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Theorem carageniuncllem1 42804
Description: The outer measure of 𝐴 ∩ (𝐺𝑛) is the sum of the outer measures of 𝐴 ∩ (𝐹𝑚). These are lines 7 to 10 of Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
carageniuncllem1.o (𝜑𝑂 ∈ OutMeas)
carageniuncllem1.s 𝑆 = (CaraGen‘𝑂)
carageniuncllem1.x 𝑋 = dom 𝑂
carageniuncllem1.a (𝜑𝐴𝑋)
carageniuncllem1.re (𝜑 → (𝑂𝐴) ∈ ℝ)
carageniuncllem1.z 𝑍 = (ℤ𝑀)
carageniuncllem1.e (𝜑𝐸:𝑍𝑆)
carageniuncllem1.g 𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))
carageniuncllem1.f 𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)))
carageniuncllem1.k (𝜑𝐾𝑍)
Assertion
Ref Expression
carageniuncllem1 (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝐾))))
Distinct variable groups:   𝐴,𝑛   𝑖,𝐸,𝑛   𝑛,𝐹   𝑛,𝐾   𝑖,𝑀,𝑛   𝑛,𝑂   𝑆,𝑖   𝑛,𝑍   𝜑,𝑖,𝑛
Allowed substitution hints:   𝐴(𝑖)   𝑆(𝑛)   𝐹(𝑖)   𝐺(𝑖,𝑛)   𝐾(𝑖)   𝑂(𝑖)   𝑋(𝑖,𝑛)   𝑍(𝑖)

Proof of Theorem carageniuncllem1
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 carageniuncllem1.k . . . 4 (𝜑𝐾𝑍)
2 carageniuncllem1.z . . . 4 𝑍 = (ℤ𝑀)
31, 2eleqtrdi 2923 . . 3 (𝜑𝐾 ∈ (ℤ𝑀))
4 eluzfz2 12914 . . 3 (𝐾 ∈ (ℤ𝑀) → 𝐾 ∈ (𝑀...𝐾))
53, 4syl 17 . 2 (𝜑𝐾 ∈ (𝑀...𝐾))
6 id 22 . 2 (𝜑𝜑)
7 oveq2 7163 . . . . . 6 (𝑘 = 𝑀 → (𝑀...𝑘) = (𝑀...𝑀))
87sumeq1d 15057 . . . . 5 (𝑘 = 𝑀 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹𝑛))))
9 fveq2 6669 . . . . . . 7 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
109ineq2d 4188 . . . . . 6 (𝑘 = 𝑀 → (𝐴 ∩ (𝐺𝑘)) = (𝐴 ∩ (𝐺𝑀)))
1110fveq2d 6673 . . . . 5 (𝑘 = 𝑀 → (𝑂‘(𝐴 ∩ (𝐺𝑘))) = (𝑂‘(𝐴 ∩ (𝐺𝑀))))
128, 11eqeq12d 2837 . . . 4 (𝑘 = 𝑀 → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘))) ↔ Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑀)))))
1312imbi2d 343 . . 3 (𝑘 = 𝑀 → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘)))) ↔ (𝜑 → Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑀))))))
14 oveq2 7163 . . . . . 6 (𝑘 = 𝑗 → (𝑀...𝑘) = (𝑀...𝑗))
1514sumeq1d 15057 . . . . 5 (𝑘 = 𝑗 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))))
16 fveq2 6669 . . . . . . 7 (𝑘 = 𝑗 → (𝐺𝑘) = (𝐺𝑗))
1716ineq2d 4188 . . . . . 6 (𝑘 = 𝑗 → (𝐴 ∩ (𝐺𝑘)) = (𝐴 ∩ (𝐺𝑗)))
1817fveq2d 6673 . . . . 5 (𝑘 = 𝑗 → (𝑂‘(𝐴 ∩ (𝐺𝑘))) = (𝑂‘(𝐴 ∩ (𝐺𝑗))))
1915, 18eqeq12d 2837 . . . 4 (𝑘 = 𝑗 → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘))) ↔ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))))
2019imbi2d 343 . . 3 (𝑘 = 𝑗 → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘)))) ↔ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗))))))
