Theorem fourierdlem81

Description: The integral of a piecewise continuous periodic function F is unchanged if the domain is shifted by its period T . In this lemma, T is assumed to be strictly positive. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	fourierdlem81.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	fourierdlem81.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	fourierdlem81.p	⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } )
	fourierdlem81.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
	fourierdlem81.t	⊢ ( 𝜑 → 𝑇 ∈ ℝ+ )
	fourierdlem81.q	⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) )
	fourierdlem81.fper	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
	fourierdlem81.s	⊢ 𝑆 = ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) )
	fourierdlem81.f	⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ )
	fourierdlem81.cncf	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
	fourierdlem81.r	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
	fourierdlem81.l	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
	fourierdlem81.g	⊢ 𝐺 = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) )
	fourierdlem81.h	⊢ 𝐻 = ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
Assertion	fourierdlem81	⊢ ( 𝜑 → ∫ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = ∫ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		fourierdlem81.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		fourierdlem81.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		fourierdlem81.p	⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } )
4		fourierdlem81.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
5		fourierdlem81.t	⊢ ( 𝜑 → 𝑇 ∈ ℝ+ )
6		fourierdlem81.q	⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) )
7		fourierdlem81.fper	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
8		fourierdlem81.s	⊢ 𝑆 = ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) )
9		fourierdlem81.f	⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ )
10		fourierdlem81.cncf	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
11		fourierdlem81.r	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) )
12		fourierdlem81.l	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
13		fourierdlem81.g	⊢ 𝐺 = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) )
14		fourierdlem81.h	⊢ 𝐻 = ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
15	3	fourierdlem2	⊢ ( 𝑀 ∈ ℕ → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) )
16	4 15	syl	⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) )
17	6 16	mpbid	⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
18	17	simprd	⊢ ( 𝜑 → ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
19	18	simpld	⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) )
20	19	simpld	⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = 𝐴 )
21	20	eqcomd	⊢ ( 𝜑 → 𝐴 = ( 𝑄 ‘ 0 ) )
22	19	simprd	⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = 𝐵 )
23	22	eqcomd	⊢ ( 𝜑 → 𝐵 = ( 𝑄 ‘ 𝑀 ) )
24	21 23	oveq12d	⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) = ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) )
25	24	itgeq1d	⊢ ( 𝜑 → ∫ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
26		0zd	⊢ ( 𝜑 → 0 ∈ ℤ )
27		nnuz	⊢ ℕ = ( ℤ≥ ‘ 1 )
28		0p1e1	⊢ ( 0 + 1 ) = 1
29	28	fveq2i	⊢ ( ℤ≥ ‘ ( 0 + 1 ) ) = ( ℤ≥ ‘ 1 )
30	27 29	eqtr4i	⊢ ℕ = ( ℤ≥ ‘ ( 0 + 1 ) )
31	4 30	eleqtrdi	⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ ( 0 + 1 ) ) )
32	17	simpld	⊢ ( 𝜑 → 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) )
33		reex	⊢ ℝ ∈ V
34	33	a1i	⊢ ( 𝜑 → ℝ ∈ V )
35		ovex	⊢ ( 0 ... 𝑀 ) ∈ V
36	35	a1i	⊢ ( 𝜑 → ( 0 ... 𝑀 ) ∈ V )
37	34 36	elmapd	⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ↔ 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) )
38	32 37	mpbid	⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ )
39	18	simprd	⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
40	39	r19.21bi	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
41	9	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝐹 : ℝ ⟶ ℂ )
42	20 1	eqeltrd	⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℝ )
43	22 2	eqeltrd	⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) ∈ ℝ )
44	42 43	iccssred	⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ⊆ ℝ )
45	44	sselda	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝑥 ∈ ℝ )
46	41 45	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ )
47	38	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ )
48		elfzofz	⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) )
49	48	adantl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) )
50	47 49	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
51		fzofzp1	⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) )
52	51	adantl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) )
53	47 52	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
54	9	feqmptd	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ℝ ↦ ( 𝐹 ‘ 𝑥 ) ) )
55	54	reseq1d	⊢ ( 𝜑 → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑥 ∈ ℝ ↦ ( 𝐹 ‘ 𝑥 ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
56	55	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑥 ∈ ℝ ↦ ( 𝐹 ‘ 𝑥 ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
57		ioossre	⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ
58	57	a1i	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ )
59	58	resmptd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑥 ∈ ℝ ↦ ( 𝐹 ‘ 𝑥 ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) )
60	56 59	eqtr2d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
61	50 53 10 12 11	iblcncfioo	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ 𝐿1 )
62	60 61	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 )
63	9	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐹 : ℝ ⟶ ℂ )
64	50 53	iccssred	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ )
65	64	sselda	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ )
66	63 65	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ )
67	50 53 62 66	ibliooicc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 )
68	26 31 38 40 46 67	itgspltprt	⊢ ( 𝜑 → ∫ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
69	8	a1i	⊢ ( 𝜑 → 𝑆 = ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ) )
70		fveq2	⊢ ( 𝑖 = 0 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 0 ) )
71	70	oveq1d	⊢ ( 𝑖 = 0 → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) = ( ( 𝑄 ‘ 0 ) + 𝑇 ) )
72	71	adantl	⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) = ( ( 𝑄 ‘ 0 ) + 𝑇 ) )
73	4	nnnn0d	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
74		nn0uz	⊢ ℕ0 = ( ℤ≥ ‘ 0 )
75	73 74	eleqtrdi	⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 0 ) )
76		eluzfz1	⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 0 ) → 0 ∈ ( 0 ... 𝑀 ) )
77	75 76	syl	⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) )
78	5	rpred	⊢ ( 𝜑 → 𝑇 ∈ ℝ )
79	42 78	readdcld	⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) + 𝑇 ) ∈ ℝ )
80	69 72 77 79	fvmptd	⊢ ( 𝜑 → ( 𝑆 ‘ 0 ) = ( ( 𝑄 ‘ 0 ) + 𝑇 ) )
81	20	oveq1d	⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) + 𝑇 ) = ( 𝐴 + 𝑇 ) )
82	80 81	eqtr2d	⊢ ( 𝜑 → ( 𝐴 + 𝑇 ) = ( 𝑆 ‘ 0 ) )
83		fveq2	⊢ ( 𝑖 = 𝑀 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑀 ) )
84	83	oveq1d	⊢ ( 𝑖 = 𝑀 → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) = ( ( 𝑄 ‘ 𝑀 ) + 𝑇 ) )
85	84	adantl	⊢ ( ( 𝜑 ∧ 𝑖 = 𝑀 ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) = ( ( 𝑄 ‘ 𝑀 ) + 𝑇 ) )
86		eluzfz2	⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 0 ) → 𝑀 ∈ ( 0 ... 𝑀 ) )
87	75 86	syl	⊢ ( 𝜑 → 𝑀 ∈ ( 0 ... 𝑀 ) )
88	43 78	readdcld	⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑀 ) + 𝑇 ) ∈ ℝ )
89	69 85 87 88	fvmptd	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑀 ) = ( ( 𝑄 ‘ 𝑀 ) + 𝑇 ) )
