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Theorem fourierdlem93 39749
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 39659 . . . . . . . . 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 479 . . . . . 6 (𝜑 → (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
87simplld 790 . . . . 5 (𝜑 → (𝑄‘0) = -π)
98eqcomd 2627 . . . 4 (𝜑 → -π = (𝑄‘0))
107simplrd 792 . . . . 5 (𝜑 → (𝑄𝑀) = π)
1110eqcomd 2627 . . . 4 (𝜑 → π = (𝑄𝑀))
129, 11oveq12d 6628 . . 3 (𝜑 → (-π[,]π) = ((𝑄‘0)[,](𝑄𝑀)))
1312itgeq1d 39505 . 2 (𝜑 → ∫(-π[,]π)(𝐹𝑡) d𝑡 = ∫((𝑄‘0)[,](𝑄𝑀))(𝐹𝑡) d𝑡)
14 0zd 11341 . . 3 (𝜑 → 0 ∈ ℤ)
15 nnuz 11675 . . . . 5 ℕ = (ℤ‘1)
162, 15syl6eleq 2708 . . . 4 (𝜑𝑀 ∈ (ℤ‘1))
17 1e0p1 11504 . . . . . 6 1 = (0 + 1)
1817a1i 11 . . . . 5 (𝜑 → 1 = (0 + 1))
1918fveq2d 6157 . . . 4 (𝜑 → (ℤ‘1) = (ℤ‘(0 + 1)))
2016, 19eleqtrd 2700 . . 3 (𝜑𝑀 ∈ (ℤ‘(0 + 1)))
213, 2, 1fourierdlem15 39672 . . . 4 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
22 pire 24131 . . . . . . 7 π ∈ ℝ
2322renegcli 10294 . . . . . 6 -π ∈ ℝ
24 iccssre 12205 . . . . . 6 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
2523, 22, 24mp2an 707 . . . . 5 (-π[,]π) ⊆ ℝ
2625a1i 11 . . . 4 (𝜑 → (-π[,]π) ⊆ ℝ)
2721, 26fssd 6019 . . 3 (𝜑𝑄:(0...𝑀)⟶ℝ)
287simprd 479 . . . 4 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2928r19.21bi 2927 . . 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 477 . . . . 5 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀)))
3312eqcomd 2627 . . . . . 6 (𝜑 → ((𝑄‘0)[,](𝑄𝑀)) = (-π[,]π))
3433adantr 481 . . . . 5 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → ((𝑄‘0)[,](𝑄𝑀)) = (-π[,]π))
3532, 34eleqtrd 2700 . . . 4 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → 𝑡 ∈ (-π[,]π))
3631, 35ffvelrnd 6321 . . 3 ((𝜑𝑡 ∈ ((𝑄‘0)[,](𝑄𝑀))) → (𝐹𝑡) ∈ ℂ)
3727adantr 481 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
38 elfzofz 12434 . . . . . 6 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
3938adantl 482 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
4037, 39ffvelrnd 6321 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
41 fzofzp1 12514 . . . . . 6 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
4241adantl 482 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
4337, 42ffvelrnd 6321 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
4430feqmptd 6211 . . . . . . . . . 10 (𝜑𝐹 = (𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)))
4544adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹 = (𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)))
4645reseq1d 5360 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
47 ioossicc 12209 . . . . . . . . . . 11 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
4847a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
4923rexri 10049 . . . . . . . . . . . . . 14 -π ∈ ℝ*
5049a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → -π ∈ ℝ*)
5122rexri 10049 . . . . . . . . . . . . . 14 π ∈ ℝ*
5251a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → π ∈ ℝ*)
5321ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
54 simplr 791 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
55 simpr 477 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
5650, 52, 53, 54, 55fourierdlem1 39658 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝑡 ∈ (-π[,]π))
5756ralrimiva 2961 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))𝑡 ∈ (-π[,]π))
58 dfss3 3577 . . . . . . . . . . 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 3597 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
6160resmptd 5416 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ (-π[,]π) ↦ (𝐹𝑡)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)))
6246, 61eqtrd 2655 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)))
6362eqcomd 2627 . . . . . 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 2698 . . . . 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 6625 . . . . . 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 2700 . . . . 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 6625 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄𝑖)))
7169, 70eleqtrd 2700 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) lim (𝑄𝑖)))
7240, 43, 65, 68, 71iblcncfioo 39527 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) ∈ 𝐿1)
7330ad2antrr 761 . . . . 5 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → 𝐹:(-π[,]π)⟶ℂ)
7473, 56ffvelrnd 6321 . . . 4 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) → (𝐹𝑡) ∈ ℂ)
7540, 43, 72, 74ibliooicc 39520 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ↦ (𝐹𝑡)) ∈ 𝐿1)
7614, 20, 27, 29, 36, 75itgspltprt 39528 . 2 (𝜑 → ∫((𝑄‘0)[,](𝑄𝑀))(𝐹𝑡) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡)
77 fvres 6169 . . . . . . . 8 (𝑡 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡) = (𝐹𝑡))
7877eqcomd 2627 . . . . . . 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 23471 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))‘𝑡) d𝑡)
81 eqid 2621 . . . . . 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 6033 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))):((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))⟶ℂ)
8448resabs1d 5392 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
8584, 64eqeltrd 2698 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
