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Theorem itgfsum 24354
Description: Take a finite sum of integrals over the same domain. (Contributed by Mario Carneiro, 24-Aug-2014.)
Hypotheses
Ref Expression
itgfsum.1 (𝜑𝐴 ∈ dom vol)
itgfsum.2 (𝜑𝐵 ∈ Fin)
itgfsum.3 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
itgfsum.4 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ 𝐿1)
Assertion
Ref Expression
itgfsum (𝜑 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 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.
StepHypRef Expression
1 ssid 3986 . 2 𝐵𝐵
2 itgfsum.2 . . 3 (𝜑𝐵 ∈ Fin)
3 sseq1 3989 . . . . . 6 (𝑡 = ∅ → (𝑡𝐵 ↔ ∅ ⊆ 𝐵))
4 itgz 24308 . . . . . . . 8 𝐴0 d𝑥 = 0
5 sumeq1 15033 . . . . . . . . . . 11 (𝑡 = ∅ → Σ𝑘𝑡 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
6 sum0 15066 . . . . . . . . . . 11 Σ𝑘 ∈ ∅ 𝐶 = 0
75, 6syl6eq 2869 . . . . . . . . . 10 (𝑡 = ∅ → Σ𝑘𝑡 𝐶 = 0)
87adantr 481 . . . . . . . . 9 ((𝑡 = ∅ ∧ 𝑥𝐴) → Σ𝑘𝑡 𝐶 = 0)
98itgeq2dv 24309 . . . . . . . 8 (𝑡 = ∅ → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = ∫𝐴0 d𝑥)
10 sumeq1 15033 . . . . . . . . 9 (𝑡 = ∅ → Σ𝑘𝑡𝐴𝐶 d𝑥 = Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥)
11 sum0 15066 . . . . . . . . 9 Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥 = 0
1210, 11syl6eq 2869 . . . . . . . 8 (𝑡 = ∅ → Σ𝑘𝑡𝐴𝐶 d𝑥 = 0)
134, 9, 123eqtr4a 2879 . . . . . . 7 (𝑡 = ∅ → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)
147mpteq2dv 5153 . . . . . . . . . 10 (𝑡 = ∅ → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝑥𝐴 ↦ 0))
15 fconstmpt 5607 . . . . . . . . . 10 (𝐴 × {0}) = (𝑥𝐴 ↦ 0)
1614, 15syl6eqr 2871 . . . . . . . . 9 (𝑡 = ∅ → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝐴 × {0}))
1716eleq1d 2894 . . . . . . . 8 (𝑡 = ∅ → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ↔ (𝐴 × {0}) ∈ 𝐿1))
1817anbi1d 629 . . . . . . 7 (𝑡 = ∅ → (((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥) ↔ ((𝐴 × {0}) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)))
1913, 18mpbiran2d 704 . . . . . 6 (𝑡 = ∅ → (((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥) ↔ (𝐴 × {0}) ∈ 𝐿1))
203, 19imbi12d 346 . . . . 5 (𝑡 = ∅ → ((𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)) ↔ (∅ ⊆ 𝐵 → (𝐴 × {0}) ∈ 𝐿1)))
2120imbi2d 342 . . . 4 (𝑡 = ∅ → ((𝜑 → (𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥))) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝐴 × {0}) ∈ 𝐿1))))
22 sseq1 3989 . . . . . 6 (𝑡 = 𝑤 → (𝑡𝐵𝑤𝐵))
23 sumeq1 15033 . . . . . . . . 9 (𝑡 = 𝑤 → Σ𝑘𝑡 𝐶 = Σ𝑘𝑤 𝐶)
2423mpteq2dv 5153 . . . . . . . 8 (𝑡 = 𝑤 → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶))
2524eleq1d 2894 . . . . . . 7 (𝑡 = 𝑤 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1))
2623adantr 481 . . . . . . . . 9 ((𝑡 = 𝑤𝑥𝐴) → Σ𝑘𝑡 𝐶 = Σ𝑘𝑤 𝐶)
2726itgeq2dv 24309 . . . . . . . 8 (𝑡 = 𝑤 → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘𝑤 𝐶 d𝑥)
28 sumeq1 15033 . . . . . . . 8 (𝑡 = 𝑤 → Σ𝑘𝑡𝐴𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)
2927, 28eqeq12d 2834 . . . . . . 7 (𝑡 = 𝑤 → (∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥))
3025, 29anbi12d 630 . . . . . 6 (𝑡 = 𝑤 → (((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥) ↔ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)))
3122, 30imbi12d 346 . . . . 5 (𝑡 = 𝑤 → ((𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)) ↔ (𝑤𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥))))
3231imbi2d 342 . . . 4 (𝑡 = 𝑤 → ((𝜑 → (𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥))) ↔ (𝜑 → (𝑤𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)))))
33 sseq1 3989 . . . . . 6 (𝑡 = (𝑤 ∪ {𝑧}) → (𝑡𝐵 ↔ (𝑤 ∪ {𝑧}) ⊆ 𝐵))
34 sumeq1 15033 . . . . . . . . 9 (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
3534mpteq2dv 5153 . . . . . . . 8 (𝑡 = (𝑤 ∪ {𝑧}) → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶))