90	22	oveq1d	⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑀 ) + 𝑇 ) = ( 𝐵 + 𝑇 ) )
91	89 90	eqtr2d	⊢ ( 𝜑 → ( 𝐵 + 𝑇 ) = ( 𝑆 ‘ 𝑀 ) )
92	82 91	oveq12d	⊢ ( 𝜑 → ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) = ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) )
93	92	itgeq1d	⊢ ( 𝜑 → ∫ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
94	38	ffvelcdmda	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
95	78	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → 𝑇 ∈ ℝ )
96	94 95	readdcld	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ∈ ℝ )
97	96 8	fmptd	⊢ ( 𝜑 → 𝑆 : ( 0 ... 𝑀 ) ⟶ ℝ )
98	78	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑇 ∈ ℝ )
99	50 53 98 40	ltadd1dd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
100	48 96	sylan2	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ∈ ℝ )
101	8	fvmpt2	⊢ ( ( 𝑖 ∈ ( 0 ... 𝑀 ) ∧ ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ∈ ℝ ) → ( 𝑆 ‘ 𝑖 ) = ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) )
102	49 100 101	syl2anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑆 ‘ 𝑖 ) = ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) )
103		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑗 ) )
104	103	oveq1d	⊢ ( 𝑖 = 𝑗 → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) = ( ( 𝑄 ‘ 𝑗 ) + 𝑇 ) )
105	104	cbvmptv	⊢ ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ) = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑗 ) + 𝑇 ) )
106	8 105	eqtri	⊢ 𝑆 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑗 ) + 𝑇 ) )
107	106	a1i	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑆 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑗 ) + 𝑇 ) ) )
108		fveq2	⊢ ( 𝑗 = ( 𝑖 + 1 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
109	108	oveq1d	⊢ ( 𝑗 = ( 𝑖 + 1 ) → ( ( 𝑄 ‘ 𝑗 ) + 𝑇 ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
110	109	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 = ( 𝑖 + 1 ) ) → ( ( 𝑄 ‘ 𝑗 ) + 𝑇 ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
111	53 98	readdcld	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ∈ ℝ )
112	107 110 52 111	fvmptd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
113	99 102 112	3brtr4d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑆 ‘ 𝑖 ) < ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
114	9	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ) → 𝐹 : ℝ ⟶ ℂ )
115	80 79	eqeltrd	⊢ ( 𝜑 → ( 𝑆 ‘ 0 ) ∈ ℝ )
116	115	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ) → ( 𝑆 ‘ 0 ) ∈ ℝ )
117	89 88	eqeltrd	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑀 ) ∈ ℝ )
118	117	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ) → ( 𝑆 ‘ 𝑀 ) ∈ ℝ )
119	116 118	iccssred	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ) → ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ⊆ ℝ )
120		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ) → 𝑥 ∈ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) )
121	119 120	sseldd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ) → 𝑥 ∈ ℝ )
122	114 121	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ )
123	102 100	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑆 ‘ 𝑖 ) ∈ ℝ )
124	112 111	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
125		ioosscn	⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ
126	125	a1i	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ )
127		eqeq1	⊢ ( 𝑤 = 𝑥 → ( 𝑤 = ( 𝑧 + 𝑇 ) ↔ 𝑥 = ( 𝑧 + 𝑇 ) ) )
128	127	rexbidv	⊢ ( 𝑤 = 𝑥 → ( ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) ↔ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) )
129		oveq1	⊢ ( 𝑧 = 𝑦 → ( 𝑧 + 𝑇 ) = ( 𝑦 + 𝑇 ) )
130	129	eqeq2d	⊢ ( 𝑧 = 𝑦 → ( 𝑥 = ( 𝑧 + 𝑇 ) ↔ 𝑥 = ( 𝑦 + 𝑇 ) ) )
131	130	cbvrexvw	⊢ ( ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ↔ ∃ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑦 + 𝑇 ) )
132	128 131	bitrdi	⊢ ( 𝑤 = 𝑥 → ( ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) ↔ ∃ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑦 + 𝑇 ) ) )
133	132	cbvrabv	⊢ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } = { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑦 + 𝑇 ) }
134		fdm	⊢ ( 𝐹 : ℝ ⟶ ℂ → dom 𝐹 = ℝ )
135	9 134	syl	⊢ ( 𝜑 → dom 𝐹 = ℝ )
136	135	feq2d	⊢ ( 𝜑 → ( 𝐹 : dom 𝐹 ⟶ ℂ ↔ 𝐹 : ℝ ⟶ ℂ ) )
137	9 136	mpbird	⊢ ( 𝜑 → 𝐹 : dom 𝐹 ⟶ ℂ )
138	137	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐹 : dom 𝐹 ⟶ ℂ )
139		elioore	⊢ ( 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑧 ∈ ℝ )
140	139	adantl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑧 ∈ ℝ )
141	78	adantr	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑇 ∈ ℝ )
142	140 141	readdcld	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑧 + 𝑇 ) ∈ ℝ )
143	142	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑧 + 𝑇 ) ∈ ℝ )
144	143	3adant3	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑧 + 𝑇 ) ) → ( 𝑧 + 𝑇 ) ∈ ℝ )
145		simp3	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑧 + 𝑇 ) ) → 𝑤 = ( 𝑧 + 𝑇 ) )
146	135	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑧 + 𝑇 ) ) → dom 𝐹 = ℝ )
147	146	3adant1r	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑧 + 𝑇 ) ) → dom 𝐹 = ℝ )
148	144 145 147	3eltr4d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑧 + 𝑇 ) ) → 𝑤 ∈ dom 𝐹 )
149	148	3exp	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑤 = ( 𝑧 + 𝑇 ) → 𝑤 ∈ dom 𝐹 ) ) )
150	149	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑤 ∈ ℂ ) → ( 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑤 = ( 𝑧 + 𝑇 ) → 𝑤 ∈ dom 𝐹 ) ) )
151	150	rexlimdv	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑤 ∈ ℂ ) → ( ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) → 𝑤 ∈ dom 𝐹 ) )
152	151	ralrimiva	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑤 ∈ ℂ ( ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) → 𝑤 ∈ dom 𝐹 ) )
153		rabss	⊢ ( { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ⊆ dom 𝐹 ↔ ∀ 𝑤 ∈ ℂ ( ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) → 𝑤 ∈ dom 𝐹 ) )
154	152 153	sylibr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ⊆ dom 𝐹 )
155		simpll	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝜑 )
156	1	rexrd	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
157	156	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐴 ∈ ℝ* )
158	2	rexrd	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
159	158	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐵 ∈ ℝ* )
160	3 4 6	fourierdlem15	⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) )
161	160	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) )
162		simplr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) )
163		ioossicc	⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
164	163	sseli	⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
165	164	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
166	157 159 161 162 165	fourierdlem1	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
167	155 166 7	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
168	126 98 133 138 154 167 10	cncfperiod	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ) ∈ ( { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } –cn→ ℂ ) )
169	128	elrab	⊢ ( 𝑥 ∈ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ↔ ( 𝑥 ∈ ℂ ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) )
170	169	simprbi	⊢ ( 𝑥 ∈ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } → ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) )
171		simpr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) → ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) )
172		nfv	⊢ Ⅎ 𝑧 ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) )
173		nfre1	⊢ Ⅎ 𝑧 ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 )
174	172 173	nfan	⊢ Ⅎ 𝑧 ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) )