8684oveq1d 6625 . . . . . . . 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 39301 . . . . . . . 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 2657 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
8966, 88eleqtrd 2700 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
9084eqcomd 2627 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
9190oveq1d 6625 . . . . . . . 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 39285 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
9391, 92eqtrd 2655 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
9469, 93eleqtrd 2700 . . . . . 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 39738 . . . . 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 761 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑋 ∈ ℝ)
10198, 100resubcld 10410 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
10299, 100resubcld 10410 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
103 simpr 477 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
104 eliccre 39170 . . . . . . . . . 10 ((((𝑄𝑖) − 𝑋) ∈ ℝ ∧ ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ ℝ)
105101, 102, 103, 104syl3anc 1323 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ∈ ℝ)
106100, 105readdcld 10021 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
107 elicc2 12188 . . . . . . . . . . . 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 692 . . . . . . . . . . 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 1072 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → ((𝑄𝑖) − 𝑋) ≤ 𝑡)
11198, 100, 105lesubadd2d 10578 . . . . . . . . 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 1073 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → 𝑡 ≤ ((𝑄‘(𝑖 + 1)) − 𝑋))
114100, 105, 99leaddsub2d 10581 . . . . . . . . 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 39168 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
117 fvres 6169 . . . . . . 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 23471 . . . . 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 2659 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
121120sumeq2dv 14375 . . 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 6618 . . . . . . 7 (𝑠 = 𝑡 → (𝑋 + 𝑠) = (𝑋 + 𝑡))
123122fveq2d 6157 . . . . . 6 (𝑠 = 𝑡 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑡)))
124123cbvitgv 23466 . . . . 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 6153 . . . . . . . . . 10 (𝑖 = 0 → (𝑄𝑖) = (𝑄‘0))
129128oveq1d 6625 . . . . . . . . 9 (𝑖 = 0 → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
130129adantl 482 . . . . . . . 8 ((𝜑𝑖 = 0) → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
1312nnzd 11433 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
13214, 131, 143jca 1240 . . . . . . . . 9 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ))
133 0le0 11062 . . . . . . . . . . 11 0 ≤ 0
134133a1i 11 . . . . . . . . . 10 (𝜑 → 0 ≤ 0)
135 0red 9993 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
1362nnred 10987 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℝ)
1372nngt0d 11016 . . . . . . . . . . 11 (𝜑 → 0 < 𝑀)
138135, 136, 137ltled 10137 . . . . . . . . . 10 (𝜑 → 0 ≤ 𝑀)
139134, 138jca 554 . . . . . . . . 9 (𝜑 → (0 ≤ 0 ∧ 0 ≤ 𝑀))
140 elfz2 12283 . . . . . . . . 9 (0 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
141132, 139, 140sylanbrc 697 . . . . . . . 8 (𝜑 → 0 ∈ (0...𝑀))
1428, 23syl6eqel 2706 . . . . . . . . 9 (𝜑 → (𝑄‘0) ∈ ℝ)
143142, 95resubcld 10410 . . . . . . . 8 (𝜑 → ((𝑄‘0) − 𝑋) ∈ ℝ)
144127, 130, 141, 143fvmptd 6250 . . . . . . 7 (𝜑 → (𝐻‘0) = ((𝑄‘0) − 𝑋))
1458oveq1d 6625 . . . . . . 7 (𝜑 → ((𝑄‘0) − 𝑋) = (-π − 𝑋))
146144, 145eqtr2d 2656 . . . . . 6 (𝜑 → (-π − 𝑋) = (𝐻‘0))
147 fveq2 6153 . . . . . . . . . 10 (𝑖 = 𝑀 → (𝑄𝑖) = (𝑄𝑀))
148147oveq1d 6625 . . . . . . . . 9 (𝑖 = 𝑀 → ((𝑄𝑖) − 𝑋) = ((𝑄𝑀) − 𝑋))
149148adantl 482 . . . . . . . 8 ((𝜑𝑖 = 𝑀) → ((𝑄𝑖) − 𝑋) = ((𝑄𝑀) − 𝑋))
15014, 131, 1313jca 1240 . . . . . . . . 9 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
151136leidd 10546 . . . . . . . . . 10 (𝜑𝑀𝑀)
152138, 151jca 554 . . . . . . . . 9 (𝜑 → (0 ≤ 𝑀𝑀𝑀))
153 elfz2 12283 . . . . . . . . 9 (𝑀 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
154150, 152, 153sylanbrc 697 . . . . . . . 8 (𝜑𝑀 ∈ (0...𝑀))
15510, 22syl6eqel 2706 . . . . . . . . 9 (𝜑 → (𝑄𝑀) ∈ ℝ)
156155, 95resubcld 10410 . . . . . . . 8 (𝜑 → ((𝑄𝑀) − 𝑋) ∈ ℝ)
157127, 149, 154, 156fvmptd 6250 . . . . . . 7 (𝜑 → (𝐻𝑀) = ((𝑄𝑀) − 𝑋))
15810oveq1d 6625 . . . . . . 7 (𝜑 → ((𝑄𝑀) − 𝑋) = (π − 𝑋))
159157, 158eqtr2d 2656 . . . . . 6 (𝜑 → (π − 𝑋) = (𝐻𝑀))
160146, 159oveq12d 6628 . . . . 5 (𝜑 → ((-π − 𝑋)[,](π − 𝑋)) = ((𝐻‘0)[,](𝐻𝑀)))
161160itgeq1d 39505 . . . 4 (𝜑 → ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐻‘0)[,](𝐻𝑀))(𝐹‘(𝑋 + 𝑡)) d𝑡)
16227ffvelrnda 6320 . . . . . . . 8 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
16395adantr 481 . . . . . . . 8 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
164162, 163resubcld 10410 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
165164, 126fmptd 6346 . . . . . 6 (𝜑𝐻:(0...𝑀)⟶ℝ)
16640, 43, 96, 29ltsub1dd 10591 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) − 𝑋) < ((𝑄‘(𝑖 + 1)) − 𝑋))
16739, 164syldan 487 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖) − 𝑋) ∈ ℝ)
168126fvmpt2 6253 . . . . . . . 8 ((𝑖 ∈ (0...𝑀) ∧ ((𝑄𝑖) − 𝑋) ∈ ℝ) → (𝐻𝑖) = ((𝑄𝑖) − 𝑋))
16939, 167, 168syl2anc 692 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) = ((𝑄𝑖) − 𝑋))
