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Theorem fourierdlem93 40836
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 40746 . . . . . . . . 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 482 . . . . . 6 (𝜑 → (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
87simplld 808 . . . . 5 (𝜑 → (𝑄‘0) = -π)
98eqcomd 2730 . . . 4 (𝜑 → -π = (𝑄‘0))
107simplrd 810 . . . . 5 (𝜑 → (𝑄𝑀) = π)
1110eqcomd 2730 . . . 4 (𝜑 → π = (𝑄𝑀))
129, 11oveq12d 6783 . . 3 (𝜑 → (-π[,]π) = ((𝑄‘0)[,](𝑄𝑀)))
1312itgeq1d 40592 . 2 (𝜑 → ∫(-π[,]π)(𝐹𝑡) d𝑡 = ∫((𝑄‘0)[,](𝑄𝑀))(𝐹𝑡) d𝑡)
14 0zd 11502 . . 3 (𝜑 → 0 ∈ ℤ)
15 nnuz 11837 . . . . 5 ℕ = (ℤ‘1)
162, 15syl6eleq 2813 . . . 4 (𝜑𝑀 ∈ (ℤ‘1))
17 1e0p1 11665 . . . . . 6 1 = (0 + 1)
1817a1i 11 . . . . 5 (𝜑 → 1 = (0 + 1))
1918fveq2d 6308 . . . 4 (𝜑 → (ℤ‘1) = (ℤ‘(0 + 1)))
2016, 19eleqtrd 2805 . . 3 (𝜑𝑀 ∈ (ℤ‘(0 + 1)))
213, 2, 1fourierdlem15 40759 . . . 4 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
22 pire 24330 . . . . . . 7 π ∈ ℝ
2322renegcli 10455 . . . . . 6 -π ∈ ℝ
24 iccssre 12369 . . . . . 6 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
2523, 22, 24mp2an 710 . . . . 5 (-π[,]π) ⊆ ℝ
2625a1i 11 . . . 4 (𝜑 → (-π[,]π) ⊆ ℝ)
2721, 26fssd 6170 . . 3 (𝜑𝑄:(0...𝑀)⟶ℝ)
287simprd 482 . . . 4 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2928r19.21bi 3034 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
30 fourierdlem93.6 . . . . 5 (𝜑𝐹:(-π[,]π)⟶ℂ)
3130adantr 472 . . . 4 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝐹:(-π[,]π)⟶ℂ)
32 simpr 479 . . . . 5 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
3312eqcomd 2730 . . . . . 6 (𝜑 → ((𝑄‘0)[,](𝑄𝑀)) = (-π[,]π))
3433adantr 472 . . . . 5 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → ((𝑄‘0)[,](𝑄𝑀)) = (-π[,]π))
3532, 34eleqtrd 2805 . . . 4 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝑡 ∈ (-π[,]π))
3631, 35ffvelrnd 6475 . . 3 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → (𝐹𝑡) ∈ ℂ)
3727adantr 472 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
38 elfzofz 12600 . . . . . 6 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
3938adantl 473 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
4037, 39ffvelrnd 6475 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
41 fzofzp1 12680 . . . . . 6 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
4241adantl 473 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
4337, 42ffvelrnd 6475 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
4430feqmptd 6363 . . . . . . . . . 10 (𝜑𝐹 = (𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)))
4544adantr 472 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹 = (𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)))
4645reseq1d 5502 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
47 ioossicc 12373 . . . . . . . . . . 11 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
4847a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
4923rexri 10210 . . . . . . . . . . . . . 14 -π ∈ ℝ*
5049a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → -π ∈ ℝ*)
5122rexri 10210 . . . . . . . . . . . . . 14 π ∈ ℝ*
5251a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → π ∈ ℝ*)
5321ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
54 simplr 809 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
55 simpr 479 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
5650, 52, 53, 54, 55fourierdlem1 40745 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑡 ∈ (-π[,]π))
5756ralrimiva 3068 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))𝑡 ∈ (-π[,]π))
58 dfss3 3698 . . . . . . . . . . 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 3719 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
6160resmptd 5562 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)))
6246, 61eqtrd 2758 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)))
6362eqcomd 2730 . . . . . 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 2803 . . . . 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 6780 . . . . . 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 2805 . . . . 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 6780 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄𝑖)))
7169, 70eleqtrd 2805 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄𝑖)))
7240, 43, 65, 68, 71iblcncfioo 40614 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) ∈ 𝐿1)
7330ad2antrr 764 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝐹:(-π[,]π)⟶ℂ)
7473, 56ffvelrnd 6475 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹𝑡) ∈ ℂ)
7540, 43, 72, 74ibliooicc 40607 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) ∈ 𝐿1)
7614, 20, 27, 29, 36, 75itgspltprt 40615 . 2 (𝜑 → ∫((𝑄‘0)[,](𝑄𝑀))(𝐹𝑡) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡)
77 fvres 6320 . . . . . . . 8 (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡) = (𝐹𝑡))
7877eqcomd 2730 . . . . . . 7 (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) → (𝐹𝑡) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡))
7978adantl 473 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹𝑡) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡))
8079itgeq2dv 23668 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡) d𝑡)
81 eqid 2724 . . . . . 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 472 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹:(-π[,]π)⟶ℂ)
8382, 59fssresd 6184 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))):((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
8448resabs1d 5538 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