175		nfv	⊢ Ⅎ 𝑧 ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
176		simp3	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → 𝑥 = ( 𝑧 + 𝑇 ) )
177	142	3adant3	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑧 + 𝑇 ) ∈ ℝ )
178	176 177	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → 𝑥 ∈ ℝ )
179	178	3adant1r	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → 𝑥 ∈ ℝ )
180	50	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
181	139	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑧 ∈ ℝ )
182	78	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑇 ∈ ℝ )
183		simpr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
184	50	rexrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* )
185	184	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* )
186	53	rexrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
187	186	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
188		elioo2	⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) → ( 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝑄 ‘ 𝑖 ) < 𝑧 ∧ 𝑧 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
189	185 187 188	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝑄 ‘ 𝑖 ) < 𝑧 ∧ 𝑧 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) )
190	183 189	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑧 ∈ ℝ ∧ ( 𝑄 ‘ 𝑖 ) < 𝑧 ∧ 𝑧 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
191	190	simp2d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑧 )
192	180 181 182 191	ltadd1dd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) < ( 𝑧 + 𝑇 ) )
193	192	3adant3	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) < ( 𝑧 + 𝑇 ) )
194	102	3ad2ant1	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑆 ‘ 𝑖 ) = ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) )
195		simp3	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → 𝑥 = ( 𝑧 + 𝑇 ) )
196	193 194 195	3brtr4d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑆 ‘ 𝑖 ) < 𝑥 )
197	53	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
198	190	simp3d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑧 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
199	181 197 182 198	ltadd1dd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑧 + 𝑇 ) < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
200	199	3adant3	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑧 + 𝑇 ) < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
201	112	3ad2ant1	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
202	200 195 201	3brtr4d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
203	179 196 202	3jca	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
204	203	3exp	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 = ( 𝑧 + 𝑇 ) → ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ) )
205	204	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 = ( 𝑧 + 𝑇 ) → ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ) )
206	174 175 205	rexlimd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) → ( ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) → ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) )
207	171 206	mpd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
208	123	rexrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑆 ‘ 𝑖 ) ∈ ℝ* )
209	208	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑆 ‘ 𝑖 ) ∈ ℝ* )
210	124	rexrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
211	210	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
212		elioo2	⊢ ( ( ( 𝑆 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) → ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) )
213	209 211 212	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) )
214	207 213	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) ) → 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
215	170 214	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ) → 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
216		elioore	⊢ ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ℝ )
217	216	recnd	⊢ ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ℂ )
218	217	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℂ )
219	184	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* )
220	186	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
221	216	adantl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ )
222	78	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑇 ∈ ℝ )
223	221 222	resubcld	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ∈ ℝ )
224	223	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ∈ ℝ )
225	102	oveq1d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) = ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) − 𝑇 ) )
226	50	recnd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℂ )
227	98	recnd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑇 ∈ ℂ )
228	226 227	pncand	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) )
229	225 228	eqtr2d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) = ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) )
230	229	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) = ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) )
231	123	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ 𝑖 ) ∈ ℝ )
232	216	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ )
233	78	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑇 ∈ ℝ )
234		simpr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
235	208	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ 𝑖 ) ∈ ℝ* )
236	210	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
237	235 236 212	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) )
238	234 237	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 ∈ ℝ ∧ ( 𝑆 ‘ 𝑖 ) < 𝑥 ∧ 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
239	238	simp2d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ 𝑖 ) < 𝑥 )
240	231 232 233 239	ltsub1dd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) < ( 𝑥 − 𝑇 ) )
241	230 240	eqbrtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑥 − 𝑇 ) )
242	124	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
243	238	simp3d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
244	232 242 233 243	ltsub1dd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) < ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) )
245	112	oveq1d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) − 𝑇 ) )
246	53	recnd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℂ )
247	246 227	pncand	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
248	245 247	eqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
249	248	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
250	244 249	breqtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
251	219 220 224 241 250	eliood	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
252	221	recnd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℂ )
253	222	recnd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑇 ∈ ℂ )
254	252 253	npcand	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑥 − 𝑇 ) + 𝑇 ) = 𝑥 )
255	254	eqcomd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 = ( ( 𝑥 − 𝑇 ) + 𝑇 ) )
256	255	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 = ( ( 𝑥 − 𝑇 ) + 𝑇 ) )
257		oveq1	⊢ ( 𝑧 = ( 𝑥 − 𝑇 ) → ( 𝑧 + 𝑇 ) = ( ( 𝑥 − 𝑇 ) + 𝑇 ) )
258	257	eqeq2d	⊢ ( 𝑧 = ( 𝑥 − 𝑇 ) → ( 𝑥 = ( 𝑧 + 𝑇 ) ↔ 𝑥 = ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) )
259	258	rspcev	⊢ ( ( ( 𝑥 − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑥 = ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) → ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) )
260	251 256 259	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑥 = ( 𝑧 + 𝑇 ) )
261	218 260 169	sylanbrc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } )
262	215 261	impbida	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ↔ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) )
263	262	eqrdv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } = ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
264	263	reseq2d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ) = ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) )
265	9	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐹 : ℝ ⟶ ℂ )
266		ioossre	⊢ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ
267	266	a1i	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ )
268	265 267	feqresmpt	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) )
269	264 268	eqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ) = ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) )
270	263	oveq1d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } –cn→ ℂ ) = ( ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
271	168 269 270	3eltr3d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
272	57 135	sseqtrrid	⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐹 )
273	272	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐹 )
274		eqid	⊢ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } = { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) }
275		simpll	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝜑 )
276	156	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐴 ∈ ℝ* )
277	158	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐵 ∈ ℝ* )
278	160	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) )
279		simplr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) )
280	163 183	sselid	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
281	276 277 278 279 280	fourierdlem1	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
282		eleq1	⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) )
283	282	anbi2d	⊢ ( 𝑥 = 𝑧 → ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ) )
284		oveq1	⊢ ( 𝑥 = 𝑧 → ( 𝑥 + 𝑇 ) = ( 𝑧 + 𝑇 ) )
285	284	fveq2d	⊢ ( 𝑥 = 𝑧 → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ ( 𝑧 + 𝑇 ) ) )
286		fveq2	⊢ ( 𝑥 = 𝑧 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑧 ) )
287	285 286	eqeq12d	⊢ ( 𝑥 = 𝑧 → ( ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝑧 + 𝑇 ) ) = ( 𝐹 ‘ 𝑧 ) ) )
288	283 287	imbi12d	⊢ ( 𝑥 = 𝑧 → ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( 𝑧 + 𝑇 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) )
289	288 7	chvarvv	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( 𝑧 + 𝑇 ) ) = ( 𝐹 ‘ 𝑧 ) )
290	275 281 289	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑧 + 𝑇 ) ) = ( 𝐹 ‘ 𝑧 ) )
291	138 126 273 227 274 154 290 12	limcperiod	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ) limℂ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) )
292	112	eqcomd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) = ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
293	269 292	oveq12d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ) limℂ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) = ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
294	291 293	eleqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
295	138 126 273 227 274 154 290 11	limcperiod	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ) limℂ ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ) )
296	102	eqcomd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) = ( 𝑆 ‘ 𝑖 ) )
297	269 296	oveq12d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑧 + 𝑇 ) } ) limℂ ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ) = ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑆 ‘ 𝑖 ) ) )
298	295 297	eleqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑆 ‘ 𝑖 ) ) )
299	123 124 271 294 298	iblcncfioo	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 )
300	9	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐹 : ℝ ⟶ ℂ )
301	123	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ 𝑖 ) ∈ ℝ )
302	124	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
303		simpr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
304		eliccre	⊢ ( ( ( 𝑆 ‘ 𝑖 ) ∈ ℝ ∧ ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ )
305	301 302 303 304	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ )
306	300 305	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ )
307	123 124 299 306	ibliooicc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 )
308	26 31 97 113 122 307	itgspltprt	⊢ ( 𝜑 → ∫ ( ( 𝑆 ‘ 0 ) [,] ( 𝑆 ‘ 𝑀 ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
309		iftrue	⊢ ( 𝑥 = ( 𝑆 ‘ 𝑖 ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = 𝑅 )
310	309	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = 𝑅 )
311		iftrue	⊢ ( 𝑥 = ( 𝑄 ‘ 𝑖 ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = 𝑅 )
312		iftrue	⊢ ( 𝑥 = ( 𝑄 ‘ 𝑖 ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = 𝑅 )
313	311 312	eqtr4d	⊢ ( 𝑥 = ( 𝑄 ‘ 𝑖 ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
314	313	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
315		iffalse	⊢ ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) )
316	315	adantr	⊢ ( ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) )
317		iftrue	⊢ ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = 𝐿 )
318	317	adantl	⊢ ( ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = 𝐿 )
319		iffalse	⊢ ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) )
320	319	adantr	⊢ ( ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) )
321		iftrue	⊢ ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = 𝐿 )
322	321	adantl	⊢ ( ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = 𝐿 )
323	320 322	eqtr2d	⊢ ( ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝐿 = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
324	316 318 323	3eqtrd	⊢ ( ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
325	324	adantll	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
326	315	ad2antlr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) )
327		iffalse	⊢ ( ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) )
328	327	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) )
329	319	ad2antlr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) )
330		iffalse	⊢ ( ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) )
331	330	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) )
332	184	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* )
333	186	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
334	65	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ℝ )
335	50	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
336	65	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → 𝑥 ∈ ℝ )
337	184	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* )
338	186	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
339		simplr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
340		iccgelb	⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ 𝑥 )
341	337 338 339 340	syl3anc	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) ≤ 𝑥 )
342		neqne	⊢ ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) → 𝑥 ≠ ( 𝑄 ‘ 𝑖 ) )
343	342	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → 𝑥 ≠ ( 𝑄 ‘ 𝑖 ) )
344	335 336 341 343	leneltd	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 )
345	344	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 )
346	53	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
347	184	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* )
348	186	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
349		simpr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
350		iccleub	⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
351	347 348 349 350	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
352	351	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