170 fveq2 6153 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
171170oveq1d 6625 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑄𝑖) − 𝑋) = ((𝑄𝑗) − 𝑋))
172171cbvmptv 4715 . . . . . . . . . 10 (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋))
173126, 172eqtri 2643 . . . . . . . . 9 𝐻 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋))
174173a1i 11 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐻 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋)))
175 fveq2 6153 . . . . . . . . . 10 (𝑗 = (𝑖 + 1) → (𝑄𝑗) = (𝑄‘(𝑖 + 1)))
176175oveq1d 6625 . . . . . . . . 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 10410 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
179174, 177, 42, 178fvmptd 6250 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) − 𝑋))
180166, 169, 1793brtr4d 4650 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) < (𝐻‘(𝑖 + 1)))
181 frn 6015 . . . . . . . . 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 6010 . . . . . . . . . 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 2698 . . . . . . . . . . . . 13 (𝜑 → (𝐻‘0) ∈ ℝ)
191190adantr 481 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻‘0) ∈ ℝ)
192157, 156eqeltrd 2698 . . . . . . . . . . . . 13 (𝜑 → (𝐻𝑀) ∈ ℝ)
193192adantr 481 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻𝑀) ∈ ℝ)
194 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)))
195 eliccre 39170 . . . . . . . . . . . 12 (((𝐻‘0) ∈ ℝ ∧ (𝐻𝑀) ∈ ℝ ∧ 𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ℝ)
196191, 193, 194, 195syl3anc 1323 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ∈ ℝ)
197189, 196readdcld 10021 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ ℝ)
198128adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 = 0) → (𝑄𝑖) = (𝑄‘0))
199198oveq1d 6625 . . . . . . . . . . . . . . 15 ((𝜑𝑖 = 0) → ((𝑄𝑖) − 𝑋) = ((𝑄‘0) − 𝑋))
200127, 199, 141, 143fvmptd 6250 . . . . . . . . . . . . . 14 (𝜑 → (𝐻‘0) = ((𝑄‘0) − 𝑋))
201200oveq2d 6626 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + (𝐻‘0)) = (𝑋 + ((𝑄‘0) − 𝑋)))
20295recnd 10020 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ ℂ)
203142recnd 10020 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) ∈ ℂ)
204202, 203pncan3d 10347 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + ((𝑄‘0) − 𝑋)) = (𝑄‘0))
205201, 204, 83eqtrrd 2660 . . . . . . . . . . . 12 (𝜑 → -π = (𝑋 + (𝐻‘0)))
206205adantr 481 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → -π = (𝑋 + (𝐻‘0)))
207 elicc2 12188 . . . . . . . . . . . . . . 15 (((𝐻‘0) ∈ ℝ ∧ (𝐻𝑀) ∈ ℝ) → (𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀)) ↔ (𝑡 ∈ ℝ ∧ (𝐻‘0) ≤ 𝑡𝑡 ≤ (𝐻𝑀))))
208191, 193, 207syl2anc 692 . . . . . . . . . . . . . 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 1072 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐻‘0) ≤ 𝑡)
211191, 196, 189, 210leadd2dd 10594 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + (𝐻‘0)) ≤ (𝑋 + 𝑡))
212206, 211eqbrtrd 4640 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → -π ≤ (𝑋 + 𝑡))
213209simp3d 1073 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → 𝑡 ≤ (𝐻𝑀))
214196, 193, 189, 213leadd2dd 10594 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ≤ (𝑋 + (𝐻𝑀)))
215157oveq2d 6626 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + (𝐻𝑀)) = (𝑋 + ((𝑄𝑀) − 𝑋)))
216155recnd 10020 . . . . . . . . . . . . . 14 (𝜑 → (𝑄𝑀) ∈ ℂ)
217202, 216pncan3d 10347 . . . . . . . . . . . . 13 (𝜑 → (𝑋 + ((𝑄𝑀) − 𝑋)) = (𝑄𝑀))
218215, 217, 103eqtrrd 2660 . . . . . . . . . . . 12 (𝜑 → π = (𝑋 + (𝐻𝑀)))
219218adantr 481 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → π = (𝑋 + (𝐻𝑀)))
220214, 219breqtrrd 4646 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ≤ π)
221187, 188, 197, 212, 220eliccd 39168 . . . . . . . . 9 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ (-π[,]π))
222 fdm 6013 . . . . . . . . . . . 12 (𝐹:(-π[,]π)⟶ℂ → dom 𝐹 = (-π[,]π))
22330, 222syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = (-π[,]π))
224223eqcomd 2627 . . . . . . . . . 10 (𝜑 → (-π[,]π) = dom 𝐹)
225224adantr 481 . . . . . . . . 9 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (-π[,]π) = dom 𝐹)
226221, 225eleqtrd 2700 . . . . . . . 8 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝑋 + 𝑡) ∈ dom 𝐹)
227 fvelrn 6313 . . . . . . . 8 ((Fun 𝐹 ∧ (𝑋 + 𝑡) ∈ dom 𝐹) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
228186, 226, 227syl2anc 692 . . . . . . 7 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
229183, 228sseldd 3588 . . . . . 6 ((𝜑𝑡 ∈ ((𝐻‘0)[,](𝐻𝑀))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
230169, 167eqeltrd 2698 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℝ)
231179, 178eqeltrd 2698 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℝ)
23282, 60fssresd 6033 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
23340rexrd 10041 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ*)
234233adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄𝑖) ∈ ℝ*)
23543rexrd 10041 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
236235adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
23795ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
238 elioore 12155 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
239238adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ℝ)
240237, 239readdcld 10021 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ℝ)
