fourierdlem82 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem82		Structured version Visualization version GIF version

Theorem fourierdlem82 43729

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 11123	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	1, 2, 3, 5	lesub1dd 11591	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) ≤ (𝐵 − 𝑋))
7	6	ditgpos 25020	. . 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 4465	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝑅)
10	9	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝑅)
11		iftrue 4465	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅)
12	11	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅)
13	10, 12	eqtr4d 2781	. . . . . . . . . 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 712	. . . . . . . . 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 4468	. . . . . . . . . . . . 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 4465	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = 𝐿)
17	15, 16	sylan9eq 2798	. . . . . . . . . . . 12 ⊢ ((¬ 𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝐿)
18	17	adantll 711	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝐿)
19		iffalse 4468	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
20		iftrue 4465	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = 𝐿)
21	19, 20	sylan9eq 2798	. . . . . . . . . . . 12 ⊢ ((¬ 𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝐿)
22	21	adantll 711	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝐿)
23	18, 22	eqtr4d 2781	. . . . . . . . . 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 4468	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))
25	24	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))
26	15	ad2antlr 724	. . . . . . . . . . 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 4468	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
28	27	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
29	19	ad2antlr 724	. . . . . . . . . . . 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 11025	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
31	30	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ∈ ℝ^*)
32	2	rexrd 11025	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
33	32	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ^*)
34	1	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
35	2	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
36		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
37		eliccre 43043	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
38	34, 35, 36, 37	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
39	38	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ)
40	1	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ∈ ℝ)
41	38	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ∈ ℝ)
42		elicc2 13144	. . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
45	44	simp2d 1142	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
46	45	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ≤ 𝑥)
47		neqne 2951	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 = 𝐴 → 𝑥 ≠ 𝐴)
48	47	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ≠ 𝐴)
49	40, 41, 46, 48	leneltd 11129	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 < 𝑥)
50	49	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 < 𝑥)
51	38	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ)
52	2	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ)
53	44	simp3d 1143	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
54	53	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ≤ 𝐵)
55		nesym 3000	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ≠ 𝑥 ↔ ¬ 𝑥 = 𝐵)
56	55	biimpri 227	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 = 𝐵 → 𝐵 ≠ 𝑥)
57	56	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ≠ 𝑥)
58	51, 52, 54, 57	leneltd 11129	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵)
59	58	adantlr 712	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵)
60	31, 33, 39, 50, 59	eliood 43036	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴(,)𝐵))
61		fvres 6793	. . . . . . . . . . . . 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 2788	. . . . . . . . . . 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 2788	. . . . . . . . . 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 810	. . . . . . . . 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 810	. . . . . . . 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 5174	. . . . . . 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 2790	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))))
69	68	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))))
70		eqeq1 2742	. . . . . . 7 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐴 ↔ (𝑋 + 𝑡) = 𝐴))
71		eqeq1 2742	. . . . . . . 8 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐵 ↔ (𝑋 + 𝑡) = 𝐵))
72		fveq2 6774	. . . . . . . 8 ⊢ (𝑥 = (𝑋 + 𝑡) → (𝐹‘𝑥) = (𝐹‘(𝑋 + 𝑡)))
73	71, 72	ifbieq2d 4485	. . . . . . 7 ⊢ (𝑥 = (𝑋 + 𝑡) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))))
74	70, 73	ifbieq2d 4485	. . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
76		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋)))
77	1, 3	resubcld 11403	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
78	77	rexrd 11025	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ^*)
79	78	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ^*)
80	2, 3	resubcld 11403	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
81	80	rexrd 11025	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ^*)
82	81	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ^*)
83		elioo2 13120	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) < 𝑡 ∧ 𝑡 < (𝐵 − 𝑋)))
86	85	simp2d 1142	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴 − 𝑋) < 𝑡)
87	3	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑋 ∈ ℝ)
88	85	simp1d 1141	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 ∈ ℝ)
89	75, 87, 88	ltsubadd2d 11573	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ((𝐴 − 𝑋) < 𝑡 ↔ 𝐴 < (𝑋 + 𝑡)))
90	86, 89	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 < (𝑋 + 𝑡))
91	75, 90	gtned 11110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐴)
92	91	neneqd 2948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐴)
93	92	iffalsed 4470	. . . . . . 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 11004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
95	85	simp3d 1143	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 < (𝐵 − 𝑋))
96	2	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐵 ∈ ℝ)
97	87, 88, 96	ltaddsub2d 11576	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ((𝑋 + 𝑡) < 𝐵 ↔ 𝑡 < (𝐵 − 𝑋)))
98	95, 97	mpbird 256	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) < 𝐵)
99	94, 98	ltned 11111	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐵)
100	99	neneqd 2948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐵)
101	100	iffalsed 4470	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))) = (𝐹‘(𝑋 + 𝑡)))
102	93, 101	eqtrd 2778	. . . . . 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 2800	. . . . 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 11123	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑡))
105	94, 96, 98	ltled 11123	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≤ 𝐵)
