Theorem itgfsum 24896

Description: Take a finite sum of integrals over the same domain. (Contributed by Mario Carneiro, 24-Aug-2014.)

Hypotheses

Ref	Expression
itgfsum.1	⊢ (𝜑 → 𝐴 ∈ dom vol)
itgfsum.2	⊢ (𝜑 → 𝐵 ∈ Fin)
itgfsum.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
itgfsum.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)

Assertion

Ref	Expression
itgfsum	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥))

Distinct variable groups:   𝑥,𝑘,𝐴   𝐵,𝑘,𝑥   𝜑,𝑘,𝑥

Allowed substitution hints:   𝐶(𝑥,𝑘)   𝑉(𝑥,𝑘)

Proof of Theorem itgfsum

Dummy variables 𝑚 𝑡 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3939	. 2 ⊢ 𝐵 ⊆ 𝐵
2		itgfsum.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 3942	. . . . . 6 ⊢ (𝑡 = ∅ → (𝑡 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		itgz 24850	. . . . . . . 8 ⊢ ∫𝐴0 d𝑥 = 0
5		sumeq1 15328	. . . . . . . . . . 11 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
6		sum0 15361	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
7	5, 6	eqtrdi 2795	. . . . . . . . . 10 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 𝐶 = 0)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝑡 = ∅ ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = 0)
9	8	itgeq2dv 24851	. . . . . . . 8 ⊢ (𝑡 = ∅ → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴0 d𝑥)
10		sumeq1 15328	. . . . . . . . 9 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥)
11		sum0 15361	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥 = 0
12	10, 11	eqtrdi 2795	. . . . . . . 8 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = 0)
13	4, 9, 12	3eqtr4a 2805	. . . . . . 7 ⊢ (𝑡 = ∅ → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥)
14	7	mpteq2dv 5172	. . . . . . . . . 10 ⊢ (𝑡 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ 0))
15		fconstmpt 5640	. . . . . . . . . 10 ⊢ (𝐴 × {0}) = (𝑥 ∈ 𝐴 ↦ 0)
16	14, 15	eqtr4di 2797	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝐴 × {0}))
17	16	eleq1d 2823	. . . . . . . 8 ⊢ (𝑡 = ∅ → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝐴 × {0}) ∈ 𝐿1))
18	17	anbi1d 629	. . . . . . 7 ⊢ (𝑡 = ∅ → (((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥) ↔ ((𝐴 × {0}) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥)))
19	13, 18	mpbiran2d 704	. . . . . 6 ⊢ (𝑡 = ∅ → (((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥) ↔ (𝐴 × {0}) ∈ 𝐿1))
20	3, 19	imbi12d 344	. . . . 5 ⊢ (𝑡 = ∅ → ((𝑡 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥)) ↔ (∅ ⊆ 𝐵 → (𝐴 × {0}) ∈ 𝐿1)))
21	20	imbi2d 340	. . . 4 ⊢ (𝑡 = ∅ → ((𝜑 → (𝑡 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥))) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝐴 × {0}) ∈ 𝐿1))))
22		sseq1 3942	. . . . . 6 ⊢ (𝑡 = 𝑤 → (𝑡 ⊆ 𝐵 ↔ 𝑤 ⊆ 𝐵))
23		sumeq1 15328	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝑤 𝐶)
24	23	mpteq2dv 5172	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶))
25	24	eleq1d 2823	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1))
26	23	adantr 480	. . . . . . . . 9 ⊢ ((𝑡 = 𝑤 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝑤 𝐶)
27	26	itgeq2dv 24851	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥)
28		sumeq1 15328	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)
29	27, 28	eqeq12d 2754	. . . . . . 7 ⊢ (𝑡 = 𝑤 → (∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥))
30	25, 29	anbi12d 630	. . . . . 6 ⊢ (𝑡 = 𝑤 → (((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥) ↔ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)))
31	22, 30	imbi12d 344	. . . . 5 ⊢ (𝑡 = 𝑤 → ((𝑡 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥)) ↔ (𝑤 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥))))
32	31	imbi2d 340	. . . 4 ⊢ (𝑡 = 𝑤 → ((𝜑 → (𝑡 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥))) ↔ (𝜑 → (𝑤 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)))))
33		sseq1 3942	. . . . . 6 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → (𝑡 ⊆ 𝐵 ↔ (𝑤 ∪ {𝑧}) ⊆ 𝐵))
34		sumeq1 15328	. . . . . . . . 9 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
35	34	mpteq2dv 5172	. . . . . . . 8 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶))
36	35	eleq1d 2823	. . . . . . 7 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1))
37	34	adantr 480	. . . . . . . . 9 ⊢ ((𝑡 = (𝑤 ∪ {𝑧}) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
38	37	itgeq2dv 24851	. . . . . . . 8 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥)
39		sumeq1 15328	. . . . . . . 8 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)
40	38, 39	eqeq12d 2754	. . . . . . 7 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → (∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))
41	36, 40	anbi12d 630	. . . . . 6 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → (((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥) ↔ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))
42	33, 41	imbi12d 344	. . . . 5 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → ((𝑡 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥)) ↔ ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))))
43	42	imbi2d 340	. . . 4 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → ((𝜑 → (𝑡 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥))) ↔ (𝜑 → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))))
44		sseq1 3942	. . . . . 6 ⊢ (𝑡 = 𝐵 → (𝑡 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
45		sumeq1 15328	. . . . . . . . 9 ⊢ (𝑡 = 𝐵 → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
46	45	mpteq2dv 5172	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶))
47	46	eleq1d 2823	. . . . . . 7 ⊢ (𝑡 = 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1))
48	45	adantr 480	. . . . . . . . 9 ⊢ ((𝑡 = 𝐵 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
49	48	itgeq2dv 24851	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥)
50		sumeq1 15328	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥)
51	49, 50	eqeq12d 2754	. . . . . . 7 ⊢ (𝑡 = 𝐵 → (∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥))
