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Theorem itgfsum 23533
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 3609 . 2 𝐵𝐵
2 itgfsum.2 . . 3 (𝜑𝐵 ∈ Fin)
3 sseq1 3611 . . . . . 6 (𝑡 = ∅ → (𝑡𝐵 ↔ ∅ ⊆ 𝐵))
4 sumeq1 14369 . . . . . . . . . . . 12 (𝑡 = ∅ → Σ𝑘𝑡 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5 sum0 14401 . . . . . . . . . . . 12 Σ𝑘 ∈ ∅ 𝐶 = 0
64, 5syl6eq 2671 . . . . . . . . . . 11 (𝑡 = ∅ → Σ𝑘𝑡 𝐶 = 0)
76mpteq2dv 4715 . . . . . . . . . 10 (𝑡 = ∅ → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝑥𝐴 ↦ 0))
8 fconstmpt 5133 . . . . . . . . . 10 (𝐴 × {0}) = (𝑥𝐴 ↦ 0)
97, 8syl6eqr 2673 . . . . . . . . 9 (𝑡 = ∅ → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝐴 × {0}))
109eleq1d 2683 . . . . . . . 8 (𝑡 = ∅ → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ↔ (𝐴 × {0}) ∈ 𝐿1))
1110anbi1d 740 . . . . . . 7 (𝑡 = ∅ → (((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥) ↔ ((𝐴 × {0}) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)))
12 itgz 23487 . . . . . . . . 9 𝐴0 d𝑥 = 0
136adantr 481 . . . . . . . . . 10 ((𝑡 = ∅ ∧ 𝑥𝐴) → Σ𝑘𝑡 𝐶 = 0)
1413itgeq2dv 23488 . . . . . . . . 9 (𝑡 = ∅ → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = ∫𝐴0 d𝑥)
15 sumeq1 14369 . . . . . . . . . 10 (𝑡 = ∅ → Σ𝑘𝑡𝐴𝐶 d𝑥 = Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥)
16 sum0 14401 . . . . . . . . . 10 Σ𝑘 ∈ ∅ ∫𝐴𝐶 d𝑥 = 0
1715, 16syl6eq 2671 . . . . . . . . 9 (𝑡 = ∅ → Σ𝑘𝑡𝐴𝐶 d𝑥 = 0)
1812, 14, 173eqtr4a 2681 . . . . . . . 8 (𝑡 = ∅ → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)
1918biantrud 528 . . . . . . 7 (𝑡 = ∅ → ((𝐴 × {0}) ∈ 𝐿1 ↔ ((𝐴 × {0}) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)))
2011, 19bitr4d 271 . . . . . 6 (𝑡 = ∅ → (((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥) ↔ (𝐴 × {0}) ∈ 𝐿1))
213, 20imbi12d 334 . . . . 5 (𝑡 = ∅ → ((𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)) ↔ (∅ ⊆ 𝐵 → (𝐴 × {0}) ∈ 𝐿1)))
2221imbi2d 330 . . . 4 (𝑡 = ∅ → ((𝜑 → (𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥))) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝐴 × {0}) ∈ 𝐿1))))
23 sseq1 3611 . . . . . 6 (𝑡 = 𝑤 → (𝑡𝐵𝑤𝐵))
24 sumeq1 14369 . . . . . . . . 9 (𝑡 = 𝑤 → Σ𝑘𝑡 𝐶 = Σ𝑘𝑤 𝐶)
2524mpteq2dv 4715 . . . . . . . 8 (𝑡 = 𝑤 → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶))
2625eleq1d 2683 . . . . . . 7 (𝑡 = 𝑤 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1))
2724adantr 481 . . . . . . . . 9 ((𝑡 = 𝑤𝑥𝐴) → Σ𝑘𝑡 𝐶 = Σ𝑘𝑤 𝐶)
2827itgeq2dv 23488 . . . . . . . 8 (𝑡 = 𝑤 → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘𝑤 𝐶 d𝑥)
29 sumeq1 14369 . . . . . . . 8 (𝑡 = 𝑤 → Σ𝑘𝑡𝐴𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)
3028, 29eqeq12d 2636 . . . . . . 7 (𝑡 = 𝑤 → (∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥))
3126, 30anbi12d 746 . . . . . 6 (𝑡 = 𝑤 → (((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥) ↔ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)))
3223, 31imbi12d 334 . . . . 5 (𝑡 = 𝑤 → ((𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)) ↔ (𝑤𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥))))
3332imbi2d 330 . . . 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)))))
34 sseq1 3611 . . . . . 6 (𝑡 = (𝑤 ∪ {𝑧}) → (𝑡𝐵 ↔ (𝑤 ∪ {𝑧}) ⊆ 𝐵))
35 sumeq1 14369 . . . . . . . . 9 (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
3635mpteq2dv 4715 . . . . . . . 8 (𝑡 = (𝑤 ∪ {𝑧}) → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶))
3736eleq1d 2683 . . . . . . 7 (𝑡 = (𝑤 ∪ {𝑧}) → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1))
3835adantr 481 . . . . . . . . 9 ((𝑡 = (𝑤 ∪ {𝑧}) ∧ 𝑥𝐴) → Σ𝑘𝑡 𝐶 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶)
3938itgeq2dv 23488 . . . . . . . 8 (𝑡 = (𝑤 ∪ {𝑧}) → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥)
40 sumeq1 14369 . . . . . . . 8 (𝑡 = (𝑤 ∪ {𝑧}) → Σ𝑘𝑡𝐴𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥)
