iblcncfioo - Mathbox for Glauco Siliprandi

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Theorem iblcncfioo 44680

Description: A continuous function 𝐹 on an open interval (𝐴(,)𝐵) with a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵 is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
iblcncfioo.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
iblcncfioo.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
iblcncfioo.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
iblcncfioo.l	⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
iblcncfioo.r	⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴))

**Assertion**
Ref	Expression
iblcncfioo	⊢ (𝜑 → 𝐹 ∈ 𝐿¹)

Proof of Theorem iblcncfioo Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iblcncfioo.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
2		cncff 24400	. . . . 5 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
4	3	feqmptd 6957	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
5		iblcncfioo.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6	5	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
7		eliooord 13379	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
8	7	simpld 495	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝐴 < 𝑥)
9	8	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑥)
10	6, 9	gtned 11345	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 𝐴)
11	10	neneqd 2945	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 𝐴)
12	11	iffalsed 4538	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
13		elioore 13350	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
14	13	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
15	7	simprd 496	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 < 𝐵)
16	15	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 < 𝐵)
17	14, 16	ltned 11346	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 𝐵)
18	17	neneqd 2945	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 𝐵)
19	18	iffalsed 4538	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
20	12, 19	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = (𝐹‘𝑥))
21	20	eqcomd 2738	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
22	21	mpteq2dva 5247	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))))
23	4, 22	eqtrd 2772	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))))
24		ioossicc 13406	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
25	24	a1i 11	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
26		ioombl 25073	. . . 4 ⊢ (𝐴(,)𝐵) ∈ dom vol
27	26	a1i 11	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
28		iftrue 4533	. . . . . . 7 ⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅)
29	28	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅)
30		limccl 25383	. . . . . . . 8 ⊢ (𝐹 lim_ℂ 𝐴) ⊆ ℂ
31		iblcncfioo.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴))
32	30, 31	sselid 3979	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
33	32	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑅 ∈ ℂ)
34	29, 33	eqeltrd 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
35	34	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
36		iffalse 4536	. . . . . . . . 9 ⊢ (¬ 𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
37	36	ad2antlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
38		iftrue 4533	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = 𝐿)
39	38	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = 𝐿)
40	37, 39	eqtrd 2772	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝐿)
41		limccl 25383	. . . . . . . . 9 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
42		iblcncfioo.l	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
43	41, 42	sselid 3979	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℂ)
44	43	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → 𝐿 ∈ ℂ)
45	40, 44	eqeltrd 2833	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
46	45	adantllr 717	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
47		simplll 773	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
48	5	rexrd 11260	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
49	47, 48	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ∈ ℝ^*)
50		iblcncfioo.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
51	50	rexrd 11260	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
52	47, 51	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ^*)
53		eliccxr 13408	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ^*)
54	53	ad3antlr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ^*)
55	49, 52, 54	3jca 1128	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*))
56	5	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ∈ ℝ)
57	5	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
58	50	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
59		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
60		eliccre 44204	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
61	57, 58, 59, 60	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
62	61	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ∈ ℝ)
63	5, 50	jca 512	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
64	63	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
65		elicc2 13385	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
66	64, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
67	59, 66	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
68	67	simp2d 1143	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
69	68	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ≤ 𝑥)
70		df-ne 2941	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝑥 = 𝐴)
71	70	biimpri 227	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝐴 → 𝑥 ≠ 𝐴)
72	71	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ≠ 𝐴)
73	56, 62, 69, 72	leneltd 11364	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 < 𝑥)
74	73	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 < 𝑥)
75		nesym 2997	. . . . . . . . . . . . 13 ⊢ (𝐵 ≠ 𝑥 ↔ ¬ 𝑥 = 𝐵)
76	75	biimpri 227	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝐵 → 𝐵 ≠ 𝑥)
77	76	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ≠ 𝑥)
78	67	simp3d 1144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
79	61, 58, 78	3jca 1128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ≤ 𝐵))
80	79	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → (𝑥 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ≤ 𝐵))
81		leltne 11299	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ≤ 𝐵) → (𝑥 < 𝐵 ↔ 𝐵 ≠ 𝑥))
82	80, 81	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → (𝑥 < 𝐵 ↔ 𝐵 ≠ 𝑥))
83	77, 82	mpbird 256	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵)
84	83	adantlr 713	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵)
85	74, 84	jca 512	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
86		elioo3g 13349	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
87	55, 85, 86	sylanbrc 583	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴(,)𝐵))
88	47, 87	jca 512	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → (𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)))
89	3	ffvelcdmda 7083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
90	20, 89	eqeltrd 2833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
91	88, 90	syl 17	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
92	46, 91	pm2.61dan 811	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
93	35, 92	pm2.61dan 811	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
94		nfv 1917	. . . . 5 ⊢ Ⅎ𝑥𝜑
95		eqid 2732	. . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
96	94, 95, 5, 50, 1, 42, 31	cncfiooicc 44596	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
97		cniccibl 25349	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) ∈ 𝐿¹)
98	5, 50, 96, 97	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) ∈ 𝐿¹)
99	25, 27, 93, 98	iblss 25313	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) ∈ 𝐿¹)
100	23, 99	eqeltrd 2833	1 ⊢ (𝜑 → 𝐹 ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3947 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 (,)cioo 13320 [,]cicc 13323 –cn→ccncf 24383 volcvol 24971 𝐿¹cibl 25125 lim_ℂ climc 25370

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-cn 22722 df-cnp 22723 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128 df-itg2 25129 df-ibl 25130 df-0p 25178 df-limc 25374

This theorem is referenced by: fourierdlem69 44877 fourierdlem73 44881 fourierdlem81 44889 fourierdlem93 44901

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