3635eleq1d 2894 . . . . . . 7 (𝑡 = (𝑤 ∪ {𝑧}) → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1))
3734adantr 481 . . . . . . . . 9 ((𝑡 = (𝑤 ∪ {𝑧}) ∧ 𝑥𝐴) → Σ𝑘𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
3837itgeq2dv 24309 . . . . . . . 8 (𝑡 = (𝑤 ∪ {𝑧}) → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥)
39 sumeq1 15033 . . . . . . . 8 (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘𝑡𝐴𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)
4038, 39eqeq12d 2834 . . . . . . 7 (𝑡 = (𝑤 ∪ {𝑧}) → (∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))
4136, 40anbi12d 630 . . . . . 6 (𝑡 = (𝑤 ∪ {𝑧}) → (((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥) ↔ ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))
4233, 41imbi12d 346 . . . . 5 (𝑡 = (𝑤 ∪ {𝑧}) → ((𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)) ↔ ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))))
4342imbi2d 342 . . . 4 (𝑡 = (𝑤 ∪ {𝑧}) → ((𝜑 → (𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥))) ↔ (𝜑 → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))))
44 sseq1 3989 . . . . . 6 (𝑡 = 𝐵 → (𝑡𝐵𝐵𝐵))
45 sumeq1 15033 . . . . . . . . 9 (𝑡 = 𝐵 → Σ𝑘𝑡 𝐶 = Σ𝑘𝐵 𝐶)
4645mpteq2dv 5153 . . . . . . . 8 (𝑡 = 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶))
4746eleq1d 2894 . . . . . . 7 (𝑡 = 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1))
4845adantr 481 . . . . . . . . 9 ((𝑡 = 𝐵𝑥𝐴) → Σ𝑘𝑡 𝐶 = Σ𝑘𝐵 𝐶)
4948itgeq2dv 24309 . . . . . . . 8 (𝑡 = 𝐵 → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘𝐵 𝐶 d𝑥)
50 sumeq1 15033 . . . . . . . 8 (𝑡 = 𝐵 → Σ𝑘𝑡𝐴𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥)
5149, 50eqeq12d 2834 . . . . . . 7 (𝑡 = 𝐵 → (∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥))
5247, 51anbi12d 630 . . . . . 6 (𝑡 = 𝐵 → (((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥) ↔ ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥)))
5344, 52imbi12d 346 . . . . 5 (𝑡 = 𝐵 → ((𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)) ↔ (𝐵𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥))))
5453imbi2d 342 . . . 4 (𝑡 = 𝐵 → ((𝜑 → (𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥))) ↔ (𝜑 → (𝐵𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥)))))
55 itgfsum.1 . . . . . 6 (𝜑𝐴 ∈ dom vol)
56 ibl0 24314 . . . . . 6 (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1)
5755, 56syl 17 . . . . 5 (𝜑 → (𝐴 × {0}) ∈ 𝐿1)
5857a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝐵 → (𝐴 × {0}) ∈ 𝐿1))
59 ssun1 4145 . . . . . . . . . 10 𝑤 ⊆ (𝑤 ∪ {𝑧})
60 sstr 3972 . . . . . . . . . 10 ((𝑤 ⊆ (𝑤 ∪ {𝑧}) ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵) → 𝑤𝐵)
6159, 60mpan 686 . . . . . . . . 9 ((𝑤 ∪ {𝑧}) ⊆ 𝐵𝑤𝐵)
6261imim1i 63 . . . . . . . 8 ((𝑤𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)))
63 nfcv 2974 . . . . . . . . . . . . . . . . . 18 𝑚𝐶
64 nfcsb1v 3904 . . . . . . . . . . . . . . . . . 18 𝑘𝑚 / 𝑘𝐶
65 csbeq1a 3894 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚𝐶 = 𝑚 / 𝑘𝐶)
6663, 64, 65cbvsumi 15042 . . . . . . . . . . . . . . . . 17 Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶
67 simprl 767 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧𝑤)
68 disjsn 4639 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑤)
6967, 68sylibr 235 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∩ {𝑧}) = ∅)
7069adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑤 ∩ {𝑧}) = ∅)
71 eqidd 2819 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
722adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
73 simprr 769 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
7472, 73ssfid 8729 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ∈ Fin)
7574adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑤 ∪ {𝑧}) ∈ Fin)
76 simplrr 774 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
7776sselda 3964 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚𝐵)
