Theorem itgfsum 25275

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 4001	. 2 ⊢ 𝐵 ⊆ 𝐵
2		itgfsum.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 4004	. . . . . 6 ⊢ (𝑡 = ∅ → (𝑡 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		itgz 25229	. . . . . . . 8 ⊢ ∫𝐴0 d𝑥 = 0
5		sumeq1 15619	. . . . . . . . . . 11 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
6		sum0 15651	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
7	5, 6	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 𝐶 = 0)
8	7	adantr 481	. . . . . . . . 9 ⊢ ((𝑡 = ∅ ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = 0)
9	8	itgeq2dv 25230	. . . . . . . 8 ⊢ (𝑡 = ∅ → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴0 d𝑥)
10		sumeq1 15619	. . . . . . . . 9 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥)
11		sum0 15651	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥 = 0
12	10, 11	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑡 = ∅ → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = 0)
13	4, 9, 12	3eqtr4a 2798	. . . . . . 7 ⊢ (𝑡 = ∅ → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥)
14	7	mpteq2dv 5244	. . . . . . . . . 10 ⊢ (𝑡 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ 0))
15		fconstmpt 5731	. . . . . . . . . 10 ⊢ (𝐴 × {0}) = (𝑥 ∈ 𝐴 ↦ 0)
16	14, 15	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝐴 × {0}))
17	16	eleq1d 2818	. . . . . . . 8 ⊢ (𝑡 = ∅ → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝐴 × {0}) ∈ 𝐿1))
18	17	anbi1d 630	. . . . . . 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 706	. . . . . 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 4004	. . . . . 6 ⊢ (𝑡 = 𝑤 → (𝑡 ⊆ 𝐵 ↔ 𝑤 ⊆ 𝐵))
23		sumeq1 15619	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝑤 𝐶)
24	23	mpteq2dv 5244	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶))
25	24	eleq1d 2818	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ∈ 𝐿1))
26	23	adantr 481	. . . . . . . . 9 ⊢ ((𝑡 = 𝑤 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝑤 𝐶)
27	26	itgeq2dv 25230	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥)
28		sumeq1 15619	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥)
29	27, 28	eqeq12d 2748	. . . . . . 7 ⊢ (𝑡 = 𝑤 → (∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥))
30	25, 29	anbi12d 631	. . . . . 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 4004	. . . . . 6 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → (𝑡 ⊆ 𝐵 ↔ (𝑤 ∪ {𝑧}) ⊆ 𝐵))
34		sumeq1 15619	. . . . . . . . 9 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
35	34	mpteq2dv 5244	. . . . . . . 8 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶))
36	35	eleq1d 2818	. . . . . . 7 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1))
37	34	adantr 481	. . . . . . . . 9 ⊢ ((𝑡 = (𝑤 ∪ {𝑧}) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
38	37	itgeq2dv 25230	. . . . . . . 8 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥)
39		sumeq1 15619	. . . . . . . 8 ⊢ (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)
40	38, 39	eqeq12d 2748	. . . . . . 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 631	. . . . . 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 4004	. . . . . 6 ⊢ (𝑡 = 𝐵 → (𝑡 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
45		sumeq1 15619	. . . . . . . . 9 ⊢ (𝑡 = 𝐵 → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
46	45	mpteq2dv 5244	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶))
47	46	eleq1d 2818	. . . . . . 7 ⊢ (𝑡 = 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ∈ 𝐿1))
48	45	adantr 481	. . . . . . . . 9 ⊢ ((𝑡 = 𝐵 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝑡 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
49	48	itgeq2dv 25230	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → ∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥)
50		sumeq1 15619	. . . . . . . 8 ⊢ (𝑡 = 𝐵 → Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥)
51	49, 50	eqeq12d 2748	. . . . . . 7 ⊢ (𝑡 = 𝐵 → (∫𝐴Σ𝑘 ∈ 𝑡 𝐶 d𝑥 = Σ𝑘 ∈ 𝑡 ∫𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘 ∈ 𝐵 𝐶 d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴𝐶 d𝑥))
52	47, 51	anbi12d 631	. . . . . 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 25235	. . . . . 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 4169	. . . . . . . . . 10 ⊢ 𝑤 ⊆ (𝑤 ∪ {𝑧})
60		sstr 3987	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ (𝑤 ∪ {𝑧}) ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵) → 𝑤 ⊆ 𝐵)
61	59, 60	mpan 688	. . . . . . . . 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 2903	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚𝐶
64		nfcsb1v 3915	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐶
65		csbeq1a 3904	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑘⦌𝐶)
66	63, 64, 65	cbvsumi 15627	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶
67		simprl 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑤)
68		disjsn 4709	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑤)
69	67, 68	sylibr 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∩ {𝑧}) = ∅)
70	69	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∩ {𝑧}) = ∅)
71		eqidd 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
72	2	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
73		simprr 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
74	72, 73	ssfid 9252	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ∈ Fin)
75	74	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∪ {𝑧}) ∈ Fin)
76		simplrr 776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