241169oveq2d 6626 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻𝑖)) = (𝑋 + ((𝑄𝑖) − 𝑋)))
242202adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
24340recnd 10020 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℂ)
244242, 243pncan3d 10347 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑄𝑖) − 𝑋)) = (𝑄𝑖))
245 eqidd 2622 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = (𝑄𝑖))
246241, 244, 2453eqtrrd 2660 . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
250248rexrd 10041 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) ∈ ℝ*)
251231rexrd 10041 . . . . . . . . . . . . . . . . . . 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 12166 . . . . . . . . . . . . . . . . . 18 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ*) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↔ (𝑡 ∈ ℝ ∧ (𝐻𝑖) < 𝑡𝑡 < (𝐻‘(𝑖 + 1)))))
254250, 252, 253syl2anc 692 . . . . . . . . . . . . . . . . 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 1072 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝐻𝑖) < 𝑡)
257248, 239, 237, 256ltadd2dd 10148 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + (𝐻𝑖)) < (𝑋 + 𝑡))
258247, 257eqbrtrd 4640 . . . . . . . . . . . . 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 1073 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑡 < (𝐻‘(𝑖 + 1)))
261239, 259, 237, 260ltadd2dd 10148 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) < (𝑋 + (𝐻‘(𝑖 + 1))))
262179oveq2d 6626 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝐻‘(𝑖 + 1))) = (𝑋 + ((𝑄‘(𝑖 + 1)) − 𝑋)))
26343recnd 10020 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
264242, 263pncan3d 10347 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑄‘(𝑖 + 1)) − 𝑋)) = (𝑄‘(𝑖 + 1)))
265262, 264eqtrd 2655 . . . . . . . . . . . . . . 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 4644 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) < (𝑄‘(𝑖 + 1)))
268234, 236, 240, 258, 267eliood 39162 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
269 eqid 2621 . . . . . . . . . . . 12 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
270268, 269fmptd 6346 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)):((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))⟶((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
271 fcompt 6360 . . . . . . . . . . 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 692 . . . . . . . . . 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 6618 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑟 → (𝑋 + 𝑡) = (𝑋 + 𝑟))
274273cbvmptv 4715 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) = (𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))
275274fveq1i 6154 . . . . . . . . . . . . . 14 ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠) = ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠)
276275fveq2i 6156 . . . . . . . . . . . . 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 4706 . . . . . . . . . . . 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 6153 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑠) = ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡))
280279fveq2d 6157 . . . . . . . . . . . . 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 4715 . . . . . . . . . . . 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 2622 . . . . . . . . . . . . . . 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 6618 . . . . . . . . . . . . . . . 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 6250 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → ((𝑟 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑟))‘𝑡) = (𝑋 + 𝑡))
287286fveq2d 6157 . . . . . . . . . . . . 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 6169 . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . 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 4709 . . . . . . . . . . 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 2659 . . . . . . . . . 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 2656 . . . . . . . . 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 2621 . . . . . . . . . . 11 (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡)) = (𝑡 ∈ ℂ ↦ (𝑋 + 𝑡))
295 ssid 3608 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
296295a1i 11 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → ℂ ⊆ ℂ)
297 id 22 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → 𝑋 ∈ ℂ)
298296, 297, 296constcncfg 39415 . . . . . . . . . . . . 13 (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
299 cncfmptid 22638 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
300295, 295, 299mp2an 707 . . . . . . . . . . . . . 14 (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ)
301300a1i 11 . . . . . . . . . . . . 13 (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
302298, 301addcncf 39417 . . . . . . . . . . . 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 39158 . . . . . . . . . . . 12 ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ⊆ ℂ
305304a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ⊆ ℂ)
306 ioosscn 39158 . . . . . . . . . . . 12 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
307306a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
308294, 303, 305, 307, 268cncfmptssg 39414 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
309308, 64cncfco 22633 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
