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Theorem fourierdlem93 42040
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 41950 . . . . . . . . 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 233 . . . . . . 7 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
76simprd 496 . . . . . 6 (𝜑 → (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
87simplld 764 . . . . 5 (𝜑 → (𝑄‘0) = -π)
98eqcomd 2800 . . . 4 (𝜑 → -π = (𝑄‘0))
107simplrd 766 . . . . 5 (𝜑 → (𝑄𝑀) = π)
1110eqcomd 2800 . . . 4 (𝜑 → π = (𝑄𝑀))
129, 11oveq12d 7037 . . 3 (𝜑 → (-π[,]π) = ((𝑄‘0)[,](𝑄𝑀)))
1312itgeq1d 41797 . 2 (𝜑 → ∫(-π[,]π)(𝐹𝑡) d𝑡 = ∫((𝑄‘0)[,](𝑄𝑀))(𝐹𝑡) d𝑡)
14 0zd 11843 . . 3 (𝜑 → 0 ∈ ℤ)
15 nnuz 12130 . . . . 5 ℕ = (ℤ‘1)
162, 15syl6eleq 2892 . . . 4 (𝜑𝑀 ∈ (ℤ‘1))
17 1e0p1 11990 . . . . . 6 1 = (0 + 1)
1817a1i 11 . . . . 5 (𝜑 → 1 = (0 + 1))
1918fveq2d 6545 . . . 4 (𝜑 → (ℤ‘1) = (ℤ‘(0 + 1)))
2016, 19eleqtrd 2884 . . 3 (𝜑𝑀 ∈ (ℤ‘(0 + 1)))
213, 2, 1fourierdlem15 41963 . . . 4 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
22 pire 24727 . . . . . . 7 π ∈ ℝ
2322renegcli 10797 . . . . . 6 -π ∈ ℝ
24 iccssre 12668 . . . . . 6 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
2523, 22, 24mp2an 688 . . . . 5 (-π[,]π) ⊆ ℝ
2625a1i 11 . . . 4 (𝜑 → (-π[,]π) ⊆ ℝ)
2721, 26fssd 6399 . . 3 (𝜑𝑄:(0...𝑀)⟶ℝ)
287simprd 496 . . . 4 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2928r19.21bi 3174 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
30 fourierdlem93.6 . . . . 5 (𝜑𝐹:(-π[,]π)⟶ℂ)
3130adantr 481 . . . 4 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝐹:(-π[,]π)⟶ℂ)
32 simpr 485 . . . . 5 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
3312eqcomd 2800 . . . . . 6 (𝜑 → ((𝑄‘0)[,](𝑄𝑀)) = (-π[,]π))
3433adantr 481 . . . . 5 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → ((𝑄‘0)[,](𝑄𝑀)) = (-π[,]π))
3532, 34eleqtrd 2884 . . . 4 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝑡 ∈ (-π[,]π))
3631, 35ffvelrnd 6720 . . 3 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → (𝐹𝑡) ∈ ℂ)
3727adantr 481 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
38 elfzofz 12903 . . . . . 6 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
3938adantl 482 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
4037, 39ffvelrnd 6720 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
41 fzofzp1 12984 . . . . . 6 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
4241adantl 482 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
4337, 42ffvelrnd 6720 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
4430feqmptd 6604 . . . . . . . . . 10 (𝜑𝐹 = (𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)))
4544adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹 = (𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)))
4645reseq1d 5736 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
47 ioossicc 12672 . . . . . . . . . . 11 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
4847a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
4923rexri 10548 . . . . . . . . . . . . . 14 -π ∈ ℝ*
5049a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → -π ∈ ℝ*)
5122rexri 10548 . . . . . . . . . . . . . 14 π ∈ ℝ*
5251a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → π ∈ ℝ*)
5321ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
54 simplr 765 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
55 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
5650, 52, 53, 54, 55fourierdlem1 41949 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑡 ∈ (-π[,]π))
5756ralrimiva 3148 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))𝑡 ∈ (-π[,]π))
58 dfss3 3880 . . . . . . . . . . 11 (((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π) ↔ ∀𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))𝑡 ∈ (-π[,]π))
5957, 58sylibr 235 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
6048, 59sstrd 3901 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
6160resmptd 5792 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)))
6246, 61eqtrd 2830 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)))
6362eqcomd 2800 . . . . . 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 2882 . . . . 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 7034 . . . . . 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 2884 . . . . 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 7034 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄𝑖)))
7169, 70eleqtrd 2884 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄𝑖)))
7240, 43, 65, 68, 71iblcncfioo 41818 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) ∈ 𝐿1)
7330ad2antrr 722 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝐹:(-π[,]π)⟶ℂ)
7473, 56ffvelrnd 6720 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹𝑡) ∈ ℂ)
7540, 43, 72, 74ibliooicc 41811 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) ∈ 𝐿1)
7614, 20, 27, 29, 36, 75itgspltprt 41819 . 2 (𝜑 → ∫((𝑄‘0)[,](𝑄𝑀))(𝐹𝑡) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡)
77 fvres 6560 . . . . . . . 8 (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡) = (𝐹𝑡))
7877eqcomd 2800 . . . . . . 7 (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) → (𝐹𝑡) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡))
7978adantl 482 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹𝑡) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡))
8079itgeq2dv 24065 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡) d𝑡)
81 eqid 2794 . . . . . 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 481 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹:(-π[,]π)⟶ℂ)
8382, 59fssresd 6416 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))):((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
8448resabs1d 5768 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
8584, 64eqeltrd 2882 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
