))
401399, 397, 400syl2anc 573 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)) ↔ ((𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)):((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))⟶ℂ ∧ ∀𝑡 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡))))
402395, 401mpbid 222 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)):((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))⟶ℂ ∧ ∀𝑡 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡)))
403402simprd 483 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑡 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡))
404 ubioc1 12431 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → (𝐻‘(𝑖 + 1)) ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
405376, 251, 180, 404syl3anc 1476 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
406 fveq2 6333 . . . . . . . . . . . . . . 15 (𝑡 = (𝐻‘(𝑖 + 1)) → ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡) = ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻‘(𝑖 + 1))))
407406eleq2d 2836 . . . . . . . . . . . . . 14 (𝑡 = (𝐻‘(𝑖 + 1)) → ((𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡) ↔ (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻‘(𝑖 + 1)))))
408407rspccva 3459 . . . . . . . . . . . . 13 ((∀𝑡 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡) ∧ (𝐻‘(𝑖 + 1)) ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻‘(𝑖 + 1))))
409403, 405, 408syl2anc 573 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻‘(𝑖 + 1))))
410 snunioo2 40249 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) = ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
411376, 251, 180, 410syl3anc 1476 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) = ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
412265eqcomd 2777 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = (𝑋 + (𝐻‘(𝑖 + 1))))
413412ad2antrr 705 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ 𝑠 = (𝐻‘(𝑖 + 1))) → (𝑄‘(𝑖 + 1)) = (𝑋 + (𝐻‘(𝑖 + 1))))
414 iftrue 4232 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐻‘(𝑖 + 1)) → if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑄‘(𝑖 + 1)))
415414adantl 467 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ 𝑠 = (𝐻‘(𝑖 + 1))) → if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑄‘(𝑖 + 1)))
416 oveq2 6803 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐻‘(𝑖 + 1)) → (𝑋 + 𝑠) = (𝑋 + (𝐻‘(𝑖 + 1))))
417416adantl 467 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ 𝑠 = (𝐻‘(𝑖 + 1))) → (𝑋 + 𝑠) = (𝑋 + (𝐻‘(𝑖 + 1))))
418413, 415, 4173eqtr4d 2815 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ 𝑠 = (𝐻‘(𝑖 + 1))) → if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
419 iffalse 4235 . . . . . . . . . . . . . . . 16 𝑠 = (𝐻‘(𝑖 + 1)) → if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))
420419adantl 467 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))
421 eqidd 2772 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
422 oveq2 6803 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (𝑋 + 𝑡) = (𝑋 + 𝑠))
423422adantl 467 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) ∧ 𝑡 = 𝑠) → (𝑋 + 𝑡) = (𝑋 + 𝑠))
424 velsn 4333 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ {(𝐻‘(𝑖 + 1))} ↔ 𝑠 = (𝐻‘(𝑖 + 1)))
425424notbii 309 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ {(𝐻‘(𝑖 + 1))} ↔ ¬ 𝑠 = (𝐻‘(𝑖 + 1)))
426 elun 3904 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ↔ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}))
427426biimpi 206 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}))
428427orcomd 860 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (𝑠 ∈ {(𝐻‘(𝑖 + 1))} ∨ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
429428ord 853 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (¬ 𝑠 ∈ {(𝐻‘(𝑖 + 1))} → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
430425, 429syl5bir 233 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (¬ 𝑠 = (𝐻‘(𝑖 + 1)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
431430imp 393 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
432431adantll 693 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
43395ad2antrr 705 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → 𝑋 ∈ ℝ)
434 elioore 12409 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
435434adantl 467 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
436 elsni 4334 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ {(𝐻‘(𝑖 + 1))} → 𝑠 = (𝐻‘(𝑖 + 1)))
437436adantl 467 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → 𝑠 = (𝐻‘(𝑖 + 1)))
438231adantr 466 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
439437, 438eqeltrd 2850 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → 𝑠 ∈ ℝ)
440435, 439jaodan 942 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻‘(𝑖 + 1))})) → 𝑠 ∈ ℝ)
441426, 440sylan2b 581 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → 𝑠 ∈ ℝ)
442433, 441readdcld 10274 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → (𝑋 + 𝑠) ∈ ℝ)
443442adantr 466 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → (𝑋 + 𝑠) ∈ ℝ)
444421, 423, 432, 443fvmptd 6432 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠) = (𝑋 + 𝑠))
445420, 444eqtrd 2805 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
446418, 445pm2.61dan 814 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
447411, 446mpteq12dva 4867 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ↦ if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) = (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)))
448411oveq2d 6811 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))))