310293, 309eqeltrd 2698 . . . . . . . 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 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
314 vex 3192 . . . . . . . . . . . . . . . . . 18 𝑟 ∈ V
315269elrnmpt 5337 . . . . . . . . . . . . . . . . . 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 1840 . . . . . . . . . . . . . . . . . 18 𝑡(𝜑𝑖 ∈ (0..^𝑀))
319 nfmpt1 4712 . . . . . . . . . . . . . . . . . . . 20 𝑡(𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
320319nfrn 5333 . . . . . . . . . . . . . . . . . . 19 𝑡ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
321320nfcri 2755 . . . . . . . . . . . . . . . . . 18 𝑡 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))
322318, 321nfan 1825 . . . . . . . . . . . . . . . . 17 𝑡((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)))
323 nfv 1840 . . . . . . . . . . . . . . . . 17 𝑡 𝑟 ∈ ℝ
324 simp3 1061 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 = (𝑋 + 𝑡))
325953ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑋 ∈ ℝ)
3262383ad2ant2 1081 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑡 ∈ ℝ)
327325, 326readdcld 10021 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑋 + 𝑡) ∈ ℝ)
328324, 327eqeltrd 2698 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 ∈ ℝ)
3293283exp 1261 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → (𝑟 = (𝑋 + 𝑡) → 𝑟 ∈ ℝ)))
330329ad2antrr 761 . . . . . . . . . . . . . . . . 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 3020 . . . . . . . . . . . . . . . 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 1840 . . . . . . . . . . . . . . . . 17 𝑡(𝑄𝑖) < 𝑟
3342583adant3 1079 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑄𝑖) < (𝑋 + 𝑡))
335 simp3 1061 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 = (𝑋 + 𝑡))
336334, 335breqtrrd 4646 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑄𝑖) < 𝑟)
3373363exp 1261 . . . . . . . . . . . . . . . . . 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 3020 . . . . . . . . . . . . . . . 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 1840 . . . . . . . . . . . . . . . . 17 𝑡 𝑟 < (𝑄‘(𝑖 + 1))
3422673adant3 1079 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → (𝑋 + 𝑡) < (𝑄‘(𝑖 + 1)))
343335, 342eqbrtrd 4640 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∧ 𝑟 = (𝑋 + 𝑡)) → 𝑟 < (𝑄‘(𝑖 + 1)))
3443433exp 1261 . . . . . . . . . . . . . . . . . 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 3020 . . . . . . . . . . . . . . . 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 39162 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
349223ineq2d 3797 . . . . . . . . . . . . . . . . 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 5383 . . . . . . . . . . . . . . . . 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 3574 . . . . . . . . . . . . . . . . 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 2665 . . . . . . . . . . . . . . 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 2701 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
358332, 347ltned 10125 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ≠ (𝑄‘(𝑖 + 1)))
359358neneqd 2795 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 = (𝑄‘(𝑖 + 1)))
360 velsn 4169 . . . . . . . . . . . . . 14 (𝑟 ∈ {(𝑄‘(𝑖 + 1))} ↔ 𝑟 = (𝑄‘(𝑖 + 1)))
361359, 360sylnibr 319 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 ∈ {(𝑄‘(𝑖 + 1))})
362357, 361eldifd 3570 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
363362ralrimiva 2961 . . . . . . . . . . 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 3577 . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠))
367202adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
368 simpr 477 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
369367, 368addcomd 10190 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑠 ∈ ℂ) → (𝑋 + 𝑠) = (𝑠 + 𝑋))
370369mpteq2dva 4709 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)))
371 eqid 2621 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋))
372371addccncf 22642 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
373202, 372syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
374370, 373eqeltrd 2698 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
375374adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
376230rexrd 10041 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℝ*)
377 iocssre 12203 . . . . . . . . . . . . . . . . . . 19 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℝ)
378376, 231, 377syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℝ)
379 ax-resscn 9945 . . . . . . . . . . . . . . . . . 18 ℝ ⊆ ℂ
380378, 379syl6ss 3599 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ⊆ ℂ)
381295a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
382202ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℂ)
383380sselda 3587 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
384382, 383addcld 10011 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℂ)
385366, 375, 380, 381, 384cncfmptssg 39414 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))–cn→ℂ))
386 eqid 2621 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
387 eqid 2621 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
388386cnfldtop 22510 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘ℂfld) ∈ Top
389 unicntop 22512 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (TopOpen‘ℂfld)