8584, 64eqeltrd 2803 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
8684oveq1d 6780 . . . . . . . 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 40289 . . . . . . . 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 2760 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
8966, 88eleqtrd 2805 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
9084eqcomd 2730 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
9190oveq1d 6780 . . . . . . . 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 40273 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
9391, 92eqtrd 2758 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
9469, 93eleqtrd 2805 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
95 fourierdlem93.5 . . . . . . 7 (𝜑𝑋 ∈ ℝ)
9695adantr 472 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
9781, 40, 43, 29, 83, 85, 89, 94, 96fourierdlem82 40825 . . . . 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 472 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑄𝑖) ∈ ℝ)
9943adantr 472 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
10095ad2antrr 764 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑋 ∈ ℝ)
10198, 100resubcld 10571 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
10299, 100resubcld 10571 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
103 simpr 479 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
104 eliccre 40148 . . . . . . . . . 10 ((((𝑄𝑖) − 𝑋) ∈ ℝ ∧ ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ ℝ)
105101, 102, 103, 104syl3anc 1439 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ ℝ)
106100, 105readdcld 10182 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
107 elicc2 12352 . . . . . . . . . . . 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 696 . . . . . . . . . . 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 1135 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄𝑖) − 𝑋) ≤ 𝑡)
11198, 100, 105lesubadd2d 10739 . . . . . . . . 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 1136 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋))
114100, 105, 99leaddsub2d 10742 . . . . . . . . 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 40146 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
117 fvres 6320 . . . . . . 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 23668 . . . . 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 2762 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
121120sumeq2dv 14553 . . 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 6773 . . . . . . 7 (𝑠 = 𝑡 → (𝑋 + 𝑠) = (𝑋 + 𝑡))
123122fveq2d 6308 . . . . . 6 (𝑠 = 𝑡 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑡)))
124123cbvitgv 23663 . . . . 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 6304 . . . . . . . . . 10 (𝑖 = 0 → (𝑄𝑖) = (𝑄‘0))
129128oveq1d 6780 . . . . . . . . 9 (𝑖 = 0 → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
130129adantl 473 . . . . . . . 8 ((𝜑𝑖 = 0) → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
1312nnzd 11594 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
13214, 131, 143jca 1379 . . . . . . . . 9 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ))
133 0le0 11223 . . . . . . . . . . 11 0 ≤ 0
134133a1i 11 . . . . . . . . . 10 (𝜑 → 0 ≤ 0)
135 0red 10154 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
1362nnred 11148 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℝ)
1372nngt0d 11177 . . . . . . . . . . 11 (𝜑 → 0 < 𝑀)
138135, 136, 137ltled 10298 . . . . . . . . . 10 (𝜑 → 0 ≤ 𝑀)
139134, 138jca 555 . . . . . . . . 9 (𝜑 → (0 ≤ 0 ∧ 0 ≤ 𝑀))
140 elfz2 12447 . . . . . . . . 9 (0 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
141132, 139, 140sylanbrc 701 . . . . . . . 8 (𝜑 → 0 ∈ (0...𝑀))
1428, 23syl6eqel 2811 . . . . . . . . 9 (𝜑 → (𝑄‘0) ∈ ℝ)
143142, 95resubcld 10571 . . . . . . . 8 (𝜑 → ((𝑄‘0) − 𝑋) ∈ ℝ)
144127, 130, 141, 143fvmptd 6402 . . . . . . 7 (𝜑 → (𝐻‘0) = ((𝑄‘0) − 𝑋))
1458oveq1d 6780 . . . . . . 7 (𝜑 → ((𝑄‘0) − 𝑋) = (-π − 𝑋))
146144, 145eqtr2d 2759 . . . . . 6 (𝜑 → (-π − 𝑋) = (𝐻‘0))
147 fveq2 6304 . . . . . . . . . 10 (𝑖 = 𝑀 → (𝑄𝑖) = (𝑄𝑀))
148147oveq1d 6780 . . . . . . . . 9 (𝑖 = 𝑀 → ((𝑄𝑖) − 𝑋) = ((𝑄𝑀) − 𝑋))
149148adantl 473 . . . . . . . 8 ((𝜑𝑖 = 𝑀) → ((𝑄𝑖) − 𝑋) = ((𝑄𝑀) − 𝑋))
15014, 131, 1313jca 1379 . . . . . . . . 9 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
151136leidd 10707 . . . . . . . . . 10 (𝜑𝑀𝑀)
152138, 151jca 555 . . . . . . . . 9 (𝜑 → (0 ≤ 𝑀𝑀𝑀))
153 elfz2 12447 . . . . . . . . 9 (𝑀 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
154150, 152, 153sylanbrc 701 . . . . . . . 8 (𝜑𝑀 ∈ (0...𝑀))
15510, 22syl6eqel 2811 . . . . . . . . 9 (𝜑 → (𝑄𝑀) ∈ ℝ)
156155, 95resubcld 10571 . . . . . . . 8 (𝜑 → ((𝑄𝑀) − 𝑋) ∈ ℝ)
157127, 149, 154, 156fvmptd 6402 . . . . . . 7 (𝜑 → (𝐻𝑀) = ((𝑄𝑀) − 𝑋))
15810oveq1d 6780 . . . . . . 7 (𝜑 → ((𝑄𝑀) − 𝑋) = (π − 𝑋))
159157, 158eqtr2d 2759 . . . . . 6 (𝜑 → (π − 𝑋) = (𝐻𝑀))
160146, 159oveq12d 6783 . . . . 5 (𝜑 → ((-π − 𝑋)[,](π − 𝑋)) = ((𝐻‘0)[,](𝐻𝑀)))
161160itgeq1d 40592 . . . 4 (𝜑 → ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐻‘0)[,](𝐻𝑀))(𝐹‘(𝑋 + 𝑡)) d𝑡)
16227ffvelrnda 6474 . . . . . . . 8 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
16395adantr 472 . . . . . . . 8 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
164162, 163resubcld 10571 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
165164, 126fmptd 6500 . . . . . 6 (𝜑𝐻:(0...𝑀)⟶ℝ)
16640, 43, 96, 29ltsub1dd 10752 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) − 𝑋) < ((𝑄‘(𝑖 + 1)) − 𝑋))