4139, 40eqeq12d 2636 . . . . . . 7 (𝑡 = (𝑤 ∪ {𝑧}) → (∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥))
4237, 41anbi12d 746 . . . . . 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)))
4334, 42imbi12d 334 . . . . 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))))
4443imbi2d 330 . . . 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)))))
45 sseq1 3611 . . . . . 6 (𝑡 = 𝐵 → (𝑡𝐵𝐵𝐵))
46 sumeq1 14369 . . . . . . . . 9 (𝑡 = 𝐵 → Σ𝑘𝑡 𝐶 = Σ𝑘𝐵 𝐶)
4746mpteq2dv 4715 . . . . . . . 8 (𝑡 = 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶))
4847eleq1d 2683 . . . . . . 7 (𝑡 = 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ↔ (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1))
4946adantr 481 . . . . . . . . 9 ((𝑡 = 𝐵𝑥𝐴) → Σ𝑘𝑡 𝐶 = Σ𝑘𝐵 𝐶)
5049itgeq2dv 23488 . . . . . . . 8 (𝑡 = 𝐵 → ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = ∫𝐴Σ𝑘𝐵 𝐶 d𝑥)
51 sumeq1 14369 . . . . . . . 8 (𝑡 = 𝐵 → Σ𝑘𝑡𝐴𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥)
5250, 51eqeq12d 2636 . . . . . . 7 (𝑡 = 𝐵 → (∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥 ↔ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥))
5348, 52anbi12d 746 . . . . . 6 (𝑡 = 𝐵 → (((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥) ↔ ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥)))
5445, 53imbi12d 334 . . . . 5 (𝑡 = 𝐵 → ((𝑡𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑡 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑡 𝐶 d𝑥 = Σ𝑘𝑡𝐴𝐶 d𝑥)) ↔ (𝐵𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥))))
5554imbi2d 330 . . . 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)))))
56 itgfsum.1 . . . . . 6 (𝜑𝐴 ∈ dom vol)
57 ibl0 23493 . . . . . 6 (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1)
5856, 57syl 17 . . . . 5 (𝜑 → (𝐴 × {0}) ∈ 𝐿1)
5958a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝐵 → (𝐴 × {0}) ∈ 𝐿1))
60 ssun1 3760 . . . . . . . . . 10 𝑤 ⊆ (𝑤 ∪ {𝑧})
61 sstr 3596 . . . . . . . . . 10 ((𝑤 ⊆ (𝑤 ∪ {𝑧}) ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵) → 𝑤𝐵)
6260, 61mpan 705 . . . . . . . . 9 ((𝑤 ∪ {𝑧}) ⊆ 𝐵𝑤𝐵)
6362imim1i 63 . . . . . . . 8 ((𝑤𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ((𝑤 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)))
64 nfcv 2761 . . . . . . . . . . . . . . . . . 18 𝑚𝐶
65 nfcsb1v 3535 . . . . . . . . . . . . . . . . . 18 𝑘𝑚 / 𝑘𝐶
66 csbeq1a 3528 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚𝐶 = 𝑚 / 𝑘𝐶)
6764, 65, 66cbvsumi 14377 . . . . . . . . . . . . . . . . 17 Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶
68 simprl 793 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧𝑤)
69 disjsn 4223 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑤)
7068, 69sylibr 224 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∩ {𝑧}) = ∅)
7170adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑤 ∩ {𝑧}) = ∅)
72 eqidd 2622 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
732adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
74 simprr 795 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
75 ssfi 8140 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ Fin ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵) → (𝑤 ∪ {𝑧}) ∈ Fin)
7673, 74, 75syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) ∈ Fin)
7776adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑤 ∪ {𝑧}) ∈ Fin)
78 simplrr 800 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑤 ∪ {𝑧}) ⊆ 𝐵)
7978sselda 3588 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚𝐵)
80 itgfsum.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ 𝐿1)
81 iblmbf 23474 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥𝐴𝐶) ∈ 𝐿1 → (𝑥𝐴𝐶) ∈ MblFn)
8280, 81syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ MblFn)