390389restid 16026 . . . . . . . . . . . . . . . . . . . 20 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
391388, 390ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
392391eqcomi 2630 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
393386, 387, 392cncfcn 22635 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
395385, 394eleqtrd 2700 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
396386cnfldtopon 22509 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
397396a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
398 resttopon 20888 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))))
400 cncnp 21007 . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 12177 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → (𝐻‘(𝑖 + 1)) ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
405376, 251, 180, 404syl3anc 1323 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
406 fveq2 6153 . . . . . . . . . . . . . . 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 2684 . . . . . . . . . . . . . 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 3297 . . . . . . . . . . . . 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 692 . . . . . . . . . . . 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 39173 . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) = ((𝐻𝑖)(,](𝐻‘(𝑖 + 1))))
412265eqcomd 2627 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = (𝑋 + (𝐻‘(𝑖 + 1))))
413412ad2antrr 761 . . . . . . . . . . . . . . 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 4069 . . . . . . . . . . . . . . . 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 6618 . . . . . . . . . . . . . . . 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 2665 . . . . . . . . . . . . . 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 4072 . . . . . . . . . . . . . . . 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 2622 . . . . . . . . . . . . . . . 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 6618 . . . . . . . . . . . . . . . . 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 4169 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ {(𝐻‘(𝑖 + 1))} ↔ 𝑠 = (𝐻‘(𝑖 + 1)))
425424notbii 310 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ {(𝐻‘(𝑖 + 1))} ↔ ¬ 𝑠 = (𝐻‘(𝑖 + 1)))
426 elun 3736 . . . . . . . . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) → (𝑠 ∈ {(𝐻‘(𝑖 + 1))} ∨ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
429428ord 392 . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
432431adantll 749 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) ∧ ¬ 𝑠 = (𝐻‘(𝑖 + 1))) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
43395ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → 𝑋 ∈ ℝ)
434 elioore 12155 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
435434adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
436 elsni 4170 . . . . . . . . . . . . . . . . . . . . . 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 2698 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻‘(𝑖 + 1))}) → 𝑠 ∈ ℝ)
440435, 439jaodan 825 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻‘(𝑖 + 1))})) → 𝑠 ∈ ℝ)
441426, 440sylan2b 492 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) → 𝑠 ∈ ℝ)
442433, 441readdcld 10021 . . . . . . . . . . . . . . . . 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 6250 . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . 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 831 . . . . . . . . . . . . 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 4697 . . . . . . . . . . . 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 6626 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)(,](𝐻‘(𝑖 + 1)))))
449448oveq1d 6625 . . . . . . . . . . . . 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 6155 . . . . . . . . . . . 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 2713 . . . . . . . . . . 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 2621 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))})) = ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻‘(𝑖 + 1))}))
453 eqid 2621 . . . . . . . . . . . 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 6019 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡)):((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))⟶ℂ)
455231recnd 10020 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻‘(𝑖 + 1)) ∈ ℂ)
456452, 386, 453, 454, 305, 455ellimc 23560 . . . . . . . . . . 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 39284 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻‘(𝑖 + 1))))
459272, 292eqtrd 2655 . . . . . . . . . 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 6625 . . . . . . . . 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 2700 . . . . . . . 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 10124 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ≠ (𝑄𝑖))
464463neneqd 2795 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 = (𝑄𝑖))
465 velsn 4169 . . . . . . . . . . . . . 14 (𝑟 ∈ {(𝑄𝑖)} ↔ 𝑟 = (𝑄𝑖))
466464, 465sylnibr 319 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → ¬ 𝑟 ∈ {(𝑄𝑖)})
467357, 466eldifd 3570 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) → 𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
468467ralrimiva 2961 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → ∀𝑟 ∈ ran (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))𝑟 ∈ (dom (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄𝑖)}))