16739, 164syldan 488 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
168126fvmpt2 6405 . . . . . . . 8 ((𝑖 ∈ (0...𝑀) ∧ ((𝑄𝑖) − 𝑋) ∈ ℝ) → (𝐻𝑖) = ((𝑄𝑖) − 𝑋))
16939, 167, 168syl2anc 696 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) = ((𝑄𝑖) − 𝑋))
170 fveq2 6304 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
171170oveq1d 6780 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑄𝑖) − 𝑋) = ((𝑄𝑗) − 𝑋))
172171cbvmptv 4858 . . . . . . . . . 10 (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋))
173126, 172eqtri 2746 . . . . . . . . 9 𝐻 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋))
174173a1i 11 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐻 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋)))
175 fveq2 6304 . . . . . . . . . 10 (𝑗 = (𝑖 + 1) → (𝑄𝑗) = (𝑄‘(𝑖 + 1)))
176175oveq1d 6780 . . . . . . . . 9 (𝑗 = (𝑖 + 1) → ((𝑄𝑗) − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
177176adantl 473 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑄𝑗) − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
17843, 96resubcld 10571 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
179174, 177, 42, 178fvmptd 6402 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) − 𝑋))
180166, 169, 1793brtr4d 4792 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) < (𝐻‘(𝑖 + 1)))
181 frn 6166 . . . . . . . . 9 (𝐹:(-π[,]π)⟶ℂ → ran 𝐹 ⊆ ℂ)
18230, 181syl 17 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℂ)
183182adantr 472 . . . . . . 7 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → ran 𝐹 ⊆ ℂ)
184 ffun 6161 . . . . . . . . . 10 (𝐹:(-π[,]π)⟶ℂ → Fun 𝐹)
18530, 184syl 17 . . . . . . . . 9 (𝜑 → Fun 𝐹)
186185adantr 472 . . . . . . . 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 472 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑋 ∈ ℝ)
190144, 143eqeltrd 2803 . . . . . . . . . . . . 13 (𝜑 → (𝐻‘0) ∈ ℝ)
191190adantr 472 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻‘0) ∈ ℝ)
192157, 156eqeltrd 2803 . . . . . . . . . . . . 13 (𝜑 → (𝐻𝑀) ∈ ℝ)
193192adantr 472 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻𝑀) ∈ ℝ)
194 simpr 479 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)))
195 eliccre 40148 . . . . . . . . . . . 12 (((𝐻‘0) ∈ ℝ ∧ (𝐻𝑀) ∈ ℝ ∧ 𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ℝ)
196191, 193, 194, 195syl3anc 1439 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ℝ)
197189, 196readdcld 10182 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ ℝ)
198128adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 = 0) → (𝑄𝑖) = (𝑄‘0))
199198oveq1d 6780 . . . . . . . . . . . . . . 15 ((𝜑𝑖 = 0) → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
200127, 199, 141, 143fvmptd 6402 . . . . . . . . . . . . . 14 (𝜑 → (𝐻‘0) = ((𝑄‘0) − 𝑋))
201200oveq2d 6781 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + (𝐻‘0)) = (𝑋 + ((𝑄‘0) − 𝑋)))
20295recnd 10181 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ ℂ)
203142recnd 10181 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) ∈ ℂ)
204202, 203pncan3d 10508 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + ((𝑄‘0) − 𝑋)) = (𝑄‘0))
205201, 204, 83eqtrrd 2763 . . . . . . . . . . . 12 (𝜑 → -π = (𝑋 + (𝐻‘0)))
206205adantr 472 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → -π = (𝑋 + (𝐻‘0)))
207 elicc2 12352 . . . . . . . . . . . . . . 15 (((𝐻‘0) ∈ ℝ ∧ (𝐻𝑀) ∈ ℝ) → (𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)) ↔ (𝑡 ∈ ℝ ∧ (𝐻‘0) ≤ 𝑡𝑡 ≤ (𝐻𝑀))))
208191, 193, 207syl2anc 696 . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻‘0) ≤ 𝑡)
211191, 196, 189, 210leadd2dd 10755 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + (𝐻‘0)) ≤ (𝑋 + 𝑡))
212206, 211eqbrtrd 4782 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → -π ≤ (𝑋 + 𝑡))
213209simp3d 1136 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ≤ (𝐻𝑀))
214196, 193, 189, 213leadd2dd 10755 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ≤ (𝑋 + (𝐻𝑀)))
215157oveq2d 6781 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + (𝐻𝑀)) = (𝑋 + ((𝑄𝑀) − 𝑋)))
216155recnd 10181 . . . . . . . . . . . . . 14 (𝜑 → (𝑄𝑀) ∈ ℂ)
217202, 216pncan3d 10508 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + ((𝑄𝑀) − 𝑋)) = (𝑄𝑀))
218215, 217, 103eqtrrd 2763 . . . . . . . . . . . 12 (𝜑 → π = (𝑋 + (𝐻𝑀)))
219218adantr 472 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → π = (𝑋 + (𝐻𝑀)))
220214, 219breqtrrd 4788 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ≤ π)
221187, 188, 197, 212, 220eliccd 40146 . . . . . . . . 9 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ (-π[,]π))
222 fdm 6164 . . . . . . . . . . . 12 (𝐹:(-π[,]π)⟶ℂ → dom 𝐹 = (-π[,]π))
22330, 222syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = (-π[,]π))
224223eqcomd 2730 . . . . . . . . . 10 (𝜑 → (-π[,]π) = dom 𝐹)
225224adantr 472 . . . . . . . . 9 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (-π[,]π) = dom 𝐹)
226221, 225eleqtrd 2805 . . . . . . . 8 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ dom 𝐹)
227 fvelrn 6467 . . . . . . . 8 ((Fun 𝐹 ∧ (𝑋 + 𝑡) ∈ dom 𝐹) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
228186, 226, 227syl2anc 696 . . . . . . 7 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
229183, 228sseldd 3710 . . . . . 6 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
230169, 167eqeltrd 2803 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℝ)
231179, 178eqeltrd 2803 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
23282, 60fssresd 6184 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
23340rexrd 10202 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ*)