78 itgfsum.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ 𝐿1)
79 iblmbf 24295 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥𝐴𝐶) ∈ 𝐿1 → (𝑥𝐴𝐶) ∈ MblFn)
8078, 79syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ MblFn)
81 itgfsum.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
8281anass1rs 651 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶𝑉)
8380, 82mbfmptcl 24164 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶 ∈ ℂ)
8483an32s 648 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
8584ralrimiva 3179 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℂ)
8685adantlr 711 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℂ)
8763nfel1 2991 . . . . . . . . . . . . . . . . . . . . . . 23 𝑚 𝐶 ∈ ℂ
8864nfel1 2991 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝑚 / 𝑘𝐶 ∈ ℂ
8965eleq1d 2894 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑚 → (𝐶 ∈ ℂ ↔ 𝑚 / 𝑘𝐶 ∈ ℂ))
9087, 88, 89cbvralw 3439 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑘𝐵 𝐶 ∈ ℂ ↔ ∀𝑚𝐵 𝑚 / 𝑘𝐶 ∈ ℂ)
9186, 90sylib 219 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → ∀𝑚𝐵 𝑚 / 𝑘𝐶 ∈ ℂ)
9291r19.21bi 3205 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑚𝐵) → 𝑚 / 𝑘𝐶 ∈ ℂ)
9377, 92syldan 591 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚 / 𝑘𝐶 ∈ ℂ)
9470, 71, 75, 93fsumsplit 15085 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 = (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + Σ𝑚 ∈ {𝑧}𝑚 / 𝑘𝐶))
95 vex 3495 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
96 csbeq1 3883 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑧𝑚 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
9796eleq1d 2894 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑧 → (𝑚 / 𝑘𝐶 ∈ ℂ ↔ 𝑧 / 𝑘𝐶 ∈ ℂ))
9873unssbd 4161 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
9995snss 4710 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝐵 ↔ {𝑧} ⊆ 𝐵)
10098, 99sylibr 235 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧𝐵)
101100adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧𝐵)
10297, 91, 101rspcdva 3622 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧 / 𝑘𝐶 ∈ ℂ)
10396sumsn 15089 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ V ∧ 𝑧 / 𝑘𝐶 ∈ ℂ) → Σ𝑚 ∈ {𝑧}𝑚 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
10495, 102, 103sylancr 587 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑚 ∈ {𝑧}𝑚 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
105104oveq2d 7161 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + Σ𝑚 ∈ {𝑧}𝑚 / 𝑘𝐶) = (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶))
10694, 105eqtrd 2853 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 = (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶))
10766, 106syl5eq 2865 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶))
108107mpteq2dva 5152 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶)))
109 nfcv 2974 . . . . . . . . . . . . . . . 16 𝑦𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶)
110 nfcsb1v 3904 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶
111 nfcv 2974 . . . . . . . . . . . . . . . . 17 𝑥 +
112 nfcsb1v 3904 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥𝑧 / 𝑘𝐶
113110, 111, 112nfov 7175 . . . . . . . . . . . . . . . 16 𝑥(𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶)
114 csbeq1a 3894 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → Σ𝑚𝑤 𝑚 / 𝑘𝐶 = 𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶)
115 csbeq1a 3894 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦𝑧 / 𝑘𝐶 = 𝑦 / 𝑥𝑧 / 𝑘𝐶)
116114, 115oveq12d 7163 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶) = (𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶))
117109, 113, 116cbvmpt 5158 . . . . . . . . . . . . . . 15 (𝑥𝐴 ↦ (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶)) = (𝑦𝐴 ↦ (𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶))