83 itgfsum.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
8483anass1rs 848 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶𝑉)
8582, 84mbfmptcl 23344 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶 ∈ ℂ)
8685an32s 845 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
8786ralrimiva 2962 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℂ)
8887adantlr 750 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℂ)
8964nfel1 2775 . . . . . . . . . . . . . . . . . . . . . . 23 𝑚 𝐶 ∈ ℂ
9065nfel1 2775 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝑚 / 𝑘𝐶 ∈ ℂ
9166eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑚 → (𝐶 ∈ ℂ ↔ 𝑚 / 𝑘𝐶 ∈ ℂ))
9289, 90, 91cbvral 3159 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑘𝐵 𝐶 ∈ ℂ ↔ ∀𝑚𝐵 𝑚 / 𝑘𝐶 ∈ ℂ)
9388, 92sylib 208 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → ∀𝑚𝐵 𝑚 / 𝑘𝐶 ∈ ℂ)
9493r19.21bi 2928 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑚𝐵) → 𝑚 / 𝑘𝐶 ∈ ℂ)
9579, 94syldan 487 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚 / 𝑘𝐶 ∈ ℂ)
9671, 72, 77, 95fsumsplit 14420 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 = (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + Σ𝑚 ∈ {𝑧}𝑚 / 𝑘𝐶))
97 vex 3193 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
9874unssbd 3775 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
9997snss 4293 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝐵 ↔ {𝑧} ⊆ 𝐵)
10098, 99sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧𝐵)
101100adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧𝐵)
102 csbeq1 3522 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑧𝑚 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
103102eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑧 → (𝑚 / 𝑘𝐶 ∈ ℂ ↔ 𝑧 / 𝑘𝐶 ∈ ℂ))
104103rspcv 3295 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝐵 → (∀𝑚𝐵 𝑚 / 𝑘𝐶 ∈ ℂ → 𝑧 / 𝑘𝐶 ∈ ℂ))
105101, 93, 104sylc 65 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧 / 𝑘𝐶 ∈ ℂ)
106102sumsn 14424 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ V ∧ 𝑧 / 𝑘𝐶 ∈ ℂ) → Σ𝑚 ∈ {𝑧}𝑚 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
10797, 105, 106sylancr 694 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑚 ∈ {𝑧}𝑚 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
108107oveq2d 6631 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + Σ𝑚 ∈ {𝑧}𝑚 / 𝑘𝐶) = (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶))
10996, 108eqtrd 2655 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 = (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶))
11067, 109syl5eq 2667 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶))
111110mpteq2dva 4714 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶)))
112 nfcv 2761 . . . . . . . . . . . . . . . 16 𝑦𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶)
113 nfcsb1v 3535 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶
114 nfcv 2761 . . . . . . . . . . . . . . . . 17 𝑥 +
115 nfcsb1v 3535 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥𝑧 / 𝑘𝐶
116113, 114, 115nfov 6641 . . . . . . . . . . . . . . . 16 𝑥(𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶)
117 csbeq1a 3528 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → Σ𝑚𝑤 𝑚 / 𝑘𝐶 = 𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶)
118 csbeq1a 3528 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦𝑧 / 𝑘𝐶 = 𝑦 / 𝑥𝑧 / 𝑘𝐶)
119117, 118oveq12d 6633 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶) = (𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶))
120112, 116, 119cbvmpt 4719 . . . . . . . . . . . . . . 15 (𝑥𝐴 ↦ (Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶)) = (𝑦𝐴 ↦ (𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶))
121111, 120syl6eq 2671 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) = (𝑦𝐴 ↦ (𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶)))
