Theorem itgfsum 25869

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 3958	. 2 ⊢ 𝐵 ⊆ 𝐵
2		itgfsum.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 3961	. . . . . 6 ⊢ (𝑡 = ∅ → (𝑡 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		itgz 25823	. . . . . . . 8 ⊢ ∫𝐴0 d𝑥 = 0
5		sumeq1 15699	. . . . . . . . . . 11 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
6		sum0 15731	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
7	5, 6	eqtrdi 2812	. . . . . . . . . 10 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 𝐶 = 0)
8	7	adantr 484	. . . . . . . . 9 ⊢ ((𝑡 = ∅ ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = 0)
9	8	itgeq2dv 25824	. . . . . . . 8 ⊢ (𝑡 = ∅ → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴0 d𝑥)
10		sumeq1 15699	. . . . . . . . 9 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥)
11		sum0 15731	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥 = 0
12	10, 11	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = 0)
13	4, 9, 12	3eqtr4a 2822	. . . . . . 7 ⊢ (𝑡 = ∅ → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥)
14	7	mpteq2dv 5193	. . . . . . . . . 10 ⊢ (𝑡 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ 0))
15		fconstmpt 5707	. . . . . . . . . 10 ⊢ (𝐴 × {0}) = (𝑥 ∈ 𝐴 ↦ 0)
16	14, 15	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝐴 × {0}))
17	16	eleq1d 2846	. . . . . . . 8 ⊢ (𝑡 = ∅ → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝐴 × {0}) ∈ 𝐿1))
18	17	anbi1d 640	. . . . . . 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 718	. . . . . 6 ⊢ (𝑡 = ∅ → (((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥) ↔ (𝐴 × {0}) ∈ 𝐿1))
20	3, 19	imbi12d 346	. . . . 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 342	. . . 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 3961	. . . . . 6 ⊢ (𝑡 = 𝑤 → (𝑡 ⊆ 𝐵 ↔ 𝑤 ⊆ 𝐵))
23		sumeq1 15699	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝑤 𝐶)
24	23	mpteq2dv 5193	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶))
25	24	eleq1d 2846	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1))
26	23	adantr 484	. . . . . . . . 9 ⊢ ((𝑡 = 𝑤 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝑤 𝐶)
27	26	itgeq2dv 25824	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥)
28		sumeq1 15699	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)
29	27, 28	eqeq12d 2777	. . . . . . 7 ⊢ (𝑡 = 𝑤 → (∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥))
30	25, 29	anbi12d 641	. . . . . 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 346	. . . . 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 342	. . . 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 3961	. . . . . 6 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → (𝑡 ⊆ 𝐵 ↔ (𝑤 ∪ {𝑧}) ⊆ 𝐵))
34		sumeq1 15699	. . . . . . . . 9 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
35	34	mpteq2dv 5193	. . . . . . . 8 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶))
36	35	eleq1d 2846	. . . . . . 7 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1))
37	34	adantr 484	. . . . . . . . 9 ⊢ ((𝑡 = (𝑤 ∪ {𝑧}) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
38	37	itgeq2dv 25824	. . . . . . . 8 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥)
39		sumeq1 15699	. . . . . . . 8 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)
40	38, 39	eqeq12d 2777	. . . . . . 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 641	. . . . . 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 346	. . . . 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 342	. . . 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 3961	. . . . . 6 ⊢ (𝑡 = 𝐵 → (𝑡 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
45		sumeq1 15699	. . . . . . . . 9 ⊢ (𝑡 = 𝐵 → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
46	45	mpteq2dv 5193	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶))
47	46	eleq1d 2846	. . . . . . 7 ⊢ (𝑡 = 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1))
48	45	adantr 484	. . . . . . . . 9 ⊢ ((𝑡 = 𝐵 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
49	48	itgeq2dv 25824	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥)
50		sumeq1 15699	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥)
51	49, 50	eqeq12d 2777	. . . . . . 7 ⊢ (𝑡 = 𝐵 → (∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥))
52	47, 51	anbi12d 641	. . . . . 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 346	. . . . 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 342	. . . 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 25829	. . . . . 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 4130	. . . . . . . . . 10 ⊢ 𝑤 ⊆ (𝑤 ∪ {𝑧})
60		sstr 3944	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ (𝑤 ∪ {𝑧}) ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵) → 𝑤 ⊆ 𝐵)
61	59, 60	mpan 700	. . . . . . . . 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		csbeq1a 3866	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑘⦌𝐶)
64		nfcv 2923	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚𝐶
65		nfcsb1v 3876	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐶
66	63, 64, 65	cbvsum 15705	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶
67		simprl 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑤)
68		disjsn 4669	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑤)
69	67, 68	sylibr 236	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∩ {𝑧}) = ∅)
70	69	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∩ {𝑧}) = ∅)
71		eqidd 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
72	2	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
73		simprr 782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
74	72, 73	ssfid 9209	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ∈ Fin)
75	74	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∪ {𝑧}) ∈ Fin)
76		simplrr 787	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