449448oveq1d 6810 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld)))
450449fveq1d 6335 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → ((((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) CnP (TopOpen‘ℂfld))‘(𝐻‘(𝑖 + 1))) = ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻‘(𝑖 + 1))))
451409, 447, 4503eltr4d 2865 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ↦ if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) ∈ ((((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) CnP (TopOpen‘ℂfld))‘(𝐻‘(𝑖 + 1))))
452 eqid 2771 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) = ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}))
453 eqid 2771 . . . . . . . . . . . 12 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ↦ if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) = (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ↦ if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)))
454270, 307fssd 6198 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)):((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))⟶ℂ)
455231recnd 10273 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℂ)
456452, 386, 453, 454, 305, 455ellimc 23856 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) lim (𝐻‘(𝑖 + 1))) ↔ (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ↦ if(𝑠 = (𝐻‘(𝑖 + 1)), (𝑄‘(𝑖 + 1)), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) ∈ ((((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) CnP (TopOpen‘ℂfld))‘(𝐻‘(𝑖 + 1)))))
457451, 456mpbird 247 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) lim (𝐻‘(𝑖 + 1))))
458365, 457, 66limccog 40367 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻‘(𝑖 + 1))))
459272, 292eqtrd 2805 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))))
460459oveq1d 6810 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻‘(𝑖 + 1))) = ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) lim (𝐻‘(𝑖 + 1))))
461458, 460eleqtrd 2852 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) lim (𝐻‘(𝑖 + 1))))
46240adantr 466 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄𝑖) ∈ ℝ)
463462, 340gtned 10377 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ≠ (𝑄𝑖))
464463neneqd 2948 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 = (𝑄𝑖))
465 velsn 4333 . . . . . . . . . . . . . 14 (𝑟 ∈ {(𝑄𝑖)} ↔ 𝑟 = (𝑄𝑖))
466464, 465sylnibr 318 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 ∈ {(𝑄𝑖)})
467357, 466eldifd 3734 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
468467ralrimiva 3115 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
469 dfss3 3741 . . . . . . . . . . 11 (ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ⊆ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}) ↔ ∀𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
470468, 469sylibr 224 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ⊆ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
471 icossre 12458 . . . . . . . . . . . . . . . . . . 19 (((𝐻𝑖) ∈ ℝ ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ*) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℝ)
472230, 251, 471syl2anc 573 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℝ)
473472, 379syl6ss 3764 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℂ)
474202ad2antrr 705 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℂ)
475473sselda 3752 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
476474, 475addcld 10264 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℂ)
477366, 375, 473, 381, 476cncfmptssg 40598 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
478 eqid 2771 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
479386, 478, 392cncfcn 22931 . . . . . . . . . . . . . . . . 17 ((((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
480473, 381, 479syl2anc 573 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
481477, 480eleqtrd 2852 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
482 resttopon 21185 . . . . . . . . . . . . . . . . 17 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))))
483397, 473, 482syl2anc 573 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))))
484 cncnp 21304 . . . . . . . . . . . . . . . 16 ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → ((𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)) ↔ ((𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)):((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))⟶ℂ ∧ ∀𝑡 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡))))
485483, 397, 484syl2anc 573 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)) ↔ ((𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)):((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))⟶ℂ ∧ ∀𝑡 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡))))
486481, 485mpbid 222 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)):((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))⟶ℂ ∧ ∀𝑡 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡)))
487486simprd 483 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑡 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡))
488 lbico1 12432 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → (𝐻𝑖) ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
489376, 251, 180, 488syl3anc 1476 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
490 fveq2 6333 . . . . . . . . . . . . . . 15 (𝑡 = (𝐻𝑖) → ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡) = ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻𝑖)))
491490eleq2d 2836 . . . . . . . . . . . . . 14 (𝑡 = (𝐻𝑖) → ((𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡) ↔ (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻𝑖))))