52	47, 51	anbi12d 630	. . . . . 6 ⊢ (𝑡 = 𝐵 → (((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥) ↔ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥)))
53	44, 52	imbi12d 344	. . . . 5 ⊢ (𝑡 = 𝐵 → ((𝑡 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥)) ↔ (𝐵 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥))))
54	53	imbi2d 340	. . . 4 ⊢ (𝑡 = 𝐵 → ((𝜑 → (𝑡 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥))) ↔ (𝜑 → (𝐵 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥)))))
55		itgfsum.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom vol)
56		ibl0 24856	. . . . . 6 ⊢ (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1)
57	55, 56	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 × {0}) ∈ 𝐿1)
58	57	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐵 → (𝐴 × {0}) ∈ 𝐿1))
59		ssun1 4102	. . . . . . . . . 10 ⊢ 𝑤 ⊆ (𝑤 ∪ {𝑧})
60		sstr 3925	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ (𝑤 ∪ {𝑧}) ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵) → 𝑤 ⊆ 𝐵)
61	59, 60	mpan 686	. . . . . . . . 9 ⊢ ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → 𝑤 ⊆ 𝐵)
62	61	imim1i 63	. . . . . . . 8 ⊢ ((𝑤 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)))
63		nfcv 2906	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚𝐶
64		nfcsb1v 3853	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐶
65		csbeq1a 3842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑘⦌𝐶)
66	63, 64, 65	cbvsumi 15337	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶
67		simprl 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑤)
68		disjsn 4644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑤)
69	67, 68	sylibr 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∩ {𝑧}) = ∅)
70	69	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∩ {𝑧}) = ∅)
71		eqidd 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
72	2	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
73		simprr 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
74	72, 73	ssfid 8971	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ∈ Fin)
75	74	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∪ {𝑧}) ∈ Fin)
76		simplrr 774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
77	76	sselda 3917	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚 ∈ 𝐵)
78		itgfsum.4	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)
79		iblmbf 24837	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
81		itgfsum.3	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
82	81	anass1rs 651	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
83	80, 82	mbfmptcl 24705	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
84	83	an32s 648	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
85	84	ralrimiva 3107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
86	85	adantlr 711	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
87	63	nfel1 2922	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑚 𝐶 ∈ ℂ
88	64	nfel1 2922	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ
89	65	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑚 → (𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ))
90	87, 88, 89	cbvralw 3363	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ ↔ ∀𝑚 ∈ 𝐵 ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
91	86, 90	sylib 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑚 ∈ 𝐵 ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
92	91	r19.21bi 3132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑚 ∈ 𝐵) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
93	77, 92	syldan 590	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
94	70, 71, 75, 93	fsumsplit 15381	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 = (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + Σ𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐶))
95		vex 3426	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
96		csbeq1 3831	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
97	96	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑧 → (⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
98	73	unssbd 4118	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
99	95	snss 4716	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
100	98, 99	sylibr 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
101	100	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑧 ∈ 𝐵)
102	97, 91, 101	rspcdva 3554	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
103	96	sumsn 15386	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ V ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
104	95, 102, 103	sylancr 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
105	104	oveq2d 7271	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + Σ𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐶) = (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
106	94, 105	eqtrd 2778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 = (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
107	66, 106	syl5eq 2791	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
108	107	mpteq2dva 5170	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) = (𝑥 ∈ 𝐴 ↦ (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
109		nfcv 2906	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶)
110		nfcsb1v 3853	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶
111		nfcv 2906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥 +
112		nfcsb1v 3853	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶
113	110, 111, 112	nfov 7285	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
114		csbeq1a 3842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
115		csbeq1a 3842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ⦋𝑧 / 𝑘⦌𝐶 = ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
116	114, 115	oveq12d 7273	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶) = (⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶))