118108, 117syl6eq 2869 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) = (𝑦𝐴 ↦ (𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶)))
119118adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) = (𝑦𝐴 ↦ (𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶)))
120 sumex 15032 . . . . . . . . . . . . . . . 16 Σ𝑚𝑤 𝑚 / 𝑘𝐶 ∈ V
121120csbex 5206 . . . . . . . . . . . . . . 15 𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 ∈ V
122121a1i 11 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) ∧ 𝑦𝐴) → 𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 ∈ V)
12363, 64, 65cbvsumi 15042 . . . . . . . . . . . . . . . . 17 Σ𝑘𝑤 𝐶 = Σ𝑚𝑤 𝑚 / 𝑘𝐶
124123mpteq2i 5149 . . . . . . . . . . . . . . . 16 (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑚𝑤 𝑚 / 𝑘𝐶)
125 nfcv 2974 . . . . . . . . . . . . . . . . 17 𝑦Σ𝑚𝑤 𝑚 / 𝑘𝐶
126125, 110, 114cbvmpt 5158 . . . . . . . . . . . . . . . 16 (𝑥𝐴 ↦ Σ𝑚𝑤 𝑚 / 𝑘𝐶) = (𝑦𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶)
127124, 126eqtri 2841 . . . . . . . . . . . . . . 15 (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑦𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶)
128 simprl 767 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1)
129127, 128eqeltrrid 2915 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑦𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶) ∈ 𝐿1)
130102elexd 3512 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧 / 𝑘𝐶 ∈ V)
131130ralrimiva 3179 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ V)
132131adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ V)
133 nfv 1906 . . . . . . . . . . . . . . . . 17 𝑦𝑧 / 𝑘𝐶 ∈ V
134112nfel1 2991 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V
135115eleq1d 2894 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝑧 / 𝑘𝐶 ∈ V ↔ 𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V))
136133, 134, 135cbvralw 3439 . . . . . . . . . . . . . . . 16 (∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ V ↔ ∀𝑦𝐴 𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
137132, 136sylib 219 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∀𝑦𝐴 𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
138137r19.21bi 3205 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) ∧ 𝑦𝐴) → 𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
139 nfcv 2974 . . . . . . . . . . . . . . . . 17 𝑦𝑧 / 𝑘𝐶
140139, 112, 115cbvmpt 5158 . . . . . . . . . . . . . . . 16 (𝑥𝐴𝑧 / 𝑘𝐶) = (𝑦𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶)
14196mpteq2dv 5153 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑧 → (𝑥𝐴𝑚 / 𝑘𝐶) = (𝑥𝐴𝑧 / 𝑘𝐶))
142141eleq1d 2894 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑧 → ((𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1 ↔ (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝐿1))
14378ralrimiva 3179 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝐿1)
144 nfv 1906 . . . . . . . . . . . . . . . . . . . 20 𝑚(𝑥𝐴𝐶) ∈ 𝐿1
145 nfcv 2974 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝐴
146145, 64nfmpt 5154 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝑥𝐴𝑚 / 𝑘𝐶)
147146nfel1 2991 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1
14865mpteq2dv 5153 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑚 → (𝑥𝐴𝐶) = (𝑥𝐴𝑚 / 𝑘𝐶))
149148eleq1d 2894 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑚 → ((𝑥𝐴𝐶) ∈ 𝐿1 ↔ (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1))
150144, 147, 149cbvralw 3439 . . . . . . . . . . . . . . . . . . 19 (∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝐿1 ↔ ∀𝑚𝐵 (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1)
151143, 150sylib 219 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑚𝐵 (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1)
152151adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑚𝐵 (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1)
