fourierdlem82 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem82 46109

Description: Integral by substitution, adding a constant to the function's argument, for a function on an open interval with finite limits ad boundary points. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem82.1	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))))
fourierdlem82.2	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem82.3	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem82.4	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem82.5	⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
fourierdlem82.6	⊢ (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
fourierdlem82.7	⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
fourierdlem82.8	⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴))
fourierdlem82.9	⊢ (𝜑 → 𝑋 ∈ ℝ)

**Assertion**
Ref	Expression
fourierdlem82	⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)

Distinct variable groups: 𝑡,𝐴,𝑥 𝑡,𝐵,𝑥 𝑥,𝐹 𝑡,𝐺 𝑥,𝐿 𝑥,𝑅 𝑡,𝑋,𝑥 𝜑,𝑡,𝑥 Allowed substitution hints: 𝑅(𝑡) 𝐹(𝑡) 𝐺(𝑥) 𝐿(𝑡)

Proof of Theorem fourierdlem82 Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem82.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		fourierdlem82.3	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		fourierdlem82.9	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ)
4		fourierdlem82.4	. . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵)
5	1, 2, 4	ltled 11438	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	1, 2, 3, 5	lesub1dd 11906	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) ≤ (𝐵 − 𝑋))
7	6	ditgpos 25911	. . 3 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐺‘(𝑋 + 𝑡)) d𝑡)
8		fourierdlem82.1	. . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))))
9		iftrue 4554	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝑅)
10	9	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝑅)
11		iftrue 4554	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅)
12	11	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅)
13	10, 12	eqtr4d 2783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
14	13	adantlr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
15		iffalse 4557	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)))
16		iftrue 4554	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = 𝐿)
17	15, 16	sylan9eq 2800	. . . . . . . . . . . 12 ⊢ ((¬ 𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝐿)
18	17	adantll 713	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝐿)
19		iffalse 4557	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
20		iftrue 4554	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = 𝐿)
21	19, 20	sylan9eq 2800	. . . . . . . . . . . 12 ⊢ ((¬ 𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝐿)
22	21	adantll 713	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝐿)
23	18, 22	eqtr4d 2783	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
24		iffalse 4557	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))
25	24	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))
26	15	ad2antlr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)))
27		iffalse 4557	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
28	27	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
29	19	ad2antlr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
30	1	rexrd 11340	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
31	30	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ∈ ℝ^*)
32	2	rexrd 11340	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
33	32	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ^*)
34	1	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
35	2	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
36		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
37		eliccre 45423	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
38	34, 35, 36, 37	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
39	38	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ)
40	1	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ∈ ℝ)
41	38	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ∈ ℝ)
42		elicc2 13472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
43	34, 35, 42	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
44	36, 43	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
45	44	simp2d 1143	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
46	45	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ≤ 𝑥)
47		neqne 2954	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 = 𝐴 → 𝑥 ≠ 𝐴)
48	47	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ≠ 𝐴)
49	40, 41, 46, 48	leneltd 11444	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 < 𝑥)
50	49	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 < 𝑥)
51	38	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ)
52	2	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ)
53	44	simp3d 1144	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
54	53	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ≤ 𝐵)
55		nesym 3003	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ≠ 𝑥 ↔ ¬ 𝑥 = 𝐵)
56	55	biimpri 228	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 = 𝐵 → 𝐵 ≠ 𝑥)
57	56	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ≠ 𝑥)
58	51, 52, 54, 57	leneltd 11444	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵)
59	58	adantlr 714	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵)
60	31, 33, 39, 50, 59	eliood 45416	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴(,)𝐵))
61		fvres 6939	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑥))
62	60, 61	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑥))
63	28, 29, 62	3eqtr4d 2790	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))
64	25, 26, 63	3eqtr4d 2790	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
65	23, 64	pm2.61dan 812	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
66	14, 65	pm2.61dan 812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
67	66	mpteq2dva 5266	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))))
