fourierdlem82 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem82 46169

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 11264	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	1, 2, 3, 5	lesub1dd 11736	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) ≤ (𝐵 − 𝑋))
7	6	ditgpos 25755	. . 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 4482	. . . . . . . . . . . 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 4482	. . . . . . . . . . . 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 2767	. . . . . . . . . 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 715	. . . . . . . . 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 4485	. . . . . . . . . . . . 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 4482	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = 𝐿)
17	15, 16	sylan9eq 2784	. . . . . . . . . . . 12 ⊢ ((¬ 𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝐿)
18	17	adantll 714	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝐿)
19		iffalse 4485	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
20		iftrue 4482	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = 𝐿)
21	19, 20	sylan9eq 2784	. . . . . . . . . . . 12 ⊢ ((¬ 𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝐿)
22	21	adantll 714	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝐿)
23	18, 22	eqtr4d 2767	. . . . . . . . . 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 4485	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 4485	. . . . . . . . . . . . 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 727	. . . . . . . . . . . 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 11165	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
31	30	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ∈ ℝ^*)
32	2	rexrd 11165	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
33	32	ad3antrrr 730	. . . . . . . . . . . . . 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 45486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
38	34, 35, 36, 37	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
39	38	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ)
40	1	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ∈ ℝ)
41	38	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ∈ ℝ)
42		elicc2 13314	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
43	34, 35, 42	syl2anc 584	. . . . . . . . . . . . . . . . . . 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 2933	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 = 𝐴 → 𝑥 ≠ 𝐴)
48	47	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ≠ 𝐴)
49	40, 41, 46, 48	leneltd 11270	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 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 2981	. . . . . . . . . . . . . . . . . 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 11270	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵)
59	58	adantlr 715	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵)
60	31, 33, 39, 50, 59	eliood 45479	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴(,)𝐵))
61		fvres 6841	. . . . . . . . . . . . 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 2774	. . . . . . . . . . 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 2774	. . . . . . . . . 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 5185	. . . . . . 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 2776	. . . . . 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 2733	. . . . . . 7 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐴 ↔ (𝑋 + 𝑡) = 𝐴))
71		eqeq1 2733	. . . . . . . 8 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐵 ↔ (𝑋 + 𝑡) = 𝐵))
72		fveq2 6822	. . . . . . . 8 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝐹‘𝑥) = (𝐹‘(𝑋 + 𝑡)))
73	71, 72	ifbieq2d 4503	. . . . . . 7 ⊢ (𝑥 = (𝑋 + 𝑡) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))))
74	70, 73	ifbieq2d 4503	. . . . . 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 11548	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
78	77	rexrd 11165	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ^*)
79	78	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ^*)
80	2, 3	resubcld 11548	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
81	80	rexrd 11165	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ^*)
82	81	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ^*)
83		elioo2 13289	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 11718	. . . . . . . . . . 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 11251	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐴)
92	91	neneqd 2930	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐴)
93	92	iffalsed 4487	. . . . . . 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 11144	. . . . . . . . . 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 11721	. . . . . . . . . . 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 11252	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐵)
100	99	neneqd 2930	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐵)
101	100	iffalsed 4487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))) = (𝐹‘(𝑋 + 𝑡)))
102	93, 101	eqtrd 2764	. . . . . 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 2786	. . . . 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 11264	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑡))
105	94, 96, 98	ltled 11264	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≤ 𝐵)
106	75, 96, 94, 104, 105	eliccd 45485	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ (𝐴[,]𝐵))
107		fourierdlem82.5	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
108	107	ffund 6656	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
109	108	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → Fun 𝐹)