117	109, 113, 116	cbvmpt 5181	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶)) = (𝑦 ∈ 𝐴 ↦ (⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶))
118	108, 117	eqtrdi 2795	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) = (𝑦 ∈ 𝐴 ↦ (⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)))
119	118	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) = (𝑦 ∈ 𝐴 ↦ (⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)))
120		sumex 15327	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 ∈ V
121	120	csbex 5230	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 ∈ V
122	121	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 ∈ V)
123	63, 64, 65	cbvsumi 15337	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶
124	123	mpteq2i 5175	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
125		nfcv 2906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶
126	125, 110, 114	cbvmpt 5181	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
127	124, 126	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
128		simprl 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1)
129	127, 128	eqeltrrid 2844	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
130	102	elexd 3442	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ V)
131	130	ralrimiva 3107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ V)
132	131	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ V)
133		nfv 1918	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦⦋𝑧 / 𝑘⦌𝐶 ∈ V
134	112	nfel1 2922	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V
135	115	eleq1d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (⦋𝑧 / 𝑘⦌𝐶 ∈ V ↔ ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V))
136	133, 134, 135	cbvralw 3363	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ V ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V)
137	132, 136	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V)
138	137	r19.21bi 3132	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V)
139		nfcv 2906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦⦋𝑧 / 𝑘⦌𝐶
140	139, 112, 115	cbvmpt 5181	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
141	96	mpteq2dv 5172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 → (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
142	141	eleq1d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1))
143	78	ralrimiva 3107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)
144		nfv 1918	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑚(𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1
145		nfcv 2906	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘𝐴
146	145, 64	nfmpt 5177	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶)
147	146	nfel1 2922	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1
148	65	mpteq2dv 5172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑚 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶))
149	148	eleq1d 2823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑚 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1))
150	144, 147, 149	cbvralw 3363	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔ ∀𝑚 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
151	143, 150	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑚 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
152	151	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑚 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
153	142, 152, 100	rspcdva 3554	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1)
154	140, 153	eqeltrrid 2844	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1)
155	154	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1)
156	122, 129, 138, 155	ibladd 24890	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → (𝑦 ∈ 𝐴 ↦ (⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)) ∈ 𝐿1)
157	119, 156	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1)
158	122, 129, 138, 155	itgadd 24894	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∫𝐴(⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶) d𝑦 = (∫𝐴⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑦 + ∫𝐴⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 d𝑦))
159	116, 109, 113	cbvitg 24845	. . . . . . . . . . . . . . 15 ⊢ ∫𝐴(Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶) d𝑥 = ∫𝐴(⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶) d𝑦
160	114, 125, 110	cbvitg 24845	. . . . . . . . . . . . . . . 16 ⊢ ∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑦
161	115, 139, 112	cbvitg 24845	. . . . . . . . . . . . . . . 16 ⊢ ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 d𝑦
162	160, 161	oveq12i 7267	. . . . . . . . . . . . . . 15 ⊢ (∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥 + ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥) = (∫𝐴⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑦 + ∫𝐴⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 d𝑦)
163	158, 159, 162	3eqtr4g 2804	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∫𝐴(Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶) d𝑥 = (∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥 + ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥))
164	106	itgeq2dv 24851	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴(Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶) d𝑥)
165	164	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴(Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶) d𝑥)
166		eqidd 2739	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
167	73	sselda 3917	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚 ∈ 𝐵)
168	92	an32s 648	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
169	152	r19.21bi 3132	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
170	168, 169	itgcl 24853	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ 𝐵) → ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
171	167, 170	syldan 590	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