77	76	sselda 3979	. . . . . . . . . . . . . . . . . . . 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 25216	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 653	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
83	80, 82	mbfmptcl 25084	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
84	83	an32s 650	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
85	84	ralrimiva 3146	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
86	85	adantlr 713	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
87	63	nfel1 2919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑚 𝐶 ∈ ℂ
88	64	nfel1 2919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ
89	65	eleq1d 2818	. . . . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑚 ∈ 𝐵 ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
92	91	r19.21bi 3248	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑚 ∈ 𝐵) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
93	77, 92	syldan 591	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
94	70, 71, 75, 93	fsumsplit 15671	. . . . . . . . . . . . . . . . . 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 3478	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑧 ∈ V
96		csbeq1 3893	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
97	96	eleq1d 2818	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑧 → (⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
98	73	unssbd 4185	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
99	95	snss 4783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
100	98, 99	sylibr 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
101	100	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑧 ∈ 𝐵)
102	97, 91, 101	rspcdva 3611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
103	96	sumsn 15676	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ V ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
104	95, 102, 103	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
105	104	oveq2d 7410	. . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = (Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
108	107	mpteq2dva 5242	. . . . . . . . . . . . . . 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 2903	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑧 / 𝑘⦌𝐶)
110		nfcsb1v 3915	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶
111		nfcv 2903	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥 +
112		nfcsb1v 3915	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶
113	110, 111, 112	nfov 7424	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 + ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
114		csbeq1a 3904	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
115		csbeq1a 3904	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ⦋𝑧 / 𝑘⦌𝐶 = ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
116	114, 115	oveq12d 7412	. . . . . . . . . . . . . . . 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 5253	. . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 15618	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 ∈ V
121	120	csbex 5305	. . . . . . . . . . . . . . 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 15627	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶
124	123	mpteq2i 5247	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
125		nfcv 2903	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶
126	125, 110, 114	cbvmpt 5253	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
127	124, 126	eqtri 2760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶)
128		simprl 769	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . 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 3494	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ V)
131	130	ralrimiva 3146	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ V)
132	131	adantr 481	. . . . . . . . . . . . . . . 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 1917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦⦋𝑧 / 𝑘⦌𝐶 ∈ V
134	112	nfel1 2919	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V
135	115	eleq1d 2818	. . . . . . . . . . . . . . . . 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 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 3248	. . . . . . . . . . . . . 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 2903	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦⦋𝑧 / 𝑘⦌𝐶
140	139, 112, 115	cbvmpt 5253	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
141	96	mpteq2dv 5244	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 → (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
142	141	eleq1d 2818	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1))
143	78	ralrimiva 3146	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)
144		nfv 1917	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑚(𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1
145		nfcv 2903	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘𝐴
146	145, 64	nfmpt 5249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶)
147	146	nfel1 2919	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1
148	65	mpteq2dv 5244	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑚 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶))
149	148	eleq1d 2818	. . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑚 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
152	151	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑚 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
153	142, 152, 100	rspcdva 3611	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1)
154	140, 153	eqeltrrid 2838	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶) ∈ 𝐿1)