106	75, 96, 94, 104, 105	eliccd 43042	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ (𝐴[,]𝐵))
107		fourierdlem82.5	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
108	107	ffund 6604	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
109	108	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → Fun 𝐹)
110	107	fdmd 6611	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = (𝐴[,]𝐵))
111	110	eqcomd 2744	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) = dom 𝐹)
112	111	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴[,]𝐵) = dom 𝐹)
113	106, 112	eleqtrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ dom 𝐹)
114		fvelrn 6954	. . . . . 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 6882	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐺‘(𝑋 + 𝑡)) = (𝐹‘(𝑋 + 𝑡)))
117	116	itgeq2dv 24946	. . 3 ⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐺‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
118	107	frnd 6608	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
119	118	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ran 𝐹 ⊆ ℂ)
120	108	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → Fun 𝐹)
121	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
122	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐵 ∈ ℝ)
123	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑋 ∈ ℝ)
124	77	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ)
125	80	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
126		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
127		eliccre 43043	. . . . . . . . . 10 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ℝ)
128	124, 125, 126, 127	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ℝ)
129	123, 128	readdcld 11004	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ)
130		elicc2 13144	. . . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑡 ∧ 𝑡 ≤ (𝐵 − 𝑋)))
133	132	simp2d 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ≤ 𝑡)
134	121, 123, 128	lesubadd2d 11574	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝐴 − 𝑋) ≤ 𝑡 ↔ 𝐴 ≤ (𝑋 + 𝑡)))
135	133, 134	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑡))
136	132	simp3d 1143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ≤ (𝐵 − 𝑋))
137	123, 128, 122	leaddsub2d 11577	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝑋 + 𝑡) ≤ 𝐵 ↔ 𝑡 ≤ (𝐵 − 𝑋)))
138	136, 137	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ≤ 𝐵)
139	121, 122, 129, 135, 138	eliccd 43042	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ (𝐴[,]𝐵))
140	111	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴[,]𝐵) = dom 𝐹)
141	139, 140	eleqtrd 2841	. . . . . 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 3922	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ)
144	77, 80, 143	itgioo 24980	. . 3 ⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
145	7, 117, 144	3eqtrrd 2783	. 2 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡)
146		nfv 1917	. . . 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 43178	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵) = (𝐹 lim_ℂ 𝐵))
150	148, 149	eleqtrrd 2842	. . . 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 43162	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴) = (𝐹 lim_ℂ 𝐴))
153	151, 152	eleqtrrd 2842	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴))
154	146, 8, 1, 2, 147, 150, 153	cncfiooicc 43435	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
155	1, 2, 5, 3, 154	itgsbtaddcnst 43523	. 2 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡 = ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠)
156	5	ditgpos 25020	. . 3 ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠)
157		fveq2 6774	. . . . 5 ⊢ (𝑠 = 𝑡 → (𝐺‘𝑠) = (𝐺‘𝑡))
158	157	cbvitgv 24941	. . . 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 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 ∈ ℝ)
161		simplr 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 ∈ (𝐴(,)𝐵))
162	30	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 ∈ ℝ^*)
163	32	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐵 ∈ ℝ^*)
164		elioo2 13120	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → (𝑡 ∈ ℝ ∧ 𝐴 < 𝑡 ∧ 𝑡 < 𝐵))
167	166	simp2d 1142	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 < 𝑡)
168		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 = 𝑡)
169	167, 168	breqtrrd 5102	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 < 𝑥)
170	160, 169	gtned 11110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐴)
171	170	neneqd 2948	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐴)
172	171	iffalsed 4470	. . . . . . 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 1141	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 ∈ ℝ)
174	168, 173	eqeltrd 2839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ∈ ℝ)
175	166	simp3d 1143	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 < 𝐵)
176	168, 175	eqbrtrd 5096	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 < 𝐵)
177	174, 176	ltned 11111	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐵)
178	177	neneqd 2948	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐵)
179	178	iffalsed 4470	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))
180	168, 161	eqeltrd 2839	. . . . . . . . 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 6774	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝐹‘𝑥) = (𝐹‘𝑡))
183	182	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → (𝐹‘𝑥) = (𝐹‘𝑡))
184	181, 183	eqtrd 2778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑡))
185	172, 179, 184	3eqtrd 2782	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = (𝐹‘𝑡))
186		ioossicc 13165	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
187		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴(,)𝐵))
188	186, 187	sselid 3919	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
189	108	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → Fun 𝐹)
190	111	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) = dom 𝐹)
191	188, 190	eleqtrd 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ dom 𝐹)
192		fvelrn 6954	. . . . . . 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 6882	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑡) = (𝐹‘𝑡))
195	194	itgeq2dv 24946	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡)
196	158, 195	eqtrid 2790	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡)
197	107	ffvelrnda 6961	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑡) ∈ ℂ)
198	1, 2, 197	itgioo 24980	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡)
199	156, 196, 198	3eqtrd 2782	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡)
200	145, 155, 199	3eqtrrd 2783	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ⊆ wss 3887 ifcif 4459 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 ↾ cres 5591 Fun wfun 6427 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 + caddc 10874 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 (,)cioo 13079 [,]cicc 13082 –cn→ccncf 24039 ∫citg 24782 ⨜cdit 25010 lim_ℂ climc 25026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-symdif 4176 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-ovol 24628 df-vol 24629 df-mbf 24783 df-itg1 24784 df-itg2 24785 df-ibl 24786 df-itg 24787 df-0p 24834 df-ditg 25011 df-limc 25030 df-dv 25031

This theorem is referenced by: fourierdlem93 43740

W3C validator