8684oveq1d 7034 . . . . . . . 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 41473 . . . . . . . 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 2832 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
8966, 88eleqtrd 2884 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
9084eqcomd 2800 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
9190oveq1d 7034 . . . . . . . 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 41457 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
9391, 92eqtrd 2830 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
9469, 93eleqtrd 2884 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
95 fourierdlem93.5 . . . . . . 7 (𝜑𝑋 ∈ ℝ)
9695adantr 481 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
9781, 40, 43, 29, 83, 85, 89, 94, 96fourierdlem82 42029 . . . . 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 481 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑄𝑖) ∈ ℝ)
9943adantr 481 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
10095ad2antrr 722 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑋 ∈ ℝ)
10198, 100resubcld 10918 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
10299, 100resubcld 10918 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
103 simpr 485 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
104 eliccre 41336 . . . . . . . . . 10 ((((𝑄𝑖) − 𝑋) ∈ ℝ ∧ ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ ℝ)
105101, 102, 103, 104syl3anc 1364 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ ℝ)
106100, 105readdcld 10519 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
107 elicc2 12651 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 233 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑡 ∈ ℝ ∧ ((𝑄𝑖) − 𝑋) ≤ 𝑡𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋)))
110109simp2d 1136 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄𝑖) − 𝑋) ≤ 𝑡)
11198, 100, 105lesubadd2d 11089 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (((𝑄𝑖) − 𝑋) ≤ 𝑡 ↔ (𝑄𝑖) ≤ (𝑋 + 𝑡)))
112110, 111mpbid 233 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑄𝑖) ≤ (𝑋 + 𝑡))
113109simp3d 1137 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋))
114100, 105, 99leaddsub2d 11092 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑋 + 𝑡) ≤ (𝑄‘(𝑖 + 1)) ↔ 𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋)))
115113, 114mpbird 258 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ≤ (𝑄‘(𝑖 + 1)))
11698, 99, 106, 112, 115eliccd 41334 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
117 fvres 6560 . . . . . . 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 24065 . . . . 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 2834 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
121120sumeq2dv 14893 . . 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 7027 . . . . . . 7 (𝑠 = 𝑡 → (𝑋 + 𝑠) = (𝑋 + 𝑡))
123122fveq2d 6545 . . . . . 6 (𝑠 = 𝑡 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑡)))
124123cbvitgv 24060 . . . . 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 6541 . . . . . . . . . 10 (𝑖 = 0 → (𝑄𝑖) = (𝑄‘0))
129128oveq1d 7034 . . . . . . . . 9 (𝑖 = 0 → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
130129adantl 482 . . . . . . . 8 ((𝜑𝑖 = 0) → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
1312nnzd 11936 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
13214, 131, 143jca 1121 . . . . . . . . 9 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ))
133 0le0 11588 . . . . . . . . . . 11 0 ≤ 0
134133a1i 11 . . . . . . . . . 10 (𝜑 → 0 ≤ 0)
135 0red 10493 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
1362nnred 11503 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℝ)
1372nngt0d 11536 . . . . . . . . . . 11 (𝜑 → 0 < 𝑀)
138135, 136, 137ltled 10637 . . . . . . . . . 10 (𝜑 → 0 ≤ 𝑀)
139134, 138jca 512 . . . . . . . . 9 (𝜑 → (0 ≤ 0 ∧ 0 ≤ 𝑀))
140 elfz2 12749 . . . . . . . . 9 (0 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
141132, 139, 140sylanbrc 583 . . . . . . . 8 (𝜑 → 0 ∈ (0...𝑀))
1428, 23syl6eqel 2890 . . . . . . . . 9 (𝜑 → (𝑄‘0) ∈ ℝ)
143142, 95resubcld 10918 . . . . . . . 8 (𝜑 → ((𝑄‘0) − 𝑋) ∈ ℝ)
144127, 130, 141, 143fvmptd 6644 . . . . . . 7 (𝜑 → (𝐻‘0) = ((𝑄‘0) − 𝑋))
1458oveq1d 7034 . . . . . . 7 (𝜑 → ((𝑄‘0) − 𝑋) = (-π − 𝑋))
146144, 145eqtr2d 2831 . . . . . 6 (𝜑 → (-π − 𝑋) = (𝐻‘0))
147 fveq2 6541 . . . . . . . . . 10 (𝑖 = 𝑀 → (𝑄𝑖) = (𝑄𝑀))
148147oveq1d 7034 . . . . . . . . 9 (𝑖 = 𝑀 → ((𝑄𝑖) − 𝑋) = ((𝑄𝑀) − 𝑋))
149148adantl 482 . . . . . . . 8 ((𝜑𝑖 = 𝑀) → ((𝑄𝑖) − 𝑋) = ((𝑄𝑀) − 𝑋))
15014, 131, 1313jca 1121 . . . . . . . . 9 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
151136leidd 11056 . . . . . . . . . 10 (𝜑𝑀𝑀)
152138, 151jca 512 . . . . . . . . 9 (𝜑 → (0 ≤ 𝑀𝑀𝑀))
153 elfz2 12749 . . . . . . . . 9 (𝑀 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
154150, 152, 153sylanbrc 583 . . . . . . . 8 (𝜑𝑀 ∈ (0...𝑀))
15510, 22syl6eqel 2890 . . . . . . . . 9 (𝜑 → (𝑄𝑀) ∈ ℝ)
156155, 95resubcld 10918 . . . . . . . 8 (𝜑 → ((𝑄𝑀) − 𝑋) ∈ ℝ)
157127, 149, 154, 156fvmptd 6644 . . . . . . 7 (𝜑 → (𝐻𝑀) = ((𝑄𝑀) − 𝑋))
15810oveq1d 7034 . . . . . . 7 (𝜑 → ((𝑄𝑀) − 𝑋) = (π − 𝑋))
159157, 158eqtr2d 2831 . . . . . 6 (𝜑 → (π − 𝑋) = (𝐻𝑀))
160146, 159oveq12d 7037 . . . . 5 (𝜑 → ((-π − 𝑋)[,](π − 𝑋)) = ((𝐻‘0)[,](𝐻𝑀)))
161160itgeq1d 41797 . . . 4 (𝜑 → ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐻‘0)[,](𝐻𝑀))(𝐹‘(𝑋 + 𝑡)) d𝑡)
16227ffvelrnda 6719 . . . . . . . 8 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
16395adantr 481 . . . . . . . 8 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
164162, 163resubcld 10918 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
165164, 126fmptd 6744 . . . . . 6 (𝜑𝐻:(0...𝑀)⟶ℝ)
16640, 43, 96, 29ltsub1dd 11102 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) − 𝑋) < ((𝑄‘(𝑖 + 1)) − 𝑋))
16739, 164syldan 591 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