234233adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) ∈ ℝ*)
23543rexrd 10202 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
236235adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
23795ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
238 elioore 12319 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
239238adantl 473 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ℝ)
240237, 239readdcld 10182 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ℝ)
241169oveq2d 6781 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻𝑖)) = (𝑋 + ((𝑄𝑖) − 𝑋)))
242202adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
24340recnd 10181 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℂ)
244242, 243pncan3d 10508 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑄𝑖) − 𝑋)) = (𝑄𝑖))
245 eqidd 2725 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = (𝑄𝑖))
246241, 244, 2453eqtrrd 2763 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = (𝑋 + (𝐻𝑖)))
247246adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) = (𝑋 + (𝐻𝑖)))
248230adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) ∈ ℝ)
249 simpr 479 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
250248rexrd 10202 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) ∈ ℝ*)
251231rexrd 10202 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℝ*)
252251adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻‘(𝑖 + 1)) ∈ ℝ*)
253 elioo2 12330 . . . . . . . . . . . . . . . . . 18 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ*) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↔ (𝑡 ∈ ℝ ∧ (𝐻𝑖) < 𝑡𝑡 < (𝐻‘(𝑖 + 1)))))
254250, 252, 253syl2anc 696 . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) < 𝑡)
257248, 239, 237, 256ltadd2dd 10309 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + (𝐻𝑖)) < (𝑋 + 𝑡))
258247, 257eqbrtrd 4782 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) < (𝑋 + 𝑡))
259231adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
260255simp3d 1136 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 < (𝐻‘(𝑖 + 1)))
261239, 259, 237, 260ltadd2dd 10309 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) < (𝑋 + (𝐻‘(𝑖 + 1))))
262179oveq2d 6781 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑋 + ((𝑄‘(𝑖 + 1)) − 𝑋)))
26343recnd 10181 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
264242, 263pncan3d 10508 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑄‘(𝑖 + 1)) − 𝑋)) = (𝑄‘(𝑖 + 1)))
265262, 264eqtrd 2758 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
266265adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
267261, 266breqtrd 4786 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) < (𝑄‘(𝑖 + 1)))
268234, 236, 240, 258, 267eliood 40140 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
269 eqid 2724 . . . . . . . . . . . 12 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
270268, 269fmptd 6500 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)):((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))⟶((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
271 fcompt 6515 . . . . . . . . . . 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 696 . . . . . . . . . 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 6773 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑟 → (𝑋 + 𝑡) = (𝑋 + 𝑟))
274273cbvmptv 4858 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))
275274fveq1i 6305 . . . . . . . . . . . . . 14 ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠) = ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠)
276275fveq2i 6307 . . . . . . . . . . . . 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 4849 . . . . . . . . . . . 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 6304 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠) = ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡))
280279fveq2d 6308 . . . . . . . . . . . . 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 4858 . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . 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 6773 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑡 → (𝑋 + 𝑟) = (𝑋 + 𝑡))
285284adantl 473 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) ∧ 𝑟 = 𝑡) → (𝑋 + 𝑟) = (𝑋 + 𝑡))
286283, 285, 249, 240fvmptd 6402 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡) = (𝑋 + 𝑡))
287286fveq2d 6308 . . . . . . . . . . . . 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 6320 . . . . . . . . . . . . . 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 2758 . . . . . . . . . . . 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 4852 . . . . . . . . . . 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 2762 . . . . . . . . . 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 2759 . . . . . . . . 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 2724 . . . . . . . . . . 11 (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡))
295 ssid 3730 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
296295a1i 11 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → ℂ ⊆ ℂ)
297 id 22 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → 𝑋 ∈ ℂ)
298296, 297, 296constcncfg 40504 . . . . . . . . . . . . 13 (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
299 cncfmptid 22837 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
300295, 295, 299mp2an 710 . . . . . . . . . . . . . 14 (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ)
301300a1i 11 . . . . . . . . . . . . 13 (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