155	154	adantr 481	. . . . . . . . . . . . . 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 25269	. . . . . . . . . . . . 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 2833	. . . . . . . . . . . 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 25273	. . . . . . . . . . . . . . 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 25224	. . . . . . . . . . . . . . 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 25224	. . . . . . . . . . . . . . . 16 ⊢ ∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑦 / 𝑥⦌Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑦
161	115, 139, 112	cbvitg 25224	. . . . . . . . . . . . . . . 16 ⊢ ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑦 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 d𝑦
162	160, 161	oveq12i 7406	. . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . 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 25230	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
167	73	sselda 3979	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚 ∈ 𝐵)
168	92	an32s 650	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐶 ∈ ℂ)
169	152	r19.21bi 3248	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ ⦋𝑚 / 𝑘⦌𝐶) ∈ 𝐿1)
170	168, 169	itgcl 25232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ 𝐵) → ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
171	167, 170	syldan 591	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
172	69, 166, 74, 171	fsumsplit 15671	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . 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 25226	. . . . . . . . . . . . . . . . . 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 3067	. . . . . . . . . . . . . . . . 17 ⊢ ∫𝐴Σ𝑘 ∈ 𝑤 𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ 𝑤 ⦋𝑚 / 𝑘⦌𝐶 d𝑥
178		nfcv 2903	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚∫𝐴𝐶 d𝑥
179	145, 64	nfitg 25223	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥
180	65	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 = 𝑚 ∧ 𝑥 ∈ 𝐴) → 𝐶 = ⦋𝑚 / 𝑘⦌𝐶)
181	180	itgeq2dv 25230	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → ∫𝐴𝐶 d𝑥 = ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥)
182	178, 179, 181	cbvsumi 15627	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ 𝑤 ∫𝐴𝐶 d𝑥 = Σ𝑚 ∈ 𝑤 ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥
183	174, 177, 182	3eqtr3g 2795	. . . . . . . . . . . . . . . 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 25232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥 ∈ ℂ)
185	184	adantr 481	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = 𝑧 ∧ 𝑥 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
187	186	itgeq2dv 25230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑧 → ∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥 = ∫𝐴⦋𝑧 / 𝑘⦌𝐶 d𝑥)
188	187	sumsn 15676	. . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . 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 25226	. . . . . . . . . . . . . 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 3067	. . . . . . . . . . . . 13 ⊢ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐶 d𝑥
197	178, 179, 181	cbvsumi 15627	. . . . . . . . . . . . 13 ⊢ Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴⦋𝑚 / 𝑘⦌𝐶 d𝑥
198	193, 196, 197	3eqtr4g 2797	. . . . . . . . . . . 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 512	. . . . . . . . . . 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 413	. . . . . . . . . 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 457	. . . . . . . . 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 414	. . . . . 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 482	. . . . 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 9150	. . 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474  ⦋csb 3890   ∪ cun 3943   ∩ cin 3944   ⊆ wss 3945  ∅c0 4319  {csn 4623   ↦ cmpt 5225   × cxp 5668  dom cdm 5670  (class class class)co 7394  Fincfn 8924  ℂcc 11092  0cc0 11094   + caddc 11097  Σcsu 15616  volcvol 24911  MblFncmbf 25062  𝐿¹cibl 25065  ∫citg 25066

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 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-inf2 9620  ax-cc 10414  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171  ax-pre-sup 11172  ax-addf 11173

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 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-of 7654  df-ofr 7655  df-om 7840  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-2o 8451  df-oadd 8454  df-omul 8455  df-er 8688  df-map 8807  df-pm 8808  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-fi 9390  df-sup 9421  df-inf 9422  df-oi 9489  df-dju 9880  df-card 9918  df-acn 9921  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-div 11856  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-n0 12457  df-z 12543  df-uz 12807  df-q 12917  df-rp 12959  df-xneg 13076  df-xadd 13077  df-xmul 13078  df-ioo 13312  df-ioc 13313  df-ico 13314  df-icc 13315  df-fz 13469  df-fzo 13612  df-fl 13741  df-mod 13819  df-seq 13951  df-exp 14012  df-hash 14275  df-cj 15030  df-re 15031  df-im 15032  df-sqrt 15166  df-abs 15167  df-clim 15416  df-rlim 15417  df-sum 15617  df-rest 17352  df-topgen 17373  df-psmet 20872  df-xmet 20873  df-met 20874  df-bl 20875  df-mopn 20876  df-top 22327  df-topon 22344  df-bases 22380  df-cmp 22822  df-ovol 24912  df-vol 24913  df-mbf 25067  df-itg1 25068  df-itg2 25069  df-ibl 25070  df-itg 25071  df-0p 25118

This theorem is referenced by:  circlemeth  33547  3factsumint1  40755  fourierdlem83  44742