153142, 152, 100rspcdva 3622 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝐿1)
154140, 153eqeltrrid 2915 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶) ∈ 𝐿1)
155154adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑦𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶) ∈ 𝐿1)
156122, 129, 138, 155ibladd 24348 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑦𝐴 ↦ (𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶)) ∈ 𝐿1)
157119, 156eqeltrd 2910 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1)
158122, 129, 138, 155itgadd 24352 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴(𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶) d𝑦 = (∫𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑦 + ∫𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶 d𝑦))
159116, 109, 113cbvitg 24303 . . . . . . . . . . . . . . 15 𝐴𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶) d𝑥 = ∫𝐴(𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶) d𝑦
160114, 125, 110cbvitg 24303 . . . . . . . . . . . . . . . 16 𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑦
161115, 139, 112cbvitg 24303 . . . . . . . . . . . . . . . 16 𝐴𝑧 / 𝑘𝐶 d𝑥 = ∫𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶 d𝑦
162160, 161oveq12i 7157 . . . . . . . . . . . . . . 15 (∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥 + ∫𝐴𝑧 / 𝑘𝐶 d𝑥) = (∫𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑦 + ∫𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶 d𝑦)
163158, 159, 1623eqtr4g 2878 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶) d𝑥 = (∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥 + ∫𝐴𝑧 / 𝑘𝐶 d𝑥))
164106itgeq2dv 24309 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶) d𝑥)
165164adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶) d𝑥)
166 eqidd 2819 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
16773sselda 3964 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚𝐵)
16892an32s 648 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚𝐵) ∧ 𝑥𝐴) → 𝑚 / 𝑘𝐶 ∈ ℂ)
169152r19.21bi 3205 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚𝐵) → (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1)
170168, 169itgcl 24311 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚𝐵) → ∫𝐴𝑚 / 𝑘𝐶 d𝑥 ∈ ℂ)
171167, 170syldan 591 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → ∫𝐴𝑚 / 𝑘𝐶 d𝑥 ∈ ℂ)
17269, 166, 74, 171fsumsplit 15085 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴𝑚 / 𝑘𝐶 d𝑥 = (Σ𝑚𝑤𝐴𝑚 / 𝑘𝐶 d𝑥 + Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥))
173172adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴𝑚 / 𝑘𝐶 d𝑥 = (Σ𝑚𝑤𝐴𝑚 / 𝑘𝐶 d𝑥 + Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥))
174 simprr 769 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)
175 itgeq2 24305 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝐴 Σ𝑘𝑤 𝐶 = Σ𝑚𝑤 𝑚 / 𝑘𝐶 → ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = ∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥)
176123a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → Σ𝑘𝑤 𝐶 = Σ𝑚𝑤 𝑚 / 𝑘𝐶)
177175, 176mprg 3149 . . . . . . . . . . . . . . . . 17 𝐴Σ𝑘𝑤 𝐶 d𝑥 = ∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥
178 nfcv 2974 . . . . . . . . . . . . . . . . . 18 𝑚𝐴𝐶 d𝑥
179145, 64nfitg 24302 . . . . . . . . . . . . . . . . . 18 𝑘𝐴𝑚 / 𝑘𝐶 d𝑥
18065adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑘 = 𝑚𝑥𝐴) → 𝐶 = 𝑚 / 𝑘𝐶)
181180itgeq2dv 24309 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → ∫𝐴𝐶 d𝑥 = ∫𝐴𝑚 / 𝑘𝐶 d𝑥)
182178, 179, 181cbvsumi 15042 . . . . . . . . . . . . . . . . 17 Σ𝑘𝑤𝐴𝐶 d𝑥 = Σ𝑚𝑤𝐴𝑚 / 𝑘𝐶 d𝑥
183174, 177, 1823eqtr3g 2876 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥 = Σ𝑚𝑤𝐴𝑚 / 𝑘𝐶 d𝑥)
184102, 153itgcl 24311 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∫𝐴𝑧 / 𝑘𝐶 d𝑥 ∈ ℂ)