302298, 301addcncf 40506 . . . . . . . . . . . 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 40136 . . . . . . . . . . . 12 ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ⊆ ℂ
305304a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ⊆ ℂ)
306 ioosscn 40136 . . . . . . . . . . . 12 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
307306a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
308294, 303, 305, 307, 268cncfmptssg 40503 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
309308, 64cncfco 22832 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
310293, 309eqeltrd 2803 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
311233adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄𝑖) ∈ ℝ*)
312235adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
313 simpr 479 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
314 vex 3307 . . . . . . . . . . . . . . . . . 18 𝑟 ∈ V
315269elrnmpt 5479 . . . . . . . . . . . . . . . . . 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 1956 . . . . . . . . . . . . . . . . . 18 𝑡(𝜑𝑖 ∈ (0..^𝑀))
319 nfmpt1 4855 . . . . . . . . . . . . . . . . . . . 20 𝑡(𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
320319nfrn 5475 . . . . . . . . . . . . . . . . . . 19 𝑡ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
321320nfcri 2860 . . . . . . . . . . . . . . . . . 18 𝑡 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
322318, 321nfan 1941 . . . . . . . . . . . . . . . . 17 𝑡((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
323 nfv 1956 . . . . . . . . . . . . . . . . 17 𝑡 𝑟 ∈ ℝ
324 simp3 1130 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 = (𝑋 + 𝑡))
325953ad2ant1 1125 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑋 ∈ ℝ)
3262383ad2ant2 1126 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑡 ∈ ℝ)
327325, 326readdcld 10182 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑋 + 𝑡) ∈ ℝ)
328324, 327eqeltrd 2803 . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . 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 3128 . . . . . . . . . . . . . . . 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 1956 . . . . . . . . . . . . . . . . 17 𝑡(𝑄𝑖) < 𝑟
3342583adant3 1124 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑄𝑖) < (𝑋 + 𝑡))
335 simp3 1130 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 = (𝑋 + 𝑡))
336334, 335breqtrrd 4788 . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . 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 3128 . . . . . . . . . . . . . . . 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 1956 . . . . . . . . . . . . . . . . 17 𝑡 𝑟 < (𝑄‘(𝑖 + 1))
3422673adant3 1124 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑋 + 𝑡) < (𝑄‘(𝑖 + 1)))
343335, 342eqbrtrd 4782 . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . 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 3128 . . . . . . . . . . . . . . . 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 40140 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
349223ineq2d 3922 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ dom 𝐹) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
350349adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ dom 𝐹) = (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ∩ (-π[,]π)))
351 dmres 5529 . . . . . . . . . . . . . . . . 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 3695 . . . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
356355adantr 472 . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
358332, 347ltned 10286 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ≠ (𝑄‘(𝑖 + 1)))
359358neneqd 2901 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 = (𝑄‘(𝑖 + 1)))
360 velsn 4301 . . . . . . . . . . . . . 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 3691 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
363362ralrimiva 3068 . . . . . . . . . . 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 3698 . . . . . . . . . . 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 2724 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠))
367202adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
368 simpr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
369367, 368addcomd 10351 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑠 ∈ ℂ) → (𝑋 + 𝑠) = (𝑠 + 𝑋))
370369mpteq2dva 4852 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)))
371 eqid 2724 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋))
372371addccncf 22841 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
373202, 372syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
374370, 373eqeltrd 2803 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
375374adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
376230rexrd 10202 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℝ*)
377 iocssre 12367 . . . . . . . . . . . . . . . . . . 19 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℝ)
378376, 231, 377syl2anc 696 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℝ)
379 ax-resscn 10106 . . . . . . . . . . . . . . . . . 18 ℝ ⊆ ℂ
380378, 379syl6ss 3721 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℂ)
381295a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
382202ad2antrr 764 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℂ)
383380sselda 3709 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
384382, 383addcld 10172 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℂ)
385366, 375, 380, 381, 384cncfmptssg 40503 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))–cn→ℂ))