68	8, 67	eqtrid 2792	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))))
69	68	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))))
70		eqeq1 2744	. . . . . . 7 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐴 ↔ (𝑋 + 𝑡) = 𝐴))
71		eqeq1 2744	. . . . . . . 8 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐵 ↔ (𝑋 + 𝑡) = 𝐵))
72		fveq2 6920	. . . . . . . 8 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝐹‘𝑥) = (𝐹‘(𝑋 + 𝑡)))
73	71, 72	ifbieq2d 4574	. . . . . . 7 ⊢ (𝑥 = (𝑋 + 𝑡) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))))
74	70, 73	ifbieq2d 4574	. . . . . 6 ⊢ (𝑥 = (𝑋 + 𝑡) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if((𝑋 + 𝑡) = 𝐴, 𝑅, if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡)))))
75	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
76		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋)))
77	1, 3	resubcld 11718	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
78	77	rexrd 11340	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ^*)
79	78	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ^*)
80	2, 3	resubcld 11718	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
81	80	rexrd 11340	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ^*)
82	81	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ^*)
83		elioo2 13448	. . . . . . . . . . . . . 14 ⊢ (((𝐴 − 𝑋) ∈ ℝ^* ∧ (𝐵 − 𝑋) ∈ ℝ^*) → (𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) < 𝑡 ∧ 𝑡 < (𝐵 − 𝑋))))
84	79, 82, 83	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) < 𝑡 ∧ 𝑡 < (𝐵 − 𝑋))))
85	76, 84	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) < 𝑡 ∧ 𝑡 < (𝐵 − 𝑋)))
86	85	simp2d 1143	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴 − 𝑋) < 𝑡)
87	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑋 ∈ ℝ)
88	85	simp1d 1142	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 ∈ ℝ)
89	75, 87, 88	ltsubadd2d 11888	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ((𝐴 − 𝑋) < 𝑡 ↔ 𝐴 < (𝑋 + 𝑡)))
90	86, 89	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 < (𝑋 + 𝑡))
91	75, 90	gtned 11425	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐴)
92	91	neneqd 2951	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐴)
93	92	iffalsed 4559	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → if((𝑋 + 𝑡) = 𝐴, 𝑅, if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡)))) = if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))))
94	87, 88	readdcld 11319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
95	85	simp3d 1144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 < (𝐵 − 𝑋))
96	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐵 ∈ ℝ)
97	87, 88, 96	ltaddsub2d 11891	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ((𝑋 + 𝑡) < 𝐵 ↔ 𝑡 < (𝐵 − 𝑋)))
98	95, 97	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) < 𝐵)
99	94, 98	ltned 11426	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐵)
100	99	neneqd 2951	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐵)
101	100	iffalsed 4559	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))) = (𝐹‘(𝑋 + 𝑡)))
102	93, 101	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → if((𝑋 + 𝑡) = 𝐴, 𝑅, if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡)))) = (𝐹‘(𝑋 + 𝑡)))
103	74, 102	sylan9eqr 2802	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) ∧ 𝑥 = (𝑋 + 𝑡)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = (𝐹‘(𝑋 + 𝑡)))
104	75, 94, 90	ltled 11438	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑡))
105	94, 96, 98	ltled 11438	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≤ 𝐵)
106	75, 96, 94, 104, 105	eliccd 45422	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ (𝐴[,]𝐵))
107		fourierdlem82.5	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
108	107	ffund 6751	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
109	108	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → Fun 𝐹)
110	107	fdmd 6757	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = (𝐴[,]𝐵))
111	110	eqcomd 2746	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) = dom 𝐹)
112	111	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴[,]𝐵) = dom 𝐹)
113	106, 112	eleqtrd 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ dom 𝐹)
114		fvelrn 7110	. . . . . 6 ⊢ ((Fun 𝐹 ∧ (𝑋 + 𝑡) ∈ dom 𝐹) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
115	109, 113, 114	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
116	69, 103, 106, 115	fvmptd 7036	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐺‘(𝑋 + 𝑡)) = (𝐹‘(𝑋 + 𝑡)))
117	116	itgeq2dv 25837	. . 3 ⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐺‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
118	107	frnd 6755	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
119	118	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ran 𝐹 ⊆ ℂ)
120	108	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → Fun 𝐹)
121	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
122	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐵 ∈ ℝ)
123	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑋 ∈ ℝ)
124	77	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ)
125	80	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
126		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
127		eliccre 45423	. . . . . . . . . 10 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ℝ)
128	124, 125, 126, 127	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ℝ)
129	123, 128	readdcld 11319	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
130		elicc2 13472	. . . . . . . . . . . 12 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ) → (𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑡 ∧ 𝑡 ≤ (𝐵 − 𝑋))))
131	124, 125, 130	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑡 ∧ 𝑡 ≤ (𝐵 − 𝑋))))
132	126, 131	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑡 ∧ 𝑡 ≤ (𝐵 − 𝑋)))
133	132	simp2d 1143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ≤ 𝑡)
134	121, 123, 128	lesubadd2d 11889	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝐴 − 𝑋) ≤ 𝑡 ↔ 𝐴 ≤ (𝑋 + 𝑡)))