185184adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴𝑧 / 𝑘𝐶 d𝑥 ∈ ℂ)
18696adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑧𝑥𝐴) → 𝑚 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
187186itgeq2dv 24309 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑧 → ∫𝐴𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑧 / 𝑘𝐶 d𝑥)
188187sumsn 15089 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ V ∧ ∫𝐴𝑧 / 𝑘𝐶 d𝑥 ∈ ℂ) → Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑧 / 𝑘𝐶 d𝑥)
18995, 185, 188sylancr 587 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑧 / 𝑘𝐶 d𝑥)
190189eqcomd 2824 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴𝑧 / 𝑘𝐶 d𝑥 = Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥)
191183, 190oveq12d 7163 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥 + ∫𝐴𝑧 / 𝑘𝐶 d𝑥) = (Σ𝑚𝑤𝐴𝑚 / 𝑘𝐶 d𝑥 + Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥))
192173, 191eqtr4d 2856 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴𝑚 / 𝑘𝐶 d𝑥 = (∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥 + ∫𝐴𝑧 / 𝑘𝐶 d𝑥))
193163, 165, 1923eqtr4d 2863 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 d𝑥 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴𝑚 / 𝑘𝐶 d𝑥)
194 itgeq2 24305 . . . . . . . . . . . . . 14 (∀𝑥𝐴 Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 → ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 d𝑥)
19566a1i 11 . . . . . . . . . . . . . 14 (𝑥𝐴 → Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶)
196194, 195mprg 3149 . . . . . . . . . . . . 13 𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 d𝑥
197178, 179, 181cbvsumi 15042 . . . . . . . . . . . . 13 Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴𝑚 / 𝑘𝐶 d𝑥
198193, 196, 1973eqtr4g 2878 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)
199157, 198jca 512 . . . . . . . . . . 11 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))
200199ex 413 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥) → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))
201200expr 457 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑧𝑤) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → (((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥) → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))))
202201a2d 29 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑧𝑤) → (((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))))
20362, 202syl5 34 . . . . . . 7 ((𝜑 ∧ ¬ 𝑧𝑤) → ((𝑤𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))))
204203expcom 414 . . . . . 6 𝑧𝑤 → (𝜑 → ((𝑤𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))))
205204adantl 482 . . . . 5 ((𝑤 ∈ Fin ∧ ¬ 𝑧𝑤) → (𝜑 → ((𝑤𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))))
206205a2d 29 . . . 4 ((𝑤 ∈ Fin ∧ ¬ 𝑧𝑤) → ((𝜑 → (𝑤𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥))) → (𝜑 → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)))))
20721, 32, 43, 54, 58, 206findcard2s 8747 . . 3 (𝐵 ∈ Fin → (𝜑 → (𝐵𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥))))
2082, 207mpcom 38 . 2 (𝜑 → (𝐵𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥)))
2091, 208mpi 20 1 (𝜑 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  csb 3880  cun 3931  cin 3932  wss 3933  c0 4288  {csn 4557  cmpt 5137   × cxp 5546  dom cdm 5548  (class class class)co 7145  Fincfn 8497  cc 10523  0cc0 10525   + caddc 10528  Σcsu 15030  volcvol 23991  MblFncmbf 24142  𝐿1cibl 24145  citg 24146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cc 9845  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-rest 16684  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923  df-ovol 23992  df-vol 23993  df-mbf 24147  df-itg1 24148  df-itg2 24149  df-ibl 24150  df-itg 24151  df-0p 24198
This theorem is referenced by:  circlemeth  31810  fourierdlem83  42351
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