122121adantr 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)))
123 sumex 14368 . . . . . . . . . . . . . . . 16 Σ𝑚𝑤 𝑚 / 𝑘𝐶 ∈ V
124123csbex 4763 . . . . . . . . . . . . . . 15 𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 ∈ V
125124a1i 11 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) ∧ 𝑦𝐴) → 𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 ∈ V)
12664, 65, 66cbvsumi 14377 . . . . . . . . . . . . . . . . 17 Σ𝑘𝑤 𝐶 = Σ𝑚𝑤 𝑚 / 𝑘𝐶
127126mpteq2i 4711 . . . . . . . . . . . . . . . 16 (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑚𝑤 𝑚 / 𝑘𝐶)
128 nfcv 2761 . . . . . . . . . . . . . . . . 17 𝑦Σ𝑚𝑤 𝑚 / 𝑘𝐶
129128, 113, 117cbvmpt 4719 . . . . . . . . . . . . . . . 16 (𝑥𝐴 ↦ Σ𝑚𝑤 𝑚 / 𝑘𝐶) = (𝑦𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶)
130127, 129eqtri 2643 . . . . . . . . . . . . . . 15 (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑦𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶)
131 simprl 793 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1)
132130, 131syl5eqelr 2703 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑦𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶) ∈ 𝐿1)
133 elex 3202 . . . . . . . . . . . . . . . . . . 19 (𝑧 / 𝑘𝐶 ∈ ℂ → 𝑧 / 𝑘𝐶 ∈ V)
134105, 133syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧 / 𝑘𝐶 ∈ V)
135134ralrimiva 2962 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ V)
136135adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ V)
137 nfv 1840 . . . . . . . . . . . . . . . . 17 𝑦𝑧 / 𝑘𝐶 ∈ V
138115nfel1 2775 . . . . . . . . . . . . . . . . 17 𝑥𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V
139118eleq1d 2683 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (𝑧 / 𝑘𝐶 ∈ V ↔ 𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V))
140137, 138, 139cbvral 3159 . . . . . . . . . . . . . . . 16 (∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ V ↔ ∀𝑦𝐴 𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
141136, 140sylib 208 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∀𝑦𝐴 𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
142141r19.21bi 2928 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) ∧ 𝑦𝐴) → 𝑦 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
143 nfcv 2761 . . . . . . . . . . . . . . . . 17 𝑦𝑧 / 𝑘𝐶
144143, 115, 118cbvmpt 4719 . . . . . . . . . . . . . . . 16 (𝑥𝐴𝑧 / 𝑘𝐶) = (𝑦𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶)
14580ralrimiva 2962 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝐿1)
146 nfv 1840 . . . . . . . . . . . . . . . . . . . 20 𝑚(𝑥𝐴𝐶) ∈ 𝐿1
147 nfcv 2761 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝐴
148147, 65nfmpt 4716 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝑥𝐴𝑚 / 𝑘𝐶)
149148nfel1 2775 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1
15066mpteq2dv 4715 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑚 → (𝑥𝐴𝐶) = (𝑥𝐴𝑚 / 𝑘𝐶))
151150eleq1d 2683 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑚 → ((𝑥𝐴𝐶) ∈ 𝐿1 ↔ (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1))
152146, 149, 151cbvral 3159 . . . . . . . . . . . . . . . . . . 19 (∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝐿1 ↔ ∀𝑚𝐵 (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1)
153145, 152sylib 208 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑚𝐵 (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1)
154153adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑚𝐵 (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1)
155102mpteq2dv 4715 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑧 → (𝑥𝐴𝑚 / 𝑘𝐶) = (𝑥𝐴𝑧 / 𝑘𝐶))
156155eleq1d 2683 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑧 → ((𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1 ↔ (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝐿1))