469 dfss3 3577 . . . . . . . . . . 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 12204 . . . . . . . . . . . . . . . . . . 19 (((𝐻𝑖) ∈ ℝ ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ*) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℝ)
472230, 251, 471syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℝ)
473472, 379syl6ss 3599 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ⊆ ℂ)
474202ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → 𝑋 ∈ ℂ)
475473sselda 3587 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → 𝑠 ∈ ℂ)
476474, 475addcld 10011 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℂ)
477366, 375, 473, 381, 476cncfmptssg 39414 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))–cn→ℂ))
478 eqid 2621 . . . . . . . . . . . . . . . . . 18 ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
479386, 478, 392cncfcn 22635 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
481477, 480eleqtrd 2700 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ (((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) Cn (TopOpen‘ℂfld)))
482 resttopon 20888 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) ∈ (TopOn‘((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))))
484 cncnp 21007 . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 12178 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → (𝐻𝑖) ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
489376, 251, 180, 488syl3anc 1323 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
490 fveq2 6153 . . . . . . . . . . . . . . 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 2684 . . . . . . . . . . . . . 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 3297 . . . . . . . . . . . . 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 692 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)) ∈ ((((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))) CnP (TopOpen‘ℂfld))‘(𝐻𝑖)))
494 uncom 3740 . . . . . . . . . . . . . 14 (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) = ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
495 snunioo 12248 . . . . . . . . . . . . . . 15 (((𝐻𝑖) ∈ ℝ* ∧ (𝐻‘(𝑖 + 1)) ∈ ℝ* ∧ (𝐻𝑖) < (𝐻‘(𝑖 + 1))) → ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
496376, 251, 180, 495syl3anc 1323 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ({(𝐻𝑖)} ∪ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
497494, 496syl5eq 2667 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) = ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1))))
498 iftrue 4069 . . . . . . . . . . . . . . . . 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 6618 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝐻𝑖) → (𝑋 + 𝑠) = (𝑋 + (𝐻𝑖)))
502501eqcomd 2627 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐻𝑖) → (𝑋 + (𝐻𝑖)) = (𝑋 + 𝑠))
503502adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → (𝑋 + (𝐻𝑖)) = (𝑋 + 𝑠))
504499, 500, 5033eqtrd 2659 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = (𝐻𝑖)) → if(𝑠 = (𝐻𝑖), (𝑄𝑖), ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠)) = (𝑋 + 𝑠))
505504adantlr 750 . . . . . . . . . . . . . 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 4072 . . . . . . . . . . . . . . . 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 2622 . . . . . . . . . . . . . . . 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 4169 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ {(𝐻𝑖)} ↔ 𝑠 = (𝐻𝑖))
511510notbii 310 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ {(𝐻𝑖)} ↔ ¬ 𝑠 = (𝐻𝑖))
512 elun 3736 . . . . . . . . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) → (𝑠 ∈ {(𝐻𝑖)} ∨ 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1)))))
515514ord 392 . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}) ∧ ¬ 𝑠 = (𝐻𝑖)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
518517adantll 749 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → 𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))))
51995ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → 𝑋 ∈ ℝ)
520 elsni 4170 . . . . . . . . . . . . . . . . . . . . . 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 2698 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝐻𝑖)}) → 𝑠 ∈ ℝ)
524435, 523jaodan 825 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ (𝑠 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝐻𝑖)})) → 𝑠 ∈ ℝ)
525512, 524sylan2b 492 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) → 𝑠 ∈ ℝ)
526519, 525readdcld 10021 . . . . . . . . . . . . . . . . 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 6250 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) ∧ ¬ 𝑠 = (𝐻𝑖)) → ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))‘𝑠) = (𝑋 + 𝑠))
529507, 528eqtrd 2655 . . . . . . . . . . . . . 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 831 . . . . . . . . . . . . 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 4697 . . . . . . . . . . . 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 6626 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) = ((TopOpen‘ℂfld) ↾t ((𝐻𝑖)[,)(𝐻‘(𝑖 + 1)))))