492491rspccva 3459 . . . . . . . . . . . . 13 ((∀𝑡 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))(𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘𝑡) ∧ (𝐻𝑖) ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻𝑖)))
493487, 489, 492syl2anc 573 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻𝑖)))
494 uncom 3908 . . . . . . . . . . . . . 14 (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) = ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
495 snunioo 12504 . . . . . . . . . . . . . . 15 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
496376, 251, 180, 495syl3anc 1476 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
497494, 496syl5eq 2817 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
498 iftrue 4232 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐻𝑖) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑄𝑖))
499498adantl 467 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑄𝑖))
500246adantr 466 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → (𝑄𝑖) = (𝑋 + (𝐻𝑖)))
501 oveq2 6803 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝐻𝑖) → (𝑋 + 𝑠) = (𝑋 + (𝐻𝑖)))
502501eqcomd 2777 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐻𝑖) → (𝑋 + (𝐻𝑖)) = (𝑋 + 𝑠))
503502adantl 467 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → (𝑋 + (𝐻𝑖)) = (𝑋 + 𝑠))
504499, 500, 5033eqtrd 2809 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
505504adantlr 694 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
506 iffalse 4235 . . . . . . . . . . . . . . . 16 𝑠 = (𝐻𝑖) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))
507506adantl 467 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))
508 eqidd 2772 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
509422adantl 467 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) ∧ 𝑡 = 𝑠) → (𝑋 + 𝑡) = (𝑋 + 𝑠))
510 velsn 4333 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ {(𝐻𝑖)} ↔ 𝑠 = (𝐻𝑖))
511510notbii 309 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ {(𝐻𝑖)} ↔ ¬ 𝑠 = (𝐻𝑖))
512 elun 3904 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ↔ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻𝑖)}))
513512biimpi 206 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻𝑖)}))
514513orcomd 860 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (𝑠 ∈ {(𝐻𝑖)} ∨ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
515514ord 853 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (¬ 𝑠 ∈ {(𝐻𝑖)} → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
516511, 515syl5bir 233 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (¬ 𝑠 = (𝐻𝑖) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
517516imp 393 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ∧ ¬ 𝑠 = (𝐻𝑖)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
518517adantll 693 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
51995ad2antrr 705 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → 𝑋 ∈ ℝ)
520 elsni 4334 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ {(𝐻𝑖)} → 𝑠 = (𝐻𝑖))
521520adantl 467 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → 𝑠 = (𝐻𝑖))
522230adantr 466 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → (𝐻𝑖) ∈ ℝ)
523521, 522eqeltrd 2850 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → 𝑠 ∈ ℝ)
524435, 523jaodan 942 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻𝑖)})) → 𝑠 ∈ ℝ)
525512, 524sylan2b 581 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → 𝑠 ∈ ℝ)
526519, 525readdcld 10274 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → (𝑋 + 𝑠) ∈ ℝ)
527526adantr 466 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → (𝑋 + 𝑠) ∈ ℝ)
528508, 509, 518, 527fvmptd 6432 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠) = (𝑋 + 𝑠))
529507, 528eqtrd 2805 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
530505, 529pm2.61dan 814 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
531497, 530mpteq12dva 4867 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ↦ if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) = (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)))
532497oveq2d 6811 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))))
533532oveq1d 6810 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld)))
534533fveq1d 6335 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → ((((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) CnP (TopOpen‘ℂfld))‘(𝐻𝑖)) = ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻𝑖)))
535493, 531, 5343eltr4d 2865 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ↦ if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) ∈ ((((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) CnP (TopOpen‘ℂfld))‘(𝐻𝑖)))
536 eqid 2771 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) = ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}))
537 eqid 2771 . . . . . . . . . . . 12 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ↦ if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) = (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ↦ if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)))
538230recnd 10273 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℂ)
539536, 386, 537, 454, 305, 538ellimc 23856 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) lim (𝐻𝑖)) ↔ (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ↦ if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))) ∈ ((((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) CnP (TopOpen‘ℂfld))‘(𝐻𝑖))))
540535, 539mpbird 247 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) lim (𝐻𝑖)))