168126fvmpt2 6648 . . . . . . . 8 ((𝑖 ∈ (0...𝑀) ∧ ((𝑄𝑖) − 𝑋) ∈ ℝ) → (𝐻𝑖) = ((𝑄𝑖) − 𝑋))
16939, 167, 168syl2anc 584 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) = ((𝑄𝑖) − 𝑋))
170 fveq2 6541 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
171170oveq1d 7034 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑄𝑖) − 𝑋) = ((𝑄𝑗) − 𝑋))
172171cbvmptv 5064 . . . . . . . . . 10 (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋))
173126, 172eqtri 2818 . . . . . . . . 9 𝐻 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋))
174173a1i 11 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐻 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋)))
175 fveq2 6541 . . . . . . . . . 10 (𝑗 = (𝑖 + 1) → (𝑄𝑗) = (𝑄‘(𝑖 + 1)))
176175oveq1d 7034 . . . . . . . . 9 (𝑗 = (𝑖 + 1) → ((𝑄𝑗) − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
177176adantl 482 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑄𝑗) − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
17843, 96resubcld 10918 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
179174, 177, 42, 178fvmptd 6644 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) − 𝑋))
180166, 169, 1793brtr4d 4996 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) < (𝐻‘(𝑖 + 1)))
181 frn 6391 . . . . . . . . 9 (𝐹:(-π[,]π)⟶ℂ → ran 𝐹 ⊆ ℂ)
18230, 181syl 17 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℂ)
183182adantr 481 . . . . . . 7 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → ran 𝐹 ⊆ ℂ)
184 ffun 6388 . . . . . . . . . 10 (𝐹:(-π[,]π)⟶ℂ → Fun 𝐹)
18530, 184syl 17 . . . . . . . . 9 (𝜑 → Fun 𝐹)
186185adantr 481 . . . . . . . 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 481 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑋 ∈ ℝ)
190144, 143eqeltrd 2882 . . . . . . . . . . . . 13 (𝜑 → (𝐻‘0) ∈ ℝ)
191190adantr 481 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻‘0) ∈ ℝ)
192157, 156eqeltrd 2882 . . . . . . . . . . . . 13 (𝜑 → (𝐻𝑀) ∈ ℝ)
193192adantr 481 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻𝑀) ∈ ℝ)
194 simpr 485 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)))
195 eliccre 41336 . . . . . . . . . . . 12 (((𝐻‘0) ∈ ℝ ∧ (𝐻𝑀) ∈ ℝ ∧ 𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ℝ)
196191, 193, 194, 195syl3anc 1364 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ℝ)
197189, 196readdcld 10519 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ ℝ)
198128adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 = 0) → (𝑄𝑖) = (𝑄‘0))
199198oveq1d 7034 . . . . . . . . . . . . . . 15 ((𝜑𝑖 = 0) → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
200127, 199, 141, 143fvmptd 6644 . . . . . . . . . . . . . 14 (𝜑 → (𝐻‘0) = ((𝑄‘0) − 𝑋))
201200oveq2d 7035 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + (𝐻‘0)) = (𝑋 + ((𝑄‘0) − 𝑋)))
20295recnd 10518 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ ℂ)
203142recnd 10518 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) ∈ ℂ)
204202, 203pncan3d 10850 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + ((𝑄‘0) − 𝑋)) = (𝑄‘0))
205201, 204, 83eqtrrd 2835 . . . . . . . . . . . 12 (𝜑 → -π = (𝑋 + (𝐻‘0)))
206205adantr 481 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → -π = (𝑋 + (𝐻‘0)))
207 elicc2 12651 . . . . . . . . . . . . . . 15 (((𝐻‘0) ∈ ℝ ∧ (𝐻𝑀) ∈ ℝ) → (𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)) ↔ (𝑡 ∈ ℝ ∧ (𝐻‘0) ≤ 𝑡𝑡 ≤ (𝐻𝑀))))
208191, 193, 207syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)) ↔ (𝑡 ∈ ℝ ∧ (𝐻‘0) ≤ 𝑡𝑡 ≤ (𝐻𝑀))))
209194, 208mpbid 233 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑡 ∈ ℝ ∧ (𝐻‘0) ≤ 𝑡𝑡 ≤ (𝐻𝑀)))
210209simp2d 1136 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻‘0) ≤ 𝑡)
211191, 196, 189, 210leadd2dd 11105 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + (𝐻‘0)) ≤ (𝑋 + 𝑡))
212206, 211eqbrtrd 4986 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → -π ≤ (𝑋 + 𝑡))
213209simp3d 1137 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ≤ (𝐻𝑀))
214196, 193, 189, 213leadd2dd 11105 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ≤ (𝑋 + (𝐻𝑀)))
215157oveq2d 7035 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + (𝐻𝑀)) = (𝑋 + ((𝑄𝑀) − 𝑋)))
216155recnd 10518 . . . . . . . . . . . . . 14 (𝜑 → (𝑄𝑀) ∈ ℂ)
217202, 216pncan3d 10850 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + ((𝑄𝑀) − 𝑋)) = (𝑄𝑀))
218215, 217, 103eqtrrd 2835 . . . . . . . . . . . 12 (𝜑 → π = (𝑋 + (𝐻𝑀)))
219218adantr 481 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → π = (𝑋 + (𝐻𝑀)))
220214, 219breqtrrd 4992 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ≤ π)
221187, 188, 197, 212, 220eliccd 41334 . . . . . . . . 9 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ (-π[,]π))
222 fdm 6393 . . . . . . . . . . . 12 (𝐹:(-π[,]π)⟶ℂ → dom 𝐹 = (-π[,]π))
22330, 222syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = (-π[,]π))
224223eqcomd 2800 . . . . . . . . . 10 (𝜑 → (-π[,]π) = dom 𝐹)
225224adantr 481 . . . . . . . . 9 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (-π[,]π) = dom 𝐹)
226221, 225eleqtrd 2884 . . . . . . . 8 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ dom 𝐹)
227 fvelrn 6712 . . . . . . . 8 ((Fun 𝐹 ∧ (𝑋 + 𝑡) ∈ dom 𝐹) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
228186, 226, 227syl2anc 584 . . . . . . 7 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
229183, 228sseldd 3892 . . . . . 6 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
230169, 167eqeltrd 2882 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℝ)
231179, 178eqeltrd 2882 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
23282, 60fssresd 6416 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
23340rexrd 10540 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ*)
234233adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) ∈ ℝ*)
23543rexrd 10540 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
236235adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