353		neqne	⊢ ( ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → 𝑥 ≠ ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
354	353	necomd	⊢ ( ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≠ 𝑥 )
355	354	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≠ 𝑥 )
356	334 346 352 355	leneltd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
357	332 333 334 345 356	eliood	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
358		fvres	⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
359	357 358	syl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
360	329 331 359	3eqtrrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
361	326 328 360	3eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
362	325 361	pm2.61dan	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
363	314 362	pm2.61dan	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
364	363	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) )
365	13 364	eqtrid	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐺 = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) )
366		eqeq1	⊢ ( 𝑥 = 𝑤 → ( 𝑥 = ( 𝑄 ‘ 𝑖 ) ↔ 𝑤 = ( 𝑄 ‘ 𝑖 ) ) )
367		eqeq1	⊢ ( 𝑥 = 𝑤 → ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
368		fveq2	⊢ ( 𝑥 = 𝑤 → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) )
369	367 368	ifbieq2d	⊢ ( 𝑥 = 𝑤 → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) )
370	366 369	ifbieq2d	⊢ ( 𝑥 = 𝑤 → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) )
371	370	cbvmptv	⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) = ( 𝑤 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) )
372	365 371	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐺 = ( 𝑤 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) ) )
373	372	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → 𝐺 = ( 𝑤 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) ) )
374		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ 𝑤 = ( 𝑥 − 𝑇 ) ) → 𝑤 = ( 𝑥 − 𝑇 ) )
375		oveq1	⊢ ( 𝑥 = ( 𝑆 ‘ 𝑖 ) → ( 𝑥 − 𝑇 ) = ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) )
376	375	ad2antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ 𝑤 = ( 𝑥 − 𝑇 ) ) → ( 𝑥 − 𝑇 ) = ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) )
377	229	eqcomd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) )
378	377	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ 𝑤 = ( 𝑥 − 𝑇 ) ) → ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) )
379	374 376 378	3eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ 𝑤 = ( 𝑥 − 𝑇 ) ) → 𝑤 = ( 𝑄 ‘ 𝑖 ) )
380	379	iftrued	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ 𝑤 = ( 𝑥 − 𝑇 ) ) → if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) = 𝑅 )
381	375	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑥 − 𝑇 ) = ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) )
382	50 53 40	ltled	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
383		lbicc2	⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
384	184 186 382 383	syl3anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
385	377 384	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
386	385	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
387	381 386	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑥 − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
388		limccl	⊢ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ⊆ ℂ
389	388 11	sselid	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ℂ )
390	389	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → 𝑅 ∈ ℂ )
391	373 380 387 390	fvmptd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) = 𝑅 )
392	310 391	eqtr4d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
393	392	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
394		iffalse	⊢ ( ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) )
395	394	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) )
396	372	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝐺 = ( 𝑤 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) ) )
397		eqeq1	⊢ ( 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) → ( 𝑤 = ( 𝑄 ‘ 𝑖 ) ↔ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) ) )
398		eqeq1	⊢ ( 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) → ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
399		fveq2	⊢ ( 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) )
400	398 399	ifbieq2d	⊢ ( 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) → if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) = if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) ) )
401	397 400	ifbieq2d	⊢ ( 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) → if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) = if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) ) ) )
402	401	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) → if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) = if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) ) ) )
403		eqeq1	⊢ ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) ↔ ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ 𝑖 ) ) )
404		iftrue	⊢ ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) ) = 𝐿 )
405	403 404	ifbieq2d	⊢ ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) ) ) = if ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , 𝐿 ) )
406	248 405	syl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) ) ) = if ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , 𝐿 ) )
407	406	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) → if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) ) ) = if ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , 𝐿 ) )
408	50 40	gtned	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≠ ( 𝑄 ‘ 𝑖 ) )
409	408	neneqd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ¬ ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ 𝑖 ) )
410	409	iffalsed	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → if ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , 𝐿 ) = 𝐿 )
411	410	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) → if ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , 𝐿 ) = 𝐿 )
412	402 407 411	3eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) → if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) = 𝐿 )
413	412	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) → if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) = 𝐿 )
414		ubicc2	⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
415	184 186 382 414	syl3anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
416	248 415	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
417	416	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
418		limccl	⊢ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ
419	418 12	sselid	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ℂ )
420	419	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝐿 ∈ ℂ )
421	396 413 417 420	fvmptd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐺 ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) = 𝐿 )
422		oveq1	⊢ ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) → ( 𝑥 − 𝑇 ) = ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) )
423	422	fveq2d	⊢ ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) → ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) = ( 𝐺 ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) )
424	423	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) = ( 𝐺 ‘ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) ) )
425		iftrue	⊢ ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = 𝐿 )
426	425	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = 𝐿 )
427	421 424 426	3eqtr4rd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
428	427	ad4ant14	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
429		iffalse	⊢ ( ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) )