21 oveq2 7163 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝑀...𝑘) = (𝑀...(𝑗 + 1)))
2221sumeq1d 15057 . . . . 5 (𝑘 = (𝑗 + 1) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))))
23 fveq2 6669 . . . . . . 7 (𝑘 = (𝑗 + 1) → (𝐺𝑘) = (𝐺‘(𝑗 + 1)))
2423ineq2d 4188 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝐴 ∩ (𝐺𝑘)) = (𝐴 ∩ (𝐺‘(𝑗 + 1))))
2524fveq2d 6673 . . . . 5 (𝑘 = (𝑗 + 1) → (𝑂‘(𝐴 ∩ (𝐺𝑘))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
2622, 25eqeq12d 2837 . . . 4 (𝑘 = (𝑗 + 1) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘))) ↔ Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1))))))
2726imbi2d 343 . . 3 (𝑘 = (𝑗 + 1) → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘)))) ↔ (𝜑 → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))))
28 oveq2 7163 . . . . . 6 (𝑘 = 𝐾 → (𝑀...𝑘) = (𝑀...𝐾))
2928sumeq1d 15057 . . . . 5 (𝑘 = 𝐾 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))))
30 fveq2 6669 . . . . . . 7 (𝑘 = 𝐾 → (𝐺𝑘) = (𝐺𝐾))
3130ineq2d 4188 . . . . . 6 (𝑘 = 𝐾 → (𝐴 ∩ (𝐺𝑘)) = (𝐴 ∩ (𝐺𝐾)))
3231fveq2d 6673 . . . . 5 (𝑘 = 𝐾 → (𝑂‘(𝐴 ∩ (𝐺𝑘))) = (𝑂‘(𝐴 ∩ (𝐺𝐾))))
3329, 32eqeq12d 2837 . . . 4 (𝑘 = 𝐾 → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘))) ↔ Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝐾)))))
3433imbi2d 343 . . 3 (𝑘 = 𝐾 → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘)))) ↔ (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝐾))))))
35 eluzel2 12247 . . . . . . . 8 (𝐾 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
363, 35syl 17 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
37 fzsn 12948 . . . . . . 7 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3836, 37syl 17 . . . . . 6 (𝜑 → (𝑀...𝑀) = {𝑀})
3938sumeq1d 15057 . . . . 5 (𝜑 → Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = Σ𝑛 ∈ {𝑀} (𝑂‘(𝐴 ∩ (𝐹𝑛))))
40 carageniuncllem1.o . . . . . . . 8 (𝜑𝑂 ∈ OutMeas)
41 carageniuncllem1.x . . . . . . . 8 𝑋 = dom 𝑂
42 carageniuncllem1.a . . . . . . . 8 (𝜑𝐴𝑋)
43 carageniuncllem1.re . . . . . . . 8 (𝜑 → (𝑂𝐴) ∈ ℝ)
44 inss1 4204 . . . . . . . . 9 (𝐴 ∩ (𝐹𝑀)) ⊆ 𝐴
4544a1i 11 . . . . . . . 8 (𝜑 → (𝐴 ∩ (𝐹𝑀)) ⊆ 𝐴)
4640, 41, 42, 43, 45omessre 42793 . . . . . . 7 (𝜑 → (𝑂‘(𝐴 ∩ (𝐹𝑀))) ∈ ℝ)
4746recnd 10668 . . . . . 6 (𝜑 → (𝑂‘(𝐴 ∩ (𝐹𝑀))) ∈ ℂ)
48 fveq2 6669 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐹𝑛) = (𝐹𝑀))
4948ineq2d 4188 . . . . . . . 8 (𝑛 = 𝑀 → (𝐴 ∩ (𝐹𝑛)) = (