172	69, 166, 74, 171	fsumsplit 15381	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 = (Σ𝑚 ∈ 𝑤 ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 + Σ𝑚 ∈ {𝑧}∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥))
173	172	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 = (Σ𝑚 ∈ 𝑤 ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 + Σ𝑚 ∈ {𝑧}∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥))
174		simprr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)
175		itgeq2 24847	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 → ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥)
176	123	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐴 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
177	175, 176	mprg 3077	. . . . . . . . . . . . . . . . 17 ⊢ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥
178		nfcv 2906	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚∫𝐴𝐶 d𝑥
179	145, 64	nfitg 24844	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥
180	65	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 = 𝑚 ∧ 𝑥 ∈ 𝐴) → 𝐶 = ⦋𝑚 / 𝑘⦌𝐶)
181	180	itgeq2dv 24851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → ∫𝐴𝐶 d𝑥 = ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥)
182	178, 179, 181	cbvsumi 15337	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥 = Σ𝑚 ∈ 𝑤 ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥
183	174, 177, 182	3eqtr3g 2802	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥 = Σ𝑚 ∈ 𝑤 ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥)
184	102, 153	itgcl 24853	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
185	184	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
186	96	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = 𝑧 ∧ 𝑥 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
187	186	itgeq2dv 24851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑧 → ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥)
188	187	sumsn 15386	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ V ∧ ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥 ∈ ℂ) → Σ𝑚 ∈ {𝑧}∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥)
189	95, 185, 188	sylancr 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → Σ𝑚 ∈ {𝑧}∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥)
190	189	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥 = Σ𝑚 ∈ {𝑧}∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥)
191	183, 190	oveq12d 7273	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → (∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥 + ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥) = (Σ𝑚 ∈ 𝑤 ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 + Σ𝑚 ∈ {𝑧}∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥))
192	173, 191	eqtr4d 2781	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 = (∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥 + ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥))
193	163, 165, 192	3eqtr4d 2788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 d𝑥 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥)
194		itgeq2 24847	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 → ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 d𝑥)
195	66	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 → Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶)
196	194, 195	mprg 3077	. . . . . . . . . . . . 13 ⊢ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 d𝑥
197	178, 179, 181	cbvsumi 15337	. . . . . . . . . . . . 13 ⊢ Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥
198	193, 196, 197	3eqtr4g 2804	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)
199	157, 198	jca 511	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))
200	199	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))
201	200	expr 456	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑤) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → (((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))))
202	201	a2d 29	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑤) → (((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))))
203	62, 202	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑤) → ((𝑤 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))))
204	203	expcom 413	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑤 → (𝜑 → ((𝑤 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))))
205	204	adantl 481	. . . . 5 ⊢ ((𝑤 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑤) → (𝜑 → ((𝑤 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))))
206	205	a2d 29	. . . 4 ⊢ ((𝑤 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑤) → ((𝜑 → (𝑤 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥))) → (𝜑 → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))))
207	21, 32, 43, 54, 58, 206	findcard2s 8910	. . 3 ⊢ (𝐵 ∈ Fin → (𝜑 → (𝐵 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥))))
208	2, 207	mpcom 38	. 2 ⊢ (𝜑 → (𝐵 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥)))
209	1, 208	mpi 20	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422  ⦋csb 3828   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558   ↦ cmpt 5153   × cxp 5578  dom cdm 5580  (class class class)co 7255  Fincfn 8691  ℂcc 10800  0cc0 10802   + caddc 10805  Σcsu 15325  volcvol 24532  MblFncmbf 24683  𝐿¹cibl 24686  ∫citg 24687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cc 10122  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-omul 8272  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-acn 9631  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ioc 13013  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-sum 15326  df-rest 17050  df-topgen 17071  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-top 21951  df-topon 21968  df-bases 22004  df-cmp 22446  df-ovol 24533  df-vol 24534  df-mbf 24688  df-itg1 24689  df-itg2 24690  df-ibl 24691  df-itg 24692  df-0p 24739

This theorem is referenced by:  circlemeth  32520  3factsumint1  39957  fourierdlem83  43620