430	429	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) )
431	372	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐺 = ( 𝑤 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) ) )
432	431	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝐺 = ( 𝑤 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) ) )
433		eqeq1	⊢ ( 𝑤 = ( 𝑥 − 𝑇 ) → ( 𝑤 = ( 𝑄 ‘ 𝑖 ) ↔ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) ) )
434		eqeq1	⊢ ( 𝑤 = ( 𝑥 − 𝑇 ) → ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
435		fveq2	⊢ ( 𝑤 = ( 𝑥 − 𝑇 ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) )
436	434 435	ifbieq2d	⊢ ( 𝑤 = ( 𝑥 − 𝑇 ) → if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) = if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) )
437	433 436	ifbieq2d	⊢ ( 𝑤 = ( 𝑥 − 𝑇 ) → if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) = if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) ) )
438	437	adantl	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑥 − 𝑇 ) ) → if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) = if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) ) )
439	305	recnd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℂ )
440	227	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑇 ∈ ℂ )
441	439 440	npcand	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑥 − 𝑇 ) + 𝑇 ) = 𝑥 )
442	441	eqcomd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 = ( ( 𝑥 − 𝑇 ) + 𝑇 ) )
443	442	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) ) → 𝑥 = ( ( 𝑥 − 𝑇 ) + 𝑇 ) )
444		oveq1	⊢ ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) → ( ( 𝑥 − 𝑇 ) + 𝑇 ) = ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) )
445	444	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) ) → ( ( 𝑥 − 𝑇 ) + 𝑇 ) = ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) )
446	296	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) = ( 𝑆 ‘ 𝑖 ) )
447	443 445 446	3eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) ) → 𝑥 = ( 𝑆 ‘ 𝑖 ) )
448	447	stoic1a	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ¬ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) )
449	448	iffalsed	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) ) = if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) )
450	449	ad2antrr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑥 − 𝑇 ) ) → if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) ) = if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) )
451	442	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 = ( ( 𝑥 − 𝑇 ) + 𝑇 ) )
452		oveq1	⊢ ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( ( 𝑥 − 𝑇 ) + 𝑇 ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
453	452	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝑥 − 𝑇 ) + 𝑇 ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
454	292	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) = ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
455	451 453 454	3eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
456	455	stoic1a	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ¬ ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
457	456	iffalsed	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) )
458	457	adantlr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) )
459	458	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑥 − 𝑇 ) ) → if ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) )
460	438 450 459	3eqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑥 − 𝑇 ) ) → if ( 𝑤 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑤 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑤 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) )
461	50	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
462	53	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
463	78	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑇 ∈ ℝ )
464	305 463	resubcld	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ∈ ℝ )
465	229	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) = ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) )
466	208	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ 𝑖 ) ∈ ℝ* )
467	210	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
468		iccgelb	⊢ ( ( ( 𝑆 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ 𝑖 ) ≤ 𝑥 )
469	466 467 303 468	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑆 ‘ 𝑖 ) ≤ 𝑥 )
470	301 305 463 469	lesub1dd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑆 ‘ 𝑖 ) − 𝑇 ) ≤ ( 𝑥 − 𝑇 ) )
471	465 470	eqbrtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑥 − 𝑇 ) )
472		iccleub	⊢ ( ( ( 𝑆 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≤ ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
473	466 467 303 472	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≤ ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
474	305 302 463 473	lesub1dd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ≤ ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) )
475	248	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑆 ‘ ( 𝑖 + 1 ) ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
476	474 475	breqtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
477	461 462 464 471 476	eliccd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
478	477	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
479	138 273	fssresd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
480	479	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
481	184	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* )
482	186	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
483	305	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ℝ )
484	98	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝑇 ∈ ℝ )
485	483 484	resubcld	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 − 𝑇 ) ∈ ℝ )
486	50	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
487	464	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑥 − 𝑇 ) ∈ ℝ )
488	471	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑥 − 𝑇 ) )
489	448	neqned	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑥 − 𝑇 ) ≠ ( 𝑄 ‘ 𝑖 ) )
490	486 487 488 489	leneltd	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑥 − 𝑇 ) )
491	490	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑥 − 𝑇 ) )
492	464	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 − 𝑇 ) ∈ ℝ )
493	53	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
494	476	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 − 𝑇 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
495		eqcom	⊢ ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑥 − 𝑇 ) )
496	455	ex	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑥 − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
497	495 496	biimtrrid	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑥 − 𝑇 ) → 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
498	497	con3dimp	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ¬ ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑥 − 𝑇 ) )
499	498	neqned	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≠ ( 𝑥 − 𝑇 ) )
500	492 493 494 499	leneltd	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 − 𝑇 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
501	500	adantlr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 − 𝑇 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
502	481 482 485 491 501	eliood	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
503	480 502	ffvelcdmd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) ∈ ℂ )
504	432 460 478 503	fvmptd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) )
505		fvres	⊢ ( ( 𝑥 − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) = ( 𝐹 ‘ ( 𝑥 − 𝑇 ) ) )
506	502 505	syl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑥 − 𝑇 ) ) = ( 𝐹 ‘ ( 𝑥 − 𝑇 ) ) )