110	107	fdmd 6662	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = (𝐴[,]𝐵))
111	110	eqcomd 2735	. . . . . . . 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 2830	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ dom 𝐹)
114		fvelrn 7010	. . . . . 6 ⊢ ((Fun 𝐹 ∧ (𝑋 + 𝑡) ∈ dom 𝐹) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
115	109, 113, 114	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
116	69, 103, 106, 115	fvmptd 6937	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐺‘(𝑋 + 𝑡)) = (𝐹‘(𝑋 + 𝑡)))
117	116	itgeq2dv 25681	. . 3 ⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐺‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
118	107	frnd 6660	. . . . . 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 45486	. . . . . . . . . 10 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ℝ)
128	124, 125, 126, 127	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ℝ)
129	123, 128	readdcld 11144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
130		elicc2 13314	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 11719	. . . . . . . . 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 11722	. . . . . . . . 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 45485	. . . . . . 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 2830	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ dom 𝐹)
142	120, 141, 114	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹)
143	119, 142	sseldd 3936	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
144	77, 80, 143	itgioo 25715	. . 3 ⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
145	7, 117, 144	3eqtrrd 2769	. 2 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡)
146		nfv 1914	. . . 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 45618	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵) = (𝐹 lim_ℂ 𝐵))
150	148, 149	eleqtrrd 2831	. . . 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 45602	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴) = (𝐹 lim_ℂ 𝐴))
153	151, 152	eleqtrrd 2831	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴))
154	146, 8, 1, 2, 147, 150, 153	cncfiooicc 45875	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
155	1, 2, 5, 3, 154	itgsbtaddcnst 45963	. 2 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡 = ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠)
156	5	ditgpos 25755	. . 3 ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠)
157		fveq2 6822	. . . . 5 ⊢ (𝑠 = 𝑡 → (𝐺‘𝑠) = (𝐺‘𝑡))
158	157	cbvitgv 25676	. . . 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 726	. . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 ∈ ℝ^*)
163	32	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐵 ∈ ℝ^*)
164		elioo2 13289	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑡 ∈ (𝐴(,)𝐵) ↔ (𝑡 ∈ ℝ ∧ 𝐴 < 𝑡 ∧ 𝑡 < 𝐵)))
165	162, 163, 164	syl2anc 584	. . . . . . . . . . . . 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 5120	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 < 𝑥)
170	160, 169	gtned 11251	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐴)
171	170	neneqd 2930	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐴)
172	171	iffalsed 4487	. . . . . . 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 2828	. . . . . . . . . 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 5114	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 < 𝐵)
177	174, 176	ltned 11252	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐵)
178	177	neneqd 2930	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐵)
179	178	iffalsed 4487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))
180	168, 161	eqeltrd 2828	. . . . . . . . 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 6822	. . . . . . . . 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 2764	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑡))
185	172, 179, 184	3eqtrd 2768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = (𝐹‘𝑡))
186		ioossicc 13336	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
187		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴(,)𝐵))
188	186, 187	sselid 3933	. . . . . 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 2830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ dom 𝐹)
192		fvelrn 7010	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑡 ∈ dom 𝐹) → (𝐹‘𝑡) ∈ ran 𝐹)
193	189, 191, 192	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ ran 𝐹)
194	159, 185, 188, 193	fvmptd 6937	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑡) = (𝐹‘𝑡))
195	194	itgeq2dv 25681	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡)
196	158, 195	eqtrid 2776	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡)
197	107	ffvelcdmda 7018	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑡) ∈ ℂ)
198	1, 2, 197	itgioo 25715	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡)
199	156, 196, 198	3eqtrd 2768	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡)
200	145, 155, 199	3eqtrrd 2769	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 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ⊆ wss 3903 ifcif 4476 class class class wbr 5092 ↦ cmpt 5173 dom cdm 5619 ran crn 5620 ↾ cres 5621 Fun wfun 6476 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 + caddc 11012 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 − cmin 11347 (,)cioo 13248 [,]cicc 13251 –cn→ccncf 24767 ∫citg 25517 ⨜cdit 25745 lim_ℂ climc 25761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-symdif 4204 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-ofr 7614 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-ovol 25363 df-vol 25364 df-mbf 25518 df-itg1 25519 df-itg2 25520 df-ibl 25521 df-itg 25522 df-0p 25569 df-ditg 25746 df-limc 25765 df-dv 25766

This theorem is referenced by: fourierdlem93 46180

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