135	133, 134	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑡))
136	132	simp3d 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ≤ (𝐵 − 𝑋))
137	123, 128, 122	leaddsub2d 11892	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝑋 + 𝑡) ≤ 𝐵 ↔ 𝑡 ≤ (𝐵 − 𝑋)))
138	136, 137	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ≤ 𝐵)
139	121, 122, 129, 135, 138	eliccd 45422	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ (𝐴[,]𝐵))
140	111	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴[,]𝐵) = dom 𝐹)
141	139, 140	eleqtrd 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ dom 𝐹)
142	120, 141, 114	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
143	119, 142	sseldd 4009	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
144	77, 80, 143	itgioo 25871	. . 3 ⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
145	7, 117, 144	3eqtrrd 2785	. 2 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡)
146		nfv 1913	. . . 4 ⊢ Ⅎ𝑥𝜑
147		fourierdlem82.6	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
148		fourierdlem82.7	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
149	1, 2, 4, 107	limcicciooub 45558	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵) = (𝐹 lim_ℂ 𝐵))
150	148, 149	eleqtrrd 2847	. . . 4 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵))
151		fourierdlem82.8	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴))
152	1, 2, 4, 107	limciccioolb 45542	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴) = (𝐹 lim_ℂ 𝐴))
153	151, 152	eleqtrrd 2847	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴))
154	146, 8, 1, 2, 147, 150, 153	cncfiooicc 45815	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
155	1, 2, 5, 3, 154	itgsbtaddcnst 45903	. 2 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡 = ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠)
156	5	ditgpos 25911	. . 3 ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠)
157		fveq2 6920	. . . . 5 ⊢ (𝑠 = 𝑡 → (𝐺‘𝑠) = (𝐺‘𝑡))
158	157	cbvitgv 25832	. . . 4 ⊢ ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐺‘𝑡) d𝑡
159	8	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)))))
160	1	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 ∈ ℝ)
161		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 ∈ (𝐴(,)𝐵))
162	30	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 ∈ ℝ^*)
163	32	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐵 ∈ ℝ^*)
164		elioo2 13448	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑡 ∈ (𝐴(,)𝐵) ↔ (𝑡 ∈ ℝ ∧ 𝐴 < 𝑡 ∧ 𝑡 < 𝐵)))
165	162, 163, 164	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → (𝑡 ∈ (𝐴(,)𝐵) ↔ (𝑡 ∈ ℝ ∧ 𝐴 < 𝑡 ∧ 𝑡 < 𝐵)))
166	161, 165	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → (𝑡 ∈ ℝ ∧ 𝐴 < 𝑡 ∧ 𝑡 < 𝐵))
167	166	simp2d 1143	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 < 𝑡)
168		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 = 𝑡)
169	167, 168	breqtrrd 5194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 < 𝑥)
170	160, 169	gtned 11425	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐴)
171	170	neneqd 2951	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐴)
172	171	iffalsed 4559	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)))
173	166	simp1d 1142	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 ∈ ℝ)
174	168, 173	eqeltrd 2844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ∈ ℝ)
175	166	simp3d 1144	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 < 𝐵)
176	168, 175	eqbrtrd 5188	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 < 𝐵)
177	174, 176	ltned 11426	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐵)
178	177	neneqd 2951	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐵)
179	178	iffalsed 4559	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))
180	168, 161	eqeltrd 2844	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ∈ (𝐴(,)𝐵))
181	180, 61	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑥))
182		fveq2 6920	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝐹‘𝑥) = (𝐹‘𝑡))
183	182	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → (𝐹‘𝑥) = (𝐹‘𝑡))
184	181, 183	eqtrd 2780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑡))
185	172, 179, 184	3eqtrd 2784	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = (𝐹‘𝑡))
186		ioossicc 13493	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
187		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴(,)𝐵))
188	186, 187	sselid 4006	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
189	108	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → Fun 𝐹)
190	111	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) = dom 𝐹)
191	188, 190	eleqtrd 2846	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ dom 𝐹)
192		fvelrn 7110	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑡 ∈ dom 𝐹) → (𝐹‘𝑡) ∈ ran 𝐹)
193	189, 191, 192	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ ran 𝐹)
194	159, 185, 188, 193	fvmptd 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑡) = (𝐹‘𝑡))
195	194	itgeq2dv 25837	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡)
196	158, 195	eqtrid 2792	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡)
197	107	ffvelcdmda 7118	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑡) ∈ ℂ)
198	1, 2, 197	itgioo 25871	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡)
199	156, 196, 198	3eqtrd 2784	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡)
200	145, 155, 199	3eqtrrd 2785	1 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ⊆ wss 3976 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ran crn 5701 ↾ cres 5702 Fun wfun 6567 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 + caddc 11187 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 (,)cioo 13407 [,]cicc 13410 –cn→ccncf 24921 ∫citg 25672 ⨜cdit 25901 lim_ℂ climc 25917

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-symdif 4272 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-ibl 25676 df-itg 25677 df-0p 25724 df-ditg 25902 df-limc 25921 df-dv 25922

This theorem is referenced by: fourierdlem93 46120

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