386 eqid 2724 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
387 eqid 2724 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
388386cnfldtop 22709 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘ℂfld) ∈ Top
389 unicntop 22711 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (TopOpen‘ℂfld)
390389restid 16217 . . . . . . . . . . . . . . . . . . . 20 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
391388, 390ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
392391eqcomi 2733 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
393386, 387, 392cncfcn 22834 . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
395385, 394eleqtrd 2805 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
396386cnfldtopon 22708 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
397396a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
398 resttopon 21088 . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))))
400 cncnp 21207 . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 12341 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → (𝐻‘(𝑖 + 1)) ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
405376, 251, 180, 404syl3anc 1439 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
406 fveq2 6304 . . . . . . . . . . . . . . 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 2789 . . . . . . . . . . . . . 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 3412 . . . . . . . . . . . . 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 696 . . . . . . . . . . . 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 40151 . . . . . . . . . . . . . 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 1439 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) = ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
412265eqcomd 2730 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = (𝑋 + (𝐻‘(𝑖 + 1))))
413412ad2antrr 764 . . . . . . . . . . . . . . 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 4200 . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 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 6773 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐻‘(𝑖 + 1)) → (𝑋 + 𝑠) = (𝑋 + (𝐻‘(𝑖 + 1))))
417416adantl 473 . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . 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 4203 . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . 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 6773 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (𝑋 + 𝑡) = (𝑋 + 𝑠))
423422adantl 473 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) ∧ 𝑡 = 𝑠) → (𝑋 + 𝑡) = (𝑋 + 𝑠))
424 velsn 4301 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ {(𝐻‘(𝑖 + 1))} ↔ 𝑠 = (𝐻‘(𝑖 + 1)))
425424notbii 309 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ {(𝐻‘(𝑖 + 1))} ↔ ¬ 𝑠 = (𝐻‘(𝑖 + 1)))
426 elun 3861 . . . . . . . . . . . . . . . . . . . . . 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 402 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (𝑠 ∈ {(𝐻‘(𝑖 + 1))} ∨ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
429428ord 391 . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
432431adantll 752 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
43395ad2antrr 764 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → 𝑋 ∈ ℝ)
434 elioore 12319 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
435434adantl 473 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
436 elsni 4302 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ {(𝐻‘(𝑖 + 1))} → 𝑠 = (𝐻‘(𝑖 + 1)))
437436adantl 473 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → 𝑠 = (𝐻‘(𝑖 + 1)))
438231adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
439437, 438eqeltrd 2803 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → 𝑠 ∈ ℝ)
440435, 439jaodan 861 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻‘(𝑖 + 1))})) → 𝑠 ∈ ℝ)
441426, 440sylan2b 493 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → 𝑠 ∈ ℝ)
442433, 441readdcld 10182 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → (𝑋 + 𝑠) ∈ ℝ)
443442adantr 472 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → (𝑋 + 𝑠) ∈ ℝ)
444421, 423, 432, 443fvmptd 6402 . . . . . . . . . . . . . . 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 2758 . . . . . . . . . . . . . 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 867 . . . . . . . . . . . . 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 4840 . . . . . . . . . . . 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 6781 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))))
449448oveq1d 6780 . . . . . . . . . . . . 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 6306 . . . . . . . . . . . 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 2818 . . . . . . . . . . 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 2724 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) = ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}))
453 eqid 2724 . . . . . . . . . . . 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 6170 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)):((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))⟶ℂ)
455231recnd 10181 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℂ)
456452, 386, 453, 454, 305, 455ellimc 23757 . . . . . . . . . . 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 40272 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻‘(𝑖 + 1))))
459272, 292eqtrd 2758 . . . . . . . . . 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 6780 . . . . . . . . 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 2805 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) lim (𝐻‘(𝑖 + 1))))