23795ad2antrr 722 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
238 elioore 12618 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
239238adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ℝ)
240237, 239readdcld 10519 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ℝ)
241169oveq2d 7035 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻𝑖)) = (𝑋 + ((𝑄𝑖) − 𝑋)))
242202adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
24340recnd 10518 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℂ)
244242, 243pncan3d 10850 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑄𝑖) − 𝑋)) = (𝑄𝑖))
245 eqidd 2795 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = (𝑄𝑖))
246241, 244, 2453eqtrrd 2835 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = (𝑋 + (𝐻𝑖)))
247246adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) = (𝑋 + (𝐻𝑖)))
248230adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) ∈ ℝ)
249 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
250248rexrd 10540 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) ∈ ℝ*)
251231rexrd 10540 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℝ*)
252251adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻‘(𝑖 + 1)) ∈ ℝ*)
253 elioo2 12629 . . . . . . . . . . . . . . . . . 18 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ*) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↔ (𝑡 ∈ ℝ ∧ (𝐻𝑖) < 𝑡𝑡 < (𝐻‘(𝑖 + 1)))))
254250, 252, 253syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↔ (𝑡 ∈ ℝ ∧ (𝐻𝑖) < 𝑡𝑡 < (𝐻‘(𝑖 + 1)))))
255249, 254mpbid 233 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑡 ∈ ℝ ∧ (𝐻𝑖) < 𝑡𝑡 < (𝐻‘(𝑖 + 1))))
256255simp2d 1136 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) < 𝑡)
257248, 239, 237, 256ltadd2dd 10648 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + (𝐻𝑖)) < (𝑋 + 𝑡))
258247, 257eqbrtrd 4986 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) < (𝑋 + 𝑡))
259231adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
260255simp3d 1137 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 < (𝐻‘(𝑖 + 1)))
261239, 259, 237, 260ltadd2dd 10648 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) < (𝑋 + (𝐻‘(𝑖 + 1))))
262179oveq2d 7035 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑋 + ((𝑄‘(𝑖 + 1)) − 𝑋)))
26343recnd 10518 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
264242, 263pncan3d 10850 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑄‘(𝑖 + 1)) − 𝑋)) = (𝑄‘(𝑖 + 1)))
265262, 264eqtrd 2830 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
266265adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
267261, 266breqtrd 4990 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) < (𝑄‘(𝑖 + 1)))
268234, 236, 240, 258, 267eliood 41328 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
269 eqid 2794 . . . . . . . . . . . 12 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
270268, 269fmptd 6744 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)):((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))⟶((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
271 fcompt 6761 . . . . . . . . . . 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 584 . . . . . . . . . 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 7027 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑟 → (𝑋 + 𝑡) = (𝑋 + 𝑟))
274273cbvmptv 5064 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))
275274fveq1i 6542 . . . . . . . . . . . . . 14 ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠) = ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠)
276275fveq2i 6544 . . . . . . . . . . . . 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 5055 . . . . . . . . . . . 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 6541 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠) = ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡))
280279fveq2d 6545 . . . . . . . . . . . . 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 5064 . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . 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 7027 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑡 → (𝑋 + 𝑟) = (𝑋 + 𝑡))
285284adantl 482 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) ∧ 𝑟 = 𝑡) → (𝑋 + 𝑟) = (𝑋 + 𝑡))
286283, 285, 249, 240fvmptd 6644 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡) = (𝑋 + 𝑡))
287286fveq2d 6545 . . . . . . . . . . . . 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 6560 . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . 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 5058 . . . . . . . . . . 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 2834 . . . . . . . . . 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 2831 . . . . . . . . 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 2794 . . . . . . . . . . 11 (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡))
295 ssid 3912 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
296295a1i 11 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → ℂ ⊆ ℂ)
297 id 22 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → 𝑋 ∈ ℂ)
298296, 297, 296constcncfg 41709 . . . . . . . . . . . . 13 (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
299 cncfmptid 23203 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
300295, 295, 299mp2an 688 . . . . . . . . . . . . . 14 (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ)
301300a1i 11 . . . . . . . . . . . . 13 (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
302298, 301addcncf 41711 . . . . . . . . . . . 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 41324 . . . . . . . . . . . 12 ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ⊆ ℂ
305304a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ⊆ ℂ)
306 ioosscn 41324 . . . . . . . . . . . 12 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
307306a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