533532oveq1d 6625 . . . . . . . . . . . . 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 6155 . . . . . . . . . . . 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 2713 . . . . . . . . . . 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 2621 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)})) = ((TopOpen‘ℂfld) ↾t (((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ∪ {(𝐻𝑖)}))
537 eqid 2621 . . . . . . . . . . . 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 10020 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐻𝑖) ∈ ℂ)
539536, 386, 537, 454, 305, 538ellimc 23560 . . . . . . . . . . 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 39284 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝑋 + 𝑡))) lim (𝐻𝑖)))
542459oveq1d 6625 . . . . . . . . 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 2700 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) lim (𝐻𝑖)))
544230, 231, 310, 461, 543iblcncfioo 39527 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)(,)(𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ 𝐿1)
54530ad2antrr 761 . . . . . . . 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 761 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
549 simplr 791 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
550 simpr 477 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))))
551169, 179oveq12d 6628 . . . . . . . . . . . 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 2700 . . . . . . . . . 10 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → 𝑡 ∈ (((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋)))
554553, 116syldan 487 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))))
555546, 547, 548, 549, 554fourierdlem1 39658 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝑋 + 𝑡) ∈ (-π[,]π))
556545, 555ffvelrnd 6321 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
557230, 231, 544, 556ibliooicc 39520 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝐻𝑖)[,](𝐻‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑡))) ∈ 𝐿1)
55814, 20, 165, 180, 229, 557itgspltprt 39528 . . . . 5 (𝜑 → ∫((𝐻‘0)[,](𝐻𝑀))(𝐹‘(𝑋 + 𝑡)) d𝑡 = Σ𝑖 ∈ (0..^𝑀)∫((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))(𝐹‘(𝑋 + 𝑡)) d𝑡)
559551itgeq1d 39505 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ∫((𝐻𝑖)[,](𝐻‘(𝑖 + 1)))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫(((𝑄𝑖) − 𝑋)[,]((𝑄‘(𝑖 + 1)) − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
560559sumeq2dv 14375 . . . . 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 2655 . . . 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 2659 . . 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 2658 . 2 (𝜑 → Σ𝑖 ∈ (0..^𝑀)∫((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))(𝐹𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠)
56413, 76, 5633eqtrd 2659 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  ifcif 4063  {csn 4153   class class class wbr 4618  cmpt 4678  dom cdm 5079  ran crn 5080  cres 5081  ccom 5083  Fun wfun 5846  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  cc 9886  cr 9887  0cc0 9888  1c1 9889   + caddc 9891  *cxr 10025   < clt 10026  cle 10027  cmin 10218  -cneg 10219  cn 10972  cz 11329  cuz 11639  (,)cioo 12125  (,]cioc 12126  [,)cico 12127  [,]cicc 12128  ...cfz 12276  ..^cfzo 12414  Σcsu 14358  πcpi 14733  t crest 16013  TopOpenctopn 16014  fldccnfld 19678  Topctop 20630  TopOnctopon 20647   Cn ccn 20951   CnP ccnp 20952  cnccncf 22602  citg 23310   lim climc 23549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cc 9209  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966  ax-addf 9967  ax-mulf 9968
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-ofr 6858  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-omul 7517  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-fi 8269  df-sup 8300  df-inf 8301  df-oi 8367  df-card 8717  df-acn 8720  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-q 11741  df-rp 11785  df-xneg 11898  df-xadd 11899  df-xmul 11900  df-ioo 12129  df-ioc 12130  df-ico 12131  df-icc 12132  df-fz 12277  df-fzo 12415  df-fl 12541  df-mod 12617  df-seq 12750  df-exp 12809  df-fac 13009  df-bc 13038  df-hash 13066  df-shft 13749  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-limsup 14144  df-clim 14161  df-rlim 14162  df-sum 14359  df-ef 14734  df-sin 14736  df-cos 14737  df-pi 14739  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-starv 15888  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-unif 15897  df-hom 15898  df-cco 15899  df-rest 16015  df-topn 16016  df-0g 16034  df-gsum 16035  df-topgen 16036  df-pt 16037  df-prds 16040  df-xrs 16094  df-qtop 16099  df-imas 16100  df-xps 16102  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-submnd 17268  df-mulg 17473  df-cntz 17682  df-cmn 18127  df-psmet 19670  df-xmet 19671  df-met 19672  df-bl 19673  df-mopn 19674  df-fbas 19675  df-fg 19676  df-cnfld 19679  df-top 20631  df-topon 20648  df-topsp 20661  df-bases 20674  df-cld 20746  df-ntr 20747  df-cls 20748  df-nei 20825  df-lp 20863  df-perf 20864  df-cn 20954  df-cnp 20955  df-haus 21042  df-cmp 21113  df-tx 21288  df-hmeo 21481  df-fil 21573  df-fm 21665  df-flim 21666  df-flf 21667  df-xms 22048  df-ms 22049  df-tms 22050  df-cncf 22604  df-ovol 23156  df-vol 23157  df-mbf 23311  df-itg1 23312  df-itg2 23313  df-ibl 23314  df-itg 23315  df-0p 23360  df-ditg 23534  df-limc 23553  df-dv 23554
This theorem is referenced by:  fourierdlem101  39757
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