46240adantr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → (𝑄𝑖) ∈ ℝ)
463462, 340gtned 10285 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ≠ (𝑄𝑖))
464463neneqd 2901 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 = (𝑄𝑖))
465 velsn 4301 . . . . . . . . . . . . . 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 3691 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
468467ralrimiva 3068 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
469 dfss3 3698 . . . . . . . . . . 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 12368 . . . . . . . . . . . . . . . . . . 19 (((𝐻𝑖) ∈ ℝ ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ*) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℝ)
472230, 251, 471syl2anc 696 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℝ)
473472, 379syl6ss 3721 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℂ)
474202ad2antrr 764 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℂ)
475473sselda 3709 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
476474, 475addcld 10172 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℂ)
477366, 375, 473, 381, 476cncfmptssg 40503 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
478 eqid 2724 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
479386, 478, 392cncfcn 22834 . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
481477, 480eleqtrd 2805 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
482 resttopon 21088 . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))))
484 cncnp 21207 . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 12342 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → (𝐻𝑖) ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
489376, 251, 180, 488syl3anc 1439 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
490 fveq2 6304 . . . . . . . . . . . . . . 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 2789 . . . . . . . . . . . . . 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 3412 . . . . . . . . . . . . 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 696 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻𝑖)))
494 uncom 3865 . . . . . . . . . . . . . 14 (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) = ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
495 snunioo 12412 . . . . . . . . . . . . . . 15 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
496376, 251, 180, 495syl3anc 1439 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
497494, 496syl5eq 2770 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
498 iftrue 4200 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐻𝑖) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑄𝑖))
499498adantl 473 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑄𝑖))
500246adantr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → (𝑄𝑖) = (𝑋 + (𝐻𝑖)))
501 oveq2 6773 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝐻𝑖) → (𝑋 + 𝑠) = (𝑋 + (𝐻𝑖)))
502501eqcomd 2730 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐻𝑖) → (𝑋 + (𝐻𝑖)) = (𝑋 + 𝑠))
503502adantl 473 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → (𝑋 + (𝐻𝑖)) = (𝑋 + 𝑠))
504499, 500, 5033eqtrd 2762 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
505504adantlr 753 . . . . . . . . . . . . . 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 4203 . . . . . . . . . . . . . . . 16 𝑠 = (𝐻𝑖) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠))
507506adantl 473 . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) ∧ 𝑡 = 𝑠) → (𝑋 + 𝑡) = (𝑋 + 𝑠))
510 velsn 4301 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ {(𝐻𝑖)} ↔ 𝑠 = (𝐻𝑖))
511510notbii 309 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ {(𝐻𝑖)} ↔ ¬ 𝑠 = (𝐻𝑖))
512 elun 3861 . . . . . . . . . . . . . . . . . . . . . 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 402 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (𝑠 ∈ {(𝐻𝑖)} ∨ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
515514ord 391 . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ∧ ¬ 𝑠 = (𝐻𝑖)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
518517adantll 752 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
51995ad2antrr 764 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → 𝑋 ∈ ℝ)
520 elsni 4302 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ {(𝐻𝑖)} → 𝑠 = (𝐻𝑖))
521520adantl 473 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → 𝑠 = (𝐻𝑖))
522230adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → (𝐻𝑖) ∈ ℝ)
523521, 522eqeltrd 2803 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → 𝑠 ∈ ℝ)
524435, 523jaodan 861 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻𝑖)})) → 𝑠 ∈ ℝ)
525512, 524sylan2b 493 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → 𝑠 ∈ ℝ)
526519, 525readdcld 10182 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → (𝑋 + 𝑠) ∈ ℝ)
527526adantr 472 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → (𝑋 + 𝑠) ∈ ℝ)
528508, 509, 518, 527fvmptd 6402 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠) = (𝑋 + 𝑠))
529507, 528eqtrd 2758 . . . . . . . . . . . . . 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 867 . . . . . . . . . . . . 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 4840 . . . . . . . . . . . 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 6781 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))))