157156rspcv 3295 . . . . . . . . . . . . . . . . 17 (𝑧𝐵 → (∀𝑚𝐵 (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1 → (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝐿1))
158100, 154, 157sylc 65 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝐿1)
159144, 158syl5eqelr 2703 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶) ∈ 𝐿1)
160159adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑦𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶) ∈ 𝐿1)
161125, 132, 142, 160ibladd 23527 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑦𝐴 ↦ (𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶)) ∈ 𝐿1)
162122, 161eqeltrd 2698 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶) ∈ 𝐿1)
163125, 132, 142, 160itgadd 23531 . . . . . . . . . . . . . . 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))
164119, 112, 116cbvitg 23482 . . . . . . . . . . . . . . 15 𝐴𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶) d𝑥 = ∫𝐴(𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑦 / 𝑥𝑧 / 𝑘𝐶) d𝑦
165117, 128, 113cbvitg 23482 . . . . . . . . . . . . . . . 16 𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑦
166118, 143, 115cbvitg 23482 . . . . . . . . . . . . . . . 16 𝐴𝑧 / 𝑘𝐶 d𝑥 = ∫𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶 d𝑦
167165, 166oveq12i 6627 . . . . . . . . . . . . . . 15 (∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥 + ∫𝐴𝑧 / 𝑘𝐶 d𝑥) = (∫𝐴𝑦 / 𝑥Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑦 + ∫𝐴𝑦 / 𝑥𝑧 / 𝑘𝐶 d𝑦)
168163, 164, 1673eqtr4g 2680 . . . . . . . . . . . . . 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))
169109itgeq2dv 23488 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑚𝑤 𝑚 / 𝑘𝐶 + 𝑧 / 𝑘𝐶) d𝑥)
170169adantr 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)
171 eqidd 2622 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → (𝑤 ∪ {𝑧}) = (𝑤 ∪ {𝑧}))
17274sselda 3588 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → 𝑚𝐵)
17394an32s 845 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚𝐵) ∧ 𝑥𝐴) → 𝑚 / 𝑘𝐶 ∈ ℂ)
174154r19.21bi 2928 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚𝐵) → (𝑥𝐴𝑚 / 𝑘𝐶) ∈ 𝐿1)
175173, 174itgcl 23490 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚𝐵) → ∫𝐴𝑚 / 𝑘𝐶 d𝑥 ∈ ℂ)
176172, 175syldan 487 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑚 ∈ (𝑤 ∪ {𝑧})) → ∫𝐴𝑚 / 𝑘𝐶 d𝑥 ∈ ℂ)
17770, 171, 76, 176fsumsplit 14420 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴𝑚 / 𝑘𝐶 d𝑥 = (Σ𝑚𝑤𝐴𝑚 / 𝑘𝐶 d𝑥 + Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥))
178177adantr 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))
179 simprr 795 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)
180 itgeq2 23484 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝐴 Σ𝑘𝑤 𝐶 = Σ𝑚𝑤 𝑚 / 𝑘𝐶 → ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = ∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥)
181126a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → Σ𝑘𝑤 𝐶 = Σ𝑚𝑤 𝑚 / 𝑘𝐶)
182180, 181mprg 2922 . . . . . . . . . . . . . . . . 17 𝐴Σ𝑘𝑤 𝐶 d𝑥 = ∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥
183 nfcv 2761 . . . . . . . . . . . . . . . . . 18 𝑚𝐴𝐶 d𝑥
184147, 65nfitg 23481 . . . . . . . . . . . . . . . . . 18 𝑘𝐴𝑚 / 𝑘𝐶 d𝑥
18566adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑘 = 𝑚𝑥𝐴) → 𝐶 = 𝑚 / 𝑘𝐶)
186185itgeq2dv 23488 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → ∫𝐴𝐶 d𝑥 = ∫𝐴𝑚 / 𝑘𝐶 d𝑥)
187183, 184, 186cbvsumi 14377 . . . . . . . . . . . . . . . . 17 Σ𝑘𝑤𝐴𝐶 d𝑥 = Σ𝑚𝑤𝐴𝑚 / 𝑘𝐶 d𝑥
188179, 182, 1873eqtr3g 2678 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴Σ𝑚𝑤 𝑚 / 𝑘𝐶 d𝑥 = Σ𝑚𝑤𝐴𝑚 / 𝑘𝐶 d𝑥)
189105, 158itgcl 23490 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) → ∫𝐴𝑧 / 𝑘𝐶 d𝑥 ∈ ℂ)
190189adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴𝑧 / 𝑘𝐶 d𝑥 ∈ ℂ)