507	504 506	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) = ( 𝐹 ‘ ( 𝑥 − 𝑇 ) ) )
508	466	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑆 ‘ 𝑖 ) ∈ ℝ* )
509	467	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* )
510	123	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑆 ‘ 𝑖 ) ∈ ℝ )
511	305	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → 𝑥 ∈ ℝ )
512	469	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑆 ‘ 𝑖 ) ≤ 𝑥 )
513		neqne	⊢ ( ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) → 𝑥 ≠ ( 𝑆 ‘ 𝑖 ) )
514	513	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → 𝑥 ≠ ( 𝑆 ‘ 𝑖 ) )
515	510 511 512 514	leneltd	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → ( 𝑆 ‘ 𝑖 ) < 𝑥 )
516	515	adantr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑆 ‘ 𝑖 ) < 𝑥 )
517	302	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
518	473	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ≤ ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
519		neqne	⊢ ( ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) → 𝑥 ≠ ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
520	519	necomd	⊢ ( ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ≠ 𝑥 )
521	520	adantl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑆 ‘ ( 𝑖 + 1 ) ) ≠ 𝑥 )
522	483 517 518 521	leneltd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 < ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
523	508 509 483 516 522	eliood	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
524		fvres	⊢ ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
525	523 524	syl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
526	441	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) )
527	526	eqcomd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) )
528	527	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) )
529	439 440	subcld	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ∈ ℂ )
530		elex	⊢ ( ( 𝑥 − 𝑇 ) ∈ ℂ → ( 𝑥 − 𝑇 ) ∈ V )
531	529 530	syl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ∈ V )
532	531	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 − 𝑇 ) ∈ V )
533		simp-4l	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝜑 )
534	156	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐴 ∈ ℝ* )
535	158	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐵 ∈ ℝ* )
536	160	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) )
537		simpr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) )
538	534 535 536 537	fourierdlem8	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
539	538	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
540	539 477	sseldd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 − 𝑇 ) ∈ ( 𝐴 [,] 𝐵 ) )
541	540	ad2antrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑥 − 𝑇 ) ∈ ( 𝐴 [,] 𝐵 ) )
542	533 541	jca	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝜑 ∧ ( 𝑥 − 𝑇 ) ∈ ( 𝐴 [,] 𝐵 ) ) )
543		eleq1	⊢ ( 𝑦 = ( 𝑥 − 𝑇 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑥 − 𝑇 ) ∈ ( 𝐴 [,] 𝐵 ) ) )
544	543	anbi2d	⊢ ( 𝑦 = ( 𝑥 − 𝑇 ) → ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝜑 ∧ ( 𝑥 − 𝑇 ) ∈ ( 𝐴 [,] 𝐵 ) ) ) )
545		oveq1	⊢ ( 𝑦 = ( 𝑥 − 𝑇 ) → ( 𝑦 + 𝑇 ) = ( ( 𝑥 − 𝑇 ) + 𝑇 ) )
546	545	fveq2d	⊢ ( 𝑦 = ( 𝑥 − 𝑇 ) → ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) )
547		fveq2	⊢ ( 𝑦 = ( 𝑥 − 𝑇 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑥 − 𝑇 ) ) )
548	546 547	eqeq12d	⊢ ( 𝑦 = ( 𝑥 − 𝑇 ) → ( ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) = ( 𝐹 ‘ ( 𝑥 − 𝑇 ) ) ) )
549	544 548	imbi12d	⊢ ( 𝑦 = ( 𝑥 − 𝑇 ) → ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝜑 ∧ ( 𝑥 − 𝑇 ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) = ( 𝐹 ‘ ( 𝑥 − 𝑇 ) ) ) ) )
550		eleq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
551	550	anbi2d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) )
552		oveq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 + 𝑇 ) = ( 𝑦 + 𝑇 ) )
553	552	fveq2d	⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) )
554		fveq2	⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) )
555	553 554	eqeq12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ 𝑦 ) ) )
556	551 555	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ 𝑦 ) ) ) )
557	556 7	chvarvv	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( 𝑦 + 𝑇 ) ) = ( 𝐹 ‘ 𝑦 ) )
558	549 557	vtoclg	⊢ ( ( 𝑥 − 𝑇 ) ∈ V → ( ( 𝜑 ∧ ( 𝑥 − 𝑇 ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) = ( 𝐹 ‘ ( 𝑥 − 𝑇 ) ) ) )
559	532 542 558	sylc	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐹 ‘ ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) = ( 𝐹 ‘ ( 𝑥 − 𝑇 ) ) )
560	525 528 559	3eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝑥 − 𝑇 ) ) )
561	507 560	eqtr4d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) = ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) )
562	430 561	eqtr4d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
563	428 562	pm2.61dan	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
564	395 563	eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
565	393 564	pm2.61dan	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
566	310 390	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ∈ ℂ )
567	566	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ∈ ℂ )
568	426 420	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ∈ ℂ )
569	568	ad4ant14	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ∈ ℂ )
570	265 267	fssresd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
571	570	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
572	571 523	ffvelcdmd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ∈ ℂ )
573	430 572	eqeltrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ∈ ℂ )
574	569 573	pm2.61dan	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ∈ ℂ )
575	395 574	eqeltrd	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ∈ ℂ )
576	567 575	pm2.61dan	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ∈ ℂ )
577		eqid	⊢ ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
578	577	fvmpt2	⊢ ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ∧ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ∈ ℂ ) → ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) = if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
579	303 576 578	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) = if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
580		nfv	⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) )
581		eqid	⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
582	580 581 50 53 10 12 11	cncfiooicc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
583	365 582	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐺 ∈ ( ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) )
584		cncff	⊢ ( 𝐺 ∈ ( ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) → 𝐺 : ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
585	583 584	syl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐺 : ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
586	585	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐺 : ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
587	586 477	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) ∈ ℂ )
588	14	fvmpt2	⊢ ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) ∈ ℂ ) → ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
589	303 587 588	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
590	565 579 589	3eqtr4rd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐻 ‘ 𝑥 ) = ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) )
591	590	itgeq2dv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐻 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) d 𝑥 )
592		ioossicc	⊢ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
593	592	sseli	⊢ ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