308294, 303, 305, 307, 268cncfmptssg 41708 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
309308, 64cncfco 23198 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
310293, 309eqeltrd 2882 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
311233adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄𝑖) ∈ ℝ*)
312235adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
313 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
314 vex 3439 . . . . . . . . . . . . . . . . . 18 𝑟 ∈ V
315269elrnmpt 5713 . . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ∃𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))𝑟 = (𝑋 + 𝑡))
318 nfv 1893 . . . . . . . . . . . . . . . . . 18 𝑡(𝜑𝑖 ∈ (0..^𝑀))
319 nfmpt1 5061 . . . . . . . . . . . . . . . . . . . 20 𝑡(𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
320319nfrn 5709 . . . . . . . . . . . . . . . . . . 19 𝑡ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
321320nfcri 2942 . . . . . . . . . . . . . . . . . 18 𝑡 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
322318, 321nfan 1882 . . . . . . . . . . . . . . . . 17 𝑡((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
323 nfv 1893 . . . . . . . . . . . . . . . . 17 𝑡 𝑟 ∈ ℝ
324 simp3 1131 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 = (𝑋 + 𝑡))
325953ad2ant1 1126 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑋 ∈ ℝ)
3262383ad2ant2 1127 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑡 ∈ ℝ)
327325, 326readdcld 10519 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑋 + 𝑡) ∈ ℝ)
328324, 327eqeltrd 2882 . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . 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 3277 . . . . . . . . . . . . . . . 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 1893 . . . . . . . . . . . . . . . . 17 𝑡(𝑄𝑖) < 𝑟
3342583adant3 1125 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑄𝑖) < (𝑋 + 𝑡))
335 simp3 1131 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 = (𝑋 + 𝑡))
336334, 335breqtrrd 4992 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 3277 . . . . . . . . . . . . . . . 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 1893 . . . . . . . . . . . . . . . . 17 𝑡 𝑟 < (𝑄‘(𝑖 + 1))
3422673adant3 1125 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑋 + 𝑡) < (𝑄‘(𝑖 + 1)))
343335, 342eqbrtrd 4986 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 3277 . . . . . . . . . . . . . . . 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 41328 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
349223ineq2d 4111 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ dom 𝐹) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
350349adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ dom 𝐹) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
351 dmres 5759 . . . . . . . . . . . . . . . . 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 3877 . . . . . . . . . . . . . . . . 17 (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π) ↔ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
35460, 353sylib 219 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
355350, 352, 3543eqtr4d 2840 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
356355adantr 481 . . . . . . . . . . . . . 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 2885 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
358332, 347ltned 10625 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ≠ (𝑄‘(𝑖 + 1)))
359358neneqd 2988 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 = (𝑄‘(𝑖 + 1)))
360 velsn 4490 . . . . . . . . . . . . . 14 (𝑟 ∈ {(𝑄‘(𝑖 + 1))} ↔ 𝑟 = (𝑄‘(𝑖 + 1)))
361359, 360sylnibr 330 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 ∈ {(𝑄‘(𝑖 + 1))})
362357, 361eldifd 3872 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
363362ralrimiva 3148 . . . . . . . . . . 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 3880 . . . . . . . . . . 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 235 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ⊆ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
366 eqid 2794 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠))
367202adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
368 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
369367, 368addcomd 10691 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑠 ∈ ℂ) → (𝑋 + 𝑠) = (𝑠 + 𝑋))
370369mpteq2dva 5058 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)))
371 eqid 2794 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋))
372371addccncf 23207 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
373202, 372syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
374370, 373eqeltrd 2882 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
375374adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
376230rexrd 10540 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℝ*)
377 iocssre 12666 . . . . . . . . . . . . . . . . . . 19 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℝ)
378376, 231, 377syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℝ)
379 ax-resscn 10443 . . . . . . . . . . . . . . . . . 18 ℝ ⊆ ℂ
380378, 379syl6ss 3903 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℂ)
381295a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
382202ad2antrr 722 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℂ)
383380sselda 3891 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
384382, 383addcld 10509 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℂ)
385366, 375, 380, 381, 384cncfmptssg 41708 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))–cn→ℂ))
386 eqid 2794 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
387 eqid 2794 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
388386cnfldtop 23075 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘ℂfld) ∈ Top
389 unicntop 23077 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (TopOpen‘ℂfld)
390389restid 16536 . . . . . . . . . . . . . . . . . . . 20 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
391388, 390ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
392391eqcomi 2803 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
393386, 387, 392cncfcn 23200 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
395385, 394eleqtrd 2884 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
396386cnfldtopon 23074 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
397396a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
398 resttopon 21453 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))))
400 cncnp 21572 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . 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 12640 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → (𝐻‘(𝑖 + 1)) ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
405376, 251, 180, 404syl3anc 1364 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
406 fveq2 6541 . . . . . . . . . . . . . . 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 2867 . . . . . . . . . . . . . 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 3556 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 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 ioounsn 12713 . . . . . . . . . . . . . 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 1364 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) = ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
412265eqcomd 2800 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = (𝑋 + (𝐻‘(𝑖 + 1))))
413412ad2antrr 722 . . . . . . . . . . . . . . 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 4389 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 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 7027 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐻‘(𝑖 + 1)) → (𝑋 + 𝑠) = (𝑋 + (𝐻‘(𝑖 + 1))))
417416adantl 482 . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . 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 4392 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . 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 7027 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (𝑋 + 𝑡) = (𝑋 + 𝑠))
423422adantl 482 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) ∧ 𝑡 = 𝑠) → (𝑋 + 𝑡) = (𝑋 + 𝑠))
424 velsn 4490 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ {(𝐻‘(𝑖 + 1))} ↔ 𝑠 = (𝐻‘(𝑖 + 1)))
425424notbii 321 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ {(𝐻‘(𝑖 + 1))} ↔ ¬ 𝑠 = (𝐻‘(𝑖 + 1)))
426 elun 4048 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ↔ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}))
427426biimpi 217 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}))
428427orcomd 866 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (𝑠 ∈ {(𝐻‘(𝑖 + 1))} ∨ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
429428ord 859 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (¬ 𝑠 ∈ {(𝐻‘(𝑖 + 1))} → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
430425, 429syl5bir 244 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (¬ 𝑠 = (𝐻‘(𝑖 + 1)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
431430imp 407 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
432431adantll 710 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
43395ad2antrr 722 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → 𝑋 ∈ ℝ)
434 elioore 12618 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
435434adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
436 elsni 4491 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ {(𝐻‘(𝑖 + 1))} → 𝑠 = (𝐻‘(𝑖 + 1)))
437436adantl 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → 𝑠 = (𝐻‘(𝑖 + 1)))
438231adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
439437, 438eqeltrd 2882 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → 𝑠 ∈ ℝ)
440435, 439jaodan 952 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻‘(𝑖 + 1))})) → 𝑠 ∈ ℝ)
441426, 440sylan2b 593 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → 𝑠 ∈ ℝ)
442433, 441readdcld 10519 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → (𝑋 + 𝑠) ∈ ℝ)
443442adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → (𝑋 + 𝑠) ∈ ℝ)
444421, 423, 432, 443fvmptd 6644 . . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . 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 5047 . . . . . . . . . . . 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 7035 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))))
449448oveq1d 7034 . . . . . . . . . . . . 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 6543 . . . . . . . . . . . 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 2897 . . . . . . . . . . 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 2794 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) = ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}))
453 eqid 2794 . . . . . . . . . . . 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 6399 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)):((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))⟶ℂ)
455231recnd 10518 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℂ)
456452, 386, 453, 454, 305, 455ellimc 24154 . . . . . . . . . . 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 258 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) lim (𝐻‘(𝑖 + 1))))
458365, 457, 66limccog 41456 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻‘(𝑖 + 1))))
459272, 292eqtrd 2830 . . . . . . . . . 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 7034 . . . . . . . . 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 2884 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) lim (𝐻‘(𝑖 + 1))))
46240adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄𝑖) ∈ ℝ)
463462, 340gtned 10624 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ≠ (𝑄𝑖))
464463neneqd 2988 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 = (𝑄𝑖))
465 velsn 4490 . . . . . . . . . . . . . 14 (𝑟 ∈ {(𝑄𝑖)} ↔ 𝑟 = (𝑄𝑖))
466464, 465sylnibr 330 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 ∈ {(𝑄𝑖)})
467357, 466eldifd 3872 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
468467ralrimiva 3148 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
469 dfss3 3880 . . . . . . . . . . 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 235 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ⊆ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
471 icossre 12667 . . . . . . . . . . . . . . . . . . 19 (((𝐻𝑖) ∈ ℝ ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ*) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℝ)
472230, 251, 471syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℝ)
473472, 379syl6ss 3903 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℂ)
474202ad2antrr 722 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℂ)
475473sselda 3891 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
476474, 475addcld 10509 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℂ)
477366, 375, 473, 381, 476cncfmptssg 41708 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
478 eqid 2794 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
479386, 478, 392cncfcn 23200 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
481477, 480eleqtrd 2884 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
482 resttopon 21453 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))))
484 cncnp 21572 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . 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 12641 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → (𝐻𝑖) ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
489376, 251, 180, 488syl3anc 1364 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
490 fveq2 6541 . . . . . . . . . . . . . . 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 2867 . . . . . . . . . . . . . 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 3556 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻𝑖)))
494 uncom 4052 . . . . . . . . . . . . . 14 (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) = ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
495 snunioo 12714 . . . . . . . . . . . . . . 15 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
496376, 251, 180, 495syl3anc 1364 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
497494, 496syl5eq 2842 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
498 iftrue 4389 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐻𝑖) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑄𝑖))
499498adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑄𝑖))
500246adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → (𝑄𝑖) = (𝑋 + (𝐻𝑖)))
501 oveq2 7027 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝐻𝑖) → (𝑋 + 𝑠) = (𝑋 + (𝐻𝑖)))
502501eqcomd 2800 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐻𝑖) → (𝑋 + (𝐻𝑖)) = (𝑋 + 𝑠))
503502adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → (𝑋 + (𝐻𝑖)) = (𝑋 + 𝑠))
504499, 500, 5033eqtrd 2834 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
505504adantlr 711 . . . . . . . . . . . . . 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 4392 . . . . . . . . . . . . . . . 16 𝑠 = (𝐻𝑖) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))
507506adantl 482 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) ∧ 𝑡 = 𝑠) → (𝑋 + 𝑡) = (𝑋 + 𝑠))
510 velsn 4490 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ {(𝐻𝑖)} ↔ 𝑠 = (𝐻𝑖))
511510notbii 321 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ {(𝐻𝑖)} ↔ ¬ 𝑠 = (𝐻𝑖))
512 elun 4048 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ↔ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻𝑖)}))
513512biimpi 217 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻𝑖)}))
514513orcomd 866 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (𝑠 ∈ {(𝐻𝑖)} ∨ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
515514ord 859 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (¬ 𝑠 ∈ {(𝐻𝑖)} → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
516511, 515syl5bir 244 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (¬ 𝑠 = (𝐻𝑖) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
517516imp 407 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ∧ ¬ 𝑠 = (𝐻𝑖)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
518517adantll 710 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
51995ad2antrr 722 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → 𝑋 ∈ ℝ)
520 elsni 4491 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ {(𝐻𝑖)} → 𝑠 = (𝐻𝑖))
521520adantl 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → 𝑠 = (𝐻𝑖))
522230adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → (𝐻𝑖) ∈ ℝ)
523521, 522eqeltrd 2882 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → 𝑠 ∈ ℝ)
524435, 523jaodan 952 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻𝑖)})) → 𝑠 ∈ ℝ)
525512, 524sylan2b 593 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → 𝑠 ∈ ℝ)
526519, 525readdcld 10519 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → (𝑋 + 𝑠) ∈ ℝ)
527526adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → (𝑋 + 𝑠) ∈ ℝ)
528508, 509, 518, 527fvmptd 6644 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠) = (𝑋 + 𝑠))
529507, 528eqtrd 2830 . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . 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 5047 . . . . . . . . . . . 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 7035 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))))
533532oveq1d 7034 . . . . . . . . . . . . 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 6543 . . . . . . . . . . . 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 2897 . . . . . . . . . . 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 2794 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) = ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}))