541470, 540, 69limccog 40367 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻𝑖)))
542459oveq1d 6810 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻𝑖)) = ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) lim (𝐻𝑖)))
543541, 542eleqtrd 2852 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) lim (𝐻𝑖)))
544230, 231, 310, 461, 543iblcncfioo 40708 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ 𝐿1)
54530ad2antrr 705 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝐹:(-π[,]π)⟶ℂ)
54649a1i 11 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → -π ∈ ℝ*)
54751a1i 11 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → π ∈ ℝ*)
54821ad2antrr 705 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
549 simplr 752 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
550 simpr 471 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))))
551169, 179oveq12d 6813 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) = (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
552551adantr 466 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) = (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
553550, 552eleqtrd 2852 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
554553, 116syldan 579 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
555546, 547, 548, 549, 554fourierdlem1 40839 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ (-π[,]π))
556545, 555ffvelrnd 6505 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
557230, 231, 544, 556ibliooicc 40701 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ 𝐿1)
55814, 20, 165, 180, 229, 557itgspltprt 40709 . . . . 5 (𝜑 → ∫((𝐻‘0)[,](𝐻𝑀))(𝐹‘(𝑋 + 𝑡)) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))(𝐹‘(𝑋 + 𝑡)) d𝑡)
559551itgeq1d 40687 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
560559sumeq2dv 14640 . . . . 5 (𝜑 → Σ𝑖 ∈ (0..^𝑀)∫((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))(𝐹‘(𝑋 + 𝑡)) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
561558, 560eqtrd 2805 . . . 4 (𝜑 → ∫((𝐻‘0)[,](𝐻𝑀))(𝐹‘(𝑋 + 𝑡)) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
562125, 161, 5613eqtrd 2809 . . 3 (𝜑 → ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠 = Σ𝑖 ∈ (0..^𝑀)∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
563121, 562eqtr4d 2808 . 2 (𝜑 → Σ𝑖 ∈ (0..^𝑀)∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠)
56413, 76, 5633eqtrd 2809 1 (𝜑 → ∫(-π[,]π)(𝐹𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  ifcif 4226  {csn 4317   class class class wbr 4787  cmpt 4864  dom cdm 5250  ran crn 5251  cres 5252  ccom 5254  Fun wfun 6024  wf 6026  cfv 6030  (class class class)co 6795  𝑚 cmap 8012  cc 10139  cr 10140  0cc0 10141  1c1 10142   + caddc 10144  *cxr 10278   < clt 10279  cle 10280  cmin 10471  -cneg 10472  cn 11225  cz 11583  cuz 11892  (,)cioo 12379  (,]cioc 12380  [,)cico 12381  [,]cicc 12382  ...cfz 12532  ..^cfzo 12672  Σcsu 14623  πcpi 15002  t crest 16288  TopOpenctopn 16289  fldccnfld 19960  Topctop 20917  TopOnctopon 20934   Cn ccn 21248   CnP ccnp 21249  cnccncf 22898  citg 23605   lim climc 23845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-inf2 8705  ax-cc 9462  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218  ax-pre-sup 10219  ax-addf 10220  ax-mulf 10221
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-disj 4756  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-of 7047  df-ofr 7048  df-om 7216  df-1st 7318  df-2nd 7319  df-supp 7450  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-2o 7717  df-oadd 7720  df-omul 7721  df-er 7899  df-map 8014  df-pm 8015  df-ixp 8066  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-fsupp 8435  df-fi 8476  df-sup 8507  df-inf 8508  df-oi 8574  df-card 8968  df-acn 8971  df-cda 9195  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-div 10890  df-nn 11226  df-2 11284  df-3 11285  df-4 11286  df-5 11287  df-6 11288  df-7 11289  df-8 11290  df-9 11291  df-n0 11499  df-z 11584  df-dec 11700  df-uz 11893  df-q 11996  df-rp 12035  df-xneg 12150  df-xadd 12151  df-xmul 12152  df-ioo 12383  df-ioc 12384  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-fl 12800  df-mod 12876  df-seq 13008  df-exp 13067  df-fac 13264  df-bc 13293  df-hash 13321  df-shft 14014  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-limsup 14409  df-clim 14426  df-rlim 14427  df-sum 14624  df-ef 15003  df-sin 15005  df-cos 15006  df-pi 15008  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-starv 16163  df-sca 16164  df-vsca 16165  df-ip 16166  df-tset 16167  df-ple 16168  df-ds 16171  df-unif 16172  df-hom 16173  df-cco 16174  df-rest 16290  df-topn 16291  df-0g 16309  df-gsum 16310  df-topgen 16311  df-pt 16312  df-prds 16315  df-xrs 16369  df-qtop 16374  df-imas 16375  df-xps 16377  df-mre 16453  df-mrc 16454  df-acs 16456  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-submnd 17543  df-mulg 17748  df-cntz 17956  df-cmn 18401  df-psmet 19952  df-xmet 19953  df-met 19954  df-bl 19955  df-mopn 19956  df-fbas 19957  df-fg 19958  df-cnfld 19961  df-top 20918  df-topon 20935  df-topsp 20957  df-bases 20970  df-cld 21043  df-ntr 21044  df-cls 21045  df-nei 21122  df-lp 21160  df-perf 21161  df-cn 21251  df-cnp 21252  df-haus 21339  df-cmp 21410  df-tx 21585  df-hmeo 21778  df-fil 21869  df-fm 21961  df-flim 21962  df-flf 21963  df-xms 22344  df-ms 22345  df-tms 22346  df-cncf 22900  df-ovol 23451  df-vol 23452  df-mbf 23606  df-itg1 23607  df-itg2 23608  df-ibl 23609  df-itg 23610  df-0p 23656  df-ditg 23830  df-limc 23849  df-dv 23850
This theorem is referenced by:  fourierdlem101  40938
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