77	76	sselda 3936	. . . . . . . . . . . . . . . . . . . 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 25809	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 665	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
83	80, 82	mbfmptcl 25678	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
84	83	an32s 662	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
85	84	ralrimiva 3153	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
86	85	adantlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
87	64	nfel1 2939	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑚 𝐶 ∈ ℂ
88	65	nfel1 2939	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ
89	63	eleq1d 2846	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑚 → (𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ))
90	87, 88, 89	cbvralw 3303	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ ↔ ∀𝑚 ∈ 𝐵 ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
91	86, 90	sylib 220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑚 ∈ 𝐵 ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
92	91	r19.21bi 3253	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑚 ∈ 𝐵) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
93	77, 92	syldan 600	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
94	70, 71, 75, 93	fsumsplit 15751	. . . . . . . . . . . . . . . . . 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 3457	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
96		csbeq1 3855	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
97	96	eleq1d 2846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑧 → (⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
98	73	unssbd 4146	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
99	95	snss 4742	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
100	98, 99	sylibr 236	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
101	100	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑧 ∈ 𝐵)
102	97, 91, 101	rspcdva 3582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
103	96	sumsn 15756	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ V ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
104	95, 102, 103	sylancr 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
105	104	oveq2d 7408	. . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . 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	eqtrid 2808	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
108	107	mpteq2dva 5192	. . . . . . . . . . . . . . 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 2923	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶)
110		nfcsb1v 3876	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶
111		nfcv 2923	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥 +
112		nfcsb1v 3876	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶
113	110, 111, 112	nfov 7422	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
114		csbeq1a 3866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
115		csbeq1a 3866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ⦋𝑧 / 𝑘⦌𝐶 = ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
116	114, 115	oveq12d 7410	. . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 15698	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 ∈ V
121	120	csbex 5260	. . . . . . . . . . . . . . 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	cbvsum 15705	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶
124	123	mpteq2i 5195	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
125		nfcv 2923	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶
126	125, 110, 114	cbvmpt 5201	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
127	124, 126	eqtri 2784	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
128		simprl 780	. . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . 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 3476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ V)
131	130	ralrimiva 3153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ V)
132	131	adantr 484	. . . . . . . . . . . . . . . 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 1933	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦⦋𝑧 / 𝑘⦌𝐶 ∈ V
134	112	nfel1 2939	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V
135	115	eleq1d 2846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (⦋𝑧 / 𝑘⦌𝐶 ∈ V ↔ ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V))
136	133, 134, 135	cbvralw 3303	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ V ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V)
137	132, 136	sylib 220	. . . . . . . . . . . . . . 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 3253	. . . . . . . . . . . . . 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 2923	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦⦋𝑧 / 𝑘⦌𝐶
140	139, 112, 115	cbvmpt 5201	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
141	96	mpteq2dv 5193	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 → (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
142	141	eleq1d 2846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1))
143	78	ralrimiva 3153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)
144		nfv 1933	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑚(𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1
145		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘𝐴
146	145, 65	nfmpt 5197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶)
147	146	nfel1 2939	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1
148	63	mpteq2dv 5193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑚 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶))
149	148	eleq1d 2846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑚 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1))
150	144, 147, 149	cbvralw 3303	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔ ∀𝑚 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
