fourierdlem82 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem82 46203

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 11409	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	1, 2, 3, 5	lesub1dd 11879	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) ≤ (𝐵 − 𝑋))
7	6	ditgpos 25891	. . 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 4531	. . . . . . . . . . . 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 4531	. . . . . . . . . . . 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 2780	. . . . . . . . . 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 4534	. . . . . . . . . . . . 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 4531	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = 𝐿)
17	15, 16	sylan9eq 2797	. . . . . . . . . . . 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 4534	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
20		iftrue 4531	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = 𝐿)
21	19, 20	sylan9eq 2797	. . . . . . . . . . . 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 2780	. . . . . . . . . 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 4534	. . . . . . . . . . . 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 4534	. . . . . . . . . . . . 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 11311	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
31	30	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ∈ ℝ^*)
32	2	rexrd 11311	. . . . . . . . . . . . . . 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 45518	. . . . . . . . . . . . . . . 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 13452	. . . . . . . . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
46	45	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ≤ 𝑥)
47		neqne 2948	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 = 𝐴 → 𝑥 ≠ 𝐴)
48	47	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ≠ 𝐴)
49	40, 41, 46, 48	leneltd 11415	. . . . . . . . . . . . . . 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 1145	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
54	53	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ≤ 𝐵)
55		nesym 2997	. . . . . . . . . . . . . . . . . 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 11415	. . . . . . . . . . . . . . 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 45511	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴(,)𝐵))
61		fvres 6925	. . . . . . . . . . . . 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 2787	. . . . . . . . . . 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 2787	. . . . . . . . . 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 813	. . . . . . . . 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 813	. . . . . . . 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 5242	. . . . . . 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 2789	. . . . . 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 2741	. . . . . . 7 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐴 ↔ (𝑋 + 𝑡) = 𝐴))
71		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐵 ↔ (𝑋 + 𝑡) = 𝐵))
72		fveq2 6906	. . . . . . . 8 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝐹‘𝑥) = (𝐹‘(𝑋 + 𝑡)))
73	71, 72	ifbieq2d 4552	. . . . . . 7 ⊢ (𝑥 = (𝑋 + 𝑡) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))))
74	70, 73	ifbieq2d 4552	. . . . . 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 11691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
78	77	rexrd 11311	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ^*)
79	78	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ^*)
80	2, 3	resubcld 11691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
81	80	rexrd 11311	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ^*)
82	81	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ^*)
83		elioo2 13428	. . . . . . . . . . . . . 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 1144	. . . . . . . . . . 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 1143	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 ∈ ℝ)
89	75, 87, 88	ltsubadd2d 11861	. . . . . . . . . . 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 11396	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐴)
92	91	neneqd 2945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐴)
93	92	iffalsed 4536	. . . . . . 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 11290	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
95	85	simp3d 1145	. . . . . . . . . . 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 11864	. . . . . . . . . . 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 11397	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐵)
100	99	neneqd 2945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐵)
101	100	iffalsed 4536	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))) = (𝐹‘(𝑋 + 𝑡)))
102	93, 101	eqtrd 2777	. . . . . 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 2799	. . . . 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 11409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑡))
105	94, 96, 98	ltled 11409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≤ 𝐵)
106	75, 96, 94, 104, 105	eliccd 45517	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ (𝐴[,]𝐵))
107		fourierdlem82.5	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
108	107	ffund 6740	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
109	108	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → Fun 𝐹)
110	107	fdmd 6746	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = (𝐴[,]𝐵))
111	110	eqcomd 2743	. . . . . . . 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 2843	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ dom 𝐹)
114		fvelrn 7096	. . . . . 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 7023	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐺‘(𝑋 + 𝑡)) = (𝐹‘(𝑋 + 𝑡)))
117	116	itgeq2dv 25817	. . 3 ⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐺‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
118	107	frnd 6744	. . . . . 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 45518	. . . . . . . . . 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 11290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
130		elicc2 13452	. . . . . . . . . . . 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 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ≤ 𝑡)
134	121, 123, 128	lesubadd2d 11862	. . . . . . . . 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 1145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ≤ (𝐵 − 𝑋))
137	123, 128, 122	leaddsub2d 11865	. . . . . . . . 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 45517	. . . . . . 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 2843	. . . . . 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 3984	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
144	77, 80, 143	itgioo 25851	. . 3 ⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
145	7, 117, 144	3eqtrrd 2782	. 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 45652	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵) = (𝐹 lim_ℂ 𝐵))
150	148, 149	eleqtrrd 2844	. . . 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 45636	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴) = (𝐹 lim_ℂ 𝐴))
153	151, 152	eleqtrrd 2844	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴))
154	146, 8, 1, 2, 147, 150, 153	cncfiooicc 45909	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
155	1, 2, 5, 3, 154	itgsbtaddcnst 45997	. 2 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡 = ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠)
156	5	ditgpos 25891	. . 3 ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠)
157		fveq2 6906	. . . . 5 ⊢ (𝑠 = 𝑡 → (𝐺‘𝑠) = (𝐺‘𝑡))
158	157	cbvitgv 25812	. . . 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 769	. . . . . . . . . . . . 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 13428	. . . . . . . . . . . . . 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 1144	. . . . . . . . . . 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 5171	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 < 𝑥)
170	160, 169	gtned 11396	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐴)
171	170	neneqd 2945	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐴)
172	171	iffalsed 4536	. . . . . . 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 1143	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 ∈ ℝ)
174	168, 173	eqeltrd 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ∈ ℝ)
175	166	simp3d 1145	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 < 𝐵)
176	168, 175	eqbrtrd 5165	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 < 𝐵)
177	174, 176	ltned 11397	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐵)
178	177	neneqd 2945	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐵)
179	178	iffalsed 4536	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))
180	168, 161	eqeltrd 2841	. . . . . . . . 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 6906	. . . . . . . . 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 2777	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑡))
185	172, 179, 184	3eqtrd 2781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = (𝐹‘𝑡))
186		ioossicc 13473	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
187		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴(,)𝐵))
188	186, 187	sselid 3981	. . . . . 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 2843	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ dom 𝐹)
192		fvelrn 7096	. . . . . . 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 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑡) = (𝐹‘𝑡))
195	194	itgeq2dv 25817	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡)
196	158, 195	eqtrid 2789	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡)
197	107	ffvelcdmda 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑡) ∈ ℂ)
198	1, 2, 197	itgioo 25851	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡)
199	156, 196, 198	3eqtrd 2781	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡)
200	145, 155, 199	3eqtrrd 2782	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 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 ↾ cres 5687 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 + caddc 11158 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 (,)cioo 13387 [,]cicc 13390 –cn→ccncf 24902 ∫citg 25653 ⨜cdit 25881 lim_ℂ climc 25897

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-symdif 4253 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-ibl 25657 df-itg 25658 df-0p 25705 df-ditg 25882 df-limc 25901 df-dv 25902

This theorem is referenced by: fourierdlem93 46214

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