594	593	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
595	593 576	sylan2	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ∈ ℂ )
596	594 595 578	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) = if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) )
597	231 239	gtned	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≠ ( 𝑆 ‘ 𝑖 ) )
598	597	neneqd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ¬ 𝑥 = ( 𝑆 ‘ 𝑖 ) )
599	598	iffalsed	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) )
600	232 243	ltned	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≠ ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
601	600	neneqd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ¬ 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) )
602	601	iffalsed	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) )
603	524	adantl	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
604	602 603	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) )
605	596 599 604	3eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
606	605	itgeq2dv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
607	579 576	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) ∈ ℂ )
608	123 124 607	itgioo	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) d 𝑥 )
609	123 124 306	itgioo	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
610	606 608 609	3eqtr3d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑆 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑆 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑖 ) (,) ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
611	591 610	eqtr2d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐻 ‘ 𝑥 ) d 𝑥 )
612	102 112	oveq12d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) = ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) )
613	612	itgeq1d	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐻 ‘ 𝑥 ) d 𝑥 = ∫ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ( 𝐻 ‘ 𝑥 ) d 𝑥 )
614		simpr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) )
615	612	eqcomd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) = ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
616	615	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) = ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
617	614 616	eleqtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → 𝑥 ∈ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) )
618	585	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → 𝐺 : ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ )
619	50	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ )
620	53	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ )
621	100	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ∈ ℝ )
622	111	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ∈ ℝ )
623		eliccre	⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ∈ ℝ ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ∈ ℝ ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → 𝑥 ∈ ℝ )
624	621 622 614 623	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → 𝑥 ∈ ℝ )
625	78	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → 𝑇 ∈ ℝ )
626	624 625	resubcld	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝑥 − 𝑇 ) ∈ ℝ )
627	228	eqcomd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) = ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) − 𝑇 ) )
628	627	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝑄 ‘ 𝑖 ) = ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) − 𝑇 ) )
629	621	rexrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ∈ ℝ* )
630	622	rexrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ∈ ℝ* )
631		iccgelb	⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ∈ ℝ* ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ≤ 𝑥 )
632	629 630 614 631	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ≤ 𝑥 )
633	621 624 625 632	lesub1dd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) − 𝑇 ) ≤ ( 𝑥 − 𝑇 ) )
634	628 633	eqbrtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑥 − 𝑇 ) )
635		iccleub	⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ∈ ℝ* ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → 𝑥 ≤ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
636	629 630 614 635	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → 𝑥 ≤ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) )
637	624 622 625 636	lesub1dd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝑥 − 𝑇 ) ≤ ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) − 𝑇 ) )
638	247	adantr	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) − 𝑇 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
639	637 638	breqtrd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝑥 − 𝑇 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) )
640	619 620 626 634 639	eliccd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝑥 − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) )
641	618 640	ffvelcdmd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) ∈ ℂ )
642	617 641 588	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
643		eqidd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ↦ ( 𝐺 ‘ ( 𝑦 − 𝑇 ) ) ) = ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ↦ ( 𝐺 ‘ ( 𝑦 − 𝑇 ) ) ) )
644		oveq1	⊢ ( 𝑦 = 𝑥 → ( 𝑦 − 𝑇 ) = ( 𝑥 − 𝑇 ) )
645	644	fveq2d	⊢ ( 𝑦 = 𝑥 → ( 𝐺 ‘ ( 𝑦 − 𝑇 ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
646	645	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) ∧ 𝑦 = 𝑥 ) → ( 𝐺 ‘ ( 𝑦 − 𝑇 ) ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
647	643 646 614 641	fvmptd	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ↦ ( 𝐺 ‘ ( 𝑦 − 𝑇 ) ) ) ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
648	642 647	eqtr4d	⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ) → ( 𝐻 ‘ 𝑥 ) = ( ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ↦ ( 𝐺 ‘ ( 𝑦 − 𝑇 ) ) ) ‘ 𝑥 ) )
649	648	itgeq2dv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ( 𝐻 ‘ 𝑥 ) d 𝑥 = ∫ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ( ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ↦ ( 𝐺 ‘ ( 𝑦 − 𝑇 ) ) ) ‘ 𝑥 ) d 𝑥 )
650	5	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑇 ∈ ℝ+ )
651	645	cbvmptv	⊢ ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ↦ ( 𝐺 ‘ ( 𝑦 − 𝑇 ) ) ) = ( 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ↦ ( 𝐺 ‘ ( 𝑥 − 𝑇 ) ) )
652	50 53 382 583 650 651	itgiccshift	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ( ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) + 𝑇 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + 𝑇 ) ) ↦ ( 𝐺 ‘ ( 𝑦 − 𝑇 ) ) ) ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐺 ‘ 𝑥 ) d 𝑥 )
653	613 649 652	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐻 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐺 ‘ 𝑥 ) d 𝑥 )
654	135	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom 𝐹 = ℝ )
655	64 654	sseqtrrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐹 )
656	50 53 138 10 655 11 12 13	itgioocnicc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐺 ∈ 𝐿1 ∧ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐺 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 ) )
657	656	simprd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐺 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
658	611 653 657	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
659	658	sumeq2dv	⊢ ( 𝜑 → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑆 ‘ 𝑖 ) [,] ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
660	93 308 659	3eqtrrd	⊢ ( 𝜑 → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = ∫ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )
661	25 68 660	3eqtrrd	⊢ ( 𝜑 → ∫ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ( 𝐹 ‘ 𝑥 ) d 𝑥 = ∫ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑥 ) d 𝑥 )