151	143, 150	sylib 220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑚 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
152	151	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑚 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
153	142, 152, 100	rspcdva 3582	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1)
154	140, 153	eqeltrrid 2866	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1)
155	154	adantr 484	. . . . . . . . . . . . . 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 25863	. . . . . . . . . . . . 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 2861	. . . . . . . . . . . 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 25867	. . . . . . . . . . . . . . 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 25818	. . . . . . . . . . . . . . 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 25818	. . . . . . . . . . . . . . . 16 ⊢ ∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑦
161	115, 139, 112	cbvitg 25818	. . . . . . . . . . . . . . . 16 ⊢ ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 d𝑦
162	160, 161	oveq12i 7404	. . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . 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 25824	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
167	73	sselda 3936	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚 ∈ 𝐵)
168	92	an32s 662	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
169	152	r19.21bi 3253	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
170	168, 169	itgcl 25826	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ 𝐵) → ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
171	167, 170	syldan 600	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
172	69, 166, 74, 171	fsumsplit 15751	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . 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 25820	. . . . . . . . . . . . . . . . . 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 3081	. . . . . . . . . . . . . . . . 17 ⊢ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥
178	63	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 = 𝑚 ∧ 𝑥 ∈ 𝐴) → 𝐶 = ⦋𝑚 / 𝑘⦌𝐶)
179	178	itgeq2dv 25824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → ∫𝐴𝐶 d𝑥 = ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥)
180		nfcv 2923	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚∫𝐴𝐶 d𝑥
181	145, 65	nfitg 25817	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥
182	179, 180, 181	cbvsum 15705	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥 = Σ𝑚 ∈ 𝑤 ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥
183	174, 177, 182	3eqtr3g 2819	. . . . . . . . . . . . . . . 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 25826	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
185	184	adantr 484	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = 𝑧 ∧ 𝑥 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
187	186	itgeq2dv 25824	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑧 → ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥)
188	187	sumsn 15756	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . 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 7410	. . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . 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 2806	. . . . . . . . . . . . 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 25820	. . . . . . . . . . . . . 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 3081	. . . . . . . . . . . . 13 ⊢ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 d𝑥
197	179, 180, 181	cbvsum 15705	. . . . . . . . . . . . 13 ⊢ Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥
198	193, 196, 197	3eqtr4g 2821	. . . . . . . . . . . 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 519	. . . . . . . . . . 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 416	. . . . . . . . . 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 460	. . . . . . . . 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 417	. . . . . 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 485	. . . . 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 9130	. . 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 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453  ⦋csb 3852   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4581   ↦ cmpt 5180   × cxp 5643  dom cdm 5645  (class class class)co 7392  Fincfn 8923  ℂcc 11068  0cc0 11070   + caddc 11073  Σcsu 15696  volcvol 25505  MblFncmbf 25656  𝐿¹cibl 25659  ∫citg 25660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-cc 10389  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148  ax-addf 11149

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-disj 5067  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-ofr 7657  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-omul 8437  df-er 8673  df-map 8805  df-pm 8806  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-fi 9354  df-sup 9385  df-inf 9386  df-oi 9455  df-dju 9856  df-card 9894  df-acn 9897  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-n0 12479  df-z 12566  df-uz 12837  df-q 12947  df-rp 12991  df-xneg 13111  df-xadd 13112  df-xmul 13113  df-ioo 13350  df-ioc 13351  df-ico 13352  df-icc 13353  df-fz 13510  df-fzo 13657  df-fl 13799  df-mod 13877  df-seq 14012  df-exp 14072  df-hash 14341  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-clim 15498  df-rlim 15499  df-sum 15697  df-rest 17434  df-topgen 17455  df-psmet 21396  df-xmet 21397  df-met 21398  df-bl 21399  df-mopn 21400  df-top 22934  df-topon 22951  df-bases 22986  df-cmp 23427  df-ovol 25506  df-vol 25507  df-mbf 25661  df-itg1 25662  df-itg2 25663  df-ibl 25664  df-itg 25665  df-0p 25712

This theorem is referenced by:  circlemeth  34898  3factsumint1  42602  fourierdlem83  46727