191102adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑧𝑥𝐴) → 𝑚 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
192191itgeq2dv 23488 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑧 → ∫𝐴𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑧 / 𝑘𝐶 d𝑥)
193192sumsn 14424 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ V ∧ ∫𝐴𝑧 / 𝑘𝐶 d𝑥 ∈ ℂ) → Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑧 / 𝑘𝐶 d𝑥)
19497, 190, 193sylancr 694 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥 = ∫𝐴𝑧 / 𝑘𝐶 d𝑥)
195194eqcomd 2627 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑤 ∧ (𝑤 ∪ {𝑧}) ⊆ 𝐵)) ∧ ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝑤 𝐶 d𝑥 = Σ𝑘𝑤𝐴𝐶 d𝑥)) → ∫𝐴𝑧 / 𝑘𝐶 d𝑥 = Σ𝑚 ∈ {𝑧}∫𝐴𝑚 / 𝑘𝐶 d𝑥)
196188, 195oveq12d 6633 . . . . . . . . . . . . . . 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))
197178, 196eqtr4d 2658 . . . . . . . . . . . . . 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))
198168, 170, 1973eqtr4d 2665 . . . . . . . . . . . . 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)
199 itgeq2 23484 . . . . . . . . . . . . . 14 (∀𝑥𝐴 Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 → ∫𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 d𝑥)
20067a1i 11 . . . . . . . . . . . . . 14 (𝑥𝐴 → Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶)
201199, 200mprg 2922 . . . . . . . . . . . . 13 𝐴Σ𝑘 ∈ (𝑤 ∪ {𝑧})𝐶 d𝑥 = ∫𝐴Σ𝑚 ∈ (𝑤 ∪ {𝑧})𝑚 / 𝑘𝐶 d𝑥
202183, 184, 186cbvsumi 14377 . . . . . . . . . . . . 13 Σ𝑘 ∈ (𝑤 ∪ {𝑧})∫𝐴𝐶 d𝑥 = Σ𝑚 ∈ (𝑤 ∪ {𝑧})∫𝐴𝑚 / 𝑘𝐶 d𝑥
203198, 201, 2023eqtr4g 2680 . . . . . . . . . . . 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)
204162, 203jca 554 . . . . . . . . . . 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))
205204ex 450 . . . . . . . . . 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)))
206205expr 642 . . . . . . . . 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))))
207206a2d 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))))
20863, 207syl5 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))))
209208expcom 451 . . . . . 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)))))
210209adantl 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)))))
211210a2d 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)))))
21222, 33, 44, 55, 59, 211findcard2s 8161 . . 3 (𝐵 ∈ Fin → (𝜑 → (𝐵𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥))))
2132, 212mpcom 38 . 2 (𝜑 → (𝐵𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥)))
2141, 213mpi 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 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  csb 3519  cun 3558  cin 3559  wss 3560  c0 3897  {csn 4155  cmpt 4683   × cxp 5082  dom cdm 5084  (class class class)co 6615  Fincfn 7915  cc 9894  0cc0 9896   + caddc 9899  Σcsu 14366  volcvol 23172  MblFncmbf 23323  𝐿1cibl 23326  citg 23327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cc 9217  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974  ax-addf 9975
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-disj 4594  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-ofr 6863  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-omul 7525  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-acn 8728  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ioc 12138  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-mod 12625  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-rlim 14170  df-sum 14367  df-rest 16023  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-bases 20690  df-cmp 21130  df-ovol 23173  df-vol 23174  df-mbf 23328  df-itg1 23329  df-itg2 23330  df-ibl 23331  df-itg 23332  df-0p 23377
This theorem is referenced by:  fourierdlem83  39743
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