533532oveq1d 6780 . . . . . . . . . . . . 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 6306 . . . . . . . . . . . 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 2818 . . . . . . . . . . 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 2724 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) = ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}))
537 eqid 2724 . . . . . . . . . . . 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 10181 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℂ)
539536, 386, 537, 454, 305, 538ellimc 23757 . . . . . . . . . . 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 40272 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻𝑖)))
542459oveq1d 6780 . . . . . . . . 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 2805 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) lim (𝐻𝑖)))
544230, 231, 310, 461, 543iblcncfioo 40614 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ 𝐿1)
54530ad2antrr 764 . . . . . . . 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 764 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
549 simplr 809 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
550 simpr 479 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))))
551169, 179oveq12d 6783 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) = (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
552551adantr 472 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) = (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
553550, 552eleqtrd 2805 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
554553, 116syldan 488 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
555546, 547, 548, 549, 554fourierdlem1 40745 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ (-π[,]π))
556545, 555ffvelrnd 6475 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
557230, 231, 544, 556ibliooicc 40607 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ 𝐿1)
55814, 20, 165, 180, 229, 557itgspltprt 40615 . . . . 5 (𝜑 → ∫((𝐻‘0)[,](𝐻𝑀))(𝐹‘(𝑋 + 𝑡)) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))(𝐹‘(𝑋 + 𝑡)) d𝑡)
559551itgeq1d 40592 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
560559sumeq2dv 14553 . . . . 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 2758 . . . 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 2762 . . 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 2761 . 2 (𝜑 → Σ𝑖 ∈ (0..^𝑀)∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠)
56413, 76, 5633eqtrd 2762 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  wo 382  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  wrex 3015  {crab 3018  Vcvv 3304  cdif 3677  cun 3678  cin 3679  wss 3680  ifcif 4194  {csn 4285   class class class wbr 4760  cmpt 4837  dom cdm 5218  ran crn 5219  cres 5220  ccom 5222  Fun wfun 5995  wf 5997  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  cc 10047  cr 10048  0cc0 10049  1c1 10050   + caddc 10052  *cxr 10186   < clt 10187  cle 10188  cmin 10379  -cneg 10380  cn 11133  cz 11490  cuz 11800  (,)cioo 12289  (,]cioc 12290  [,)cico 12291  [,]cicc 12292  ...cfz 12440  ..^cfzo 12580  Σcsu 14536  πcpi 14917  t crest 16204  TopOpenctopn 16205  fldccnfld 19869  Topctop 20821  TopOnctopon 20838   Cn ccn 21151   CnP ccnp 21152  cnccncf 22801  citg 23507   lim climc 23746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cc 9370  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127  ax-addf 10128  ax-mulf 10129
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-disj 4729  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-of 7014  df-ofr 7015  df-om 7183  df-1st 7285  df-2nd 7286  df-supp 7416  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-omul 7685  df-er 7862  df-map 7976  df-pm 7977  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fsupp 8392  df-fi 8433  df-sup 8464  df-inf 8465  df-oi 8531  df-card 8878  df-acn 8881  df-cda 9103  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-dec 11607  df-uz 11801  df-q 11903  df-rp 11947  df-xneg 12060  df-xadd 12061  df-xmul 12062  df-ioo 12293  df-ioc 12294  df-ico 12295  df-icc 12296  df-fz 12441  df-fzo 12581  df-fl 12708  df-mod 12784  df-seq 12917  df-exp 12976  df-fac 13176  df-bc 13205  df-hash 13233  df-shft 13927  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-limsup 14322  df-clim 14339  df-rlim 14340  df-sum 14537  df-ef 14918  df-sin 14920  df-cos 14921  df-pi 14923  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-mulr 16078  df-starv 16079  df-sca 16080  df-vsca 16081  df-ip 16082  df-tset 16083  df-ple 16084  df-ds 16087  df-unif 16088  df-hom 16089  df-cco 16090  df-rest 16206  df-topn 16207  df-0g 16225  df-gsum 16226  df-topgen 16227  df-pt 16228  df-prds 16231  df-xrs 16285  df-qtop 16290  df-imas 16291  df-xps 16293  df-mre 16369  df-mrc 16370  df-acs 16372  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-submnd 17458  df-mulg 17663  df-cntz 17871  df-cmn 18316  df-psmet 19861  df-xmet 19862  df-met 19863  df-bl 19864  df-mopn 19865  df-fbas 19866  df-fg 19867  df-cnfld 19870  df-top 20822  df-topon 20839  df-topsp 20860  df-bases 20873  df-cld 20946  df-ntr 20947  df-cls 20948  df-nei 21025  df-lp 21063  df-perf 21064  df-cn 21154  df-cnp 21155  df-haus 21242  df-cmp 21313  df-tx 21488  df-hmeo 21681  df-fil 21772  df-fm 21864  df-flim 21865  df-flf 21866  df-xms 22247  df-ms 22248  df-tms 22249  df-cncf 22803  df-ovol 23354  df-vol 23355  df-mbf 23508  df-itg1 23509  df-itg2 23510  df-ibl 23511  df-itg 23512  df-0p 23557  df-ditg 23731  df-limc 23750  df-dv 23751
This theorem is referenced by:  fourierdlem101  40844
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