537 eqid 2794 . . . . . . . . . . . 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 10518 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℂ)
539536, 386, 537, 454, 305, 538ellimc 24154 . . . . . . . . . . 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 258 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) lim (𝐻𝑖)))
541470, 540, 69limccog 41456 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻𝑖)))
542459oveq1d 7034 . . . . . . . . 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 2884 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) lim (𝐻𝑖)))
544230, 231, 310, 461, 543iblcncfioo 41818 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ 𝐿1)
54530ad2antrr 722 . . . . . . . 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 722 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
549 simplr 765 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
550 simpr 485 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))))
551169, 179oveq12d 7037 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) = (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
552551adantr 481 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) = (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
553550, 552eleqtrd 2884 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
554553, 116syldan 591 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
555546, 547, 548, 549, 554fourierdlem1 41949 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ (-π[,]π))
556545, 555ffvelrnd 6720 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
557230, 231, 544, 556ibliooicc 41811 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ 𝐿1)
55814, 20, 165, 180, 229, 557itgspltprt 41819 . . . . 5 (𝜑 → ∫((𝐻‘0)[,](𝐻𝑀))(𝐹‘(𝑋 + 𝑡)) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))(𝐹‘(𝑋 + 𝑡)) d𝑡)
559551itgeq1d 41797 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
560559sumeq2dv 14893 . . . . 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 2830 . . . 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 2834 . . 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 2833 . 2 (𝜑 → Σ𝑖 ∈ (0..^𝑀)∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠)
56413, 76, 5633eqtrd 2834 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 207  wa 396  wo 842  w3a 1080   = wceq 1522  wcel 2080  wral 3104  wrex 3105  {crab 3108  Vcvv 3436  cdif 3858  cun 3859  cin 3860  wss 3861  ifcif 4383  {csn 4474   class class class wbr 4964  cmpt 5043  dom cdm 5446  ran crn 5447  cres 5448  ccom 5450  Fun wfun 6222  wf 6224  cfv 6228  (class class class)co 7019  𝑚 cmap 8259  cc 10384  cr 10385  0cc0 10386  1c1 10387   + caddc 10389  *cxr 10523   < clt 10524  cle 10525  cmin 10719  -cneg 10720  cn 11488  cz 11831  cuz 12093  (,)cioo 12588  (,]cioc 12589  [,)cico 12590  [,]cicc 12591  ...cfz 12742  ..^cfzo 12883  Σcsu 14876  πcpi 15253  t crest 16523  TopOpenctopn 16524  fldccnfld 20227  Topctop 21185  TopOnctopon 21202   Cn ccn 21516   CnP ccnp 21517  cnccncf 23167  citg 23902   lim climc 24143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-inf2 8953  ax-cc 9706  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463  ax-pre-sup 10464  ax-addf 10465  ax-mulf 10466
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-symdif 4141  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-int 4785  df-iun 4829  df-iin 4830  df-disj 4933  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-se 5406  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-isom 6237  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-of 7270  df-ofr 7271  df-om 7440  df-1st 7548  df-2nd 7549  df-supp 7685  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-1o 7956  df-2o 7957  df-oadd 7960  df-omul 7961  df-er 8142  df-map 8261  df-pm 8262  df-ixp 8314  df-en 8361  df-dom 8362  df-sdom 8363  df-fin 8364  df-fsupp 8683  df-fi 8724  df-sup 8755  df-inf 8756  df-oi 8823  df-dju 9179  df-card 9217  df-acn 9220  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-div 11148  df-nn 11489  df-2 11550  df-3 11551  df-4 11552  df-5 11553  df-6 11554  df-7 11555  df-8 11556  df-9 11557  df-n0 11748  df-z 11832  df-dec 11949  df-uz 12094  df-q 12198  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-ioo 12592  df-ioc 12593  df-ico 12594  df-icc 12595  df-fz 12743  df-fzo 12884  df-fl 13012  df-mod 13088  df-seq 13220  df-exp 13280  df-fac 13484  df-bc 13513  df-hash 13541  df-shft 14260  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-limsup 14662  df-clim 14679  df-rlim 14680  df-sum 14877  df-ef 15254  df-sin 15256  df-cos 15257  df-pi 15259  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-starv 16409  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ds 16416  df-unif 16417  df-hom 16418  df-cco 16419  df-rest 16525  df-topn 16526  df-0g 16544  df-gsum 16545  df-topgen 16546  df-pt 16547  df-prds 16550  df-xrs 16604  df-qtop 16609  df-imas 16610  df-xps 16612  df-mre 16686  df-mrc 16687  df-acs 16689  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-submnd 17775  df-mulg 17982  df-cntz 18188  df-cmn 18635  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-mopn 20223  df-fbas 20224  df-fg 20225  df-cnfld 20228  df-top 21186  df-topon 21203  df-topsp 21225  df-bases 21238  df-cld 21311  df-ntr 21312  df-cls 21313  df-nei 21390  df-lp 21428  df-perf 21429  df-cn 21519  df-cnp 21520  df-haus 21607  df-cmp 21679  df-tx 21854  df-hmeo 22047  df-fil 22138  df-fm 22230  df-flim 22231  df-flf 22232  df-xms 22613  df-ms 22614  df-tms 22615  df-cncf 23169  df-ovol 23748  df-vol 23749  df-mbf 23903  df-itg1 23904  df-itg2 23905  df-ibl 23906  df-itg 23907  df-0p 23954  df-ditg 24128  df-limc 24147  df-dv 24148
This theorem is referenced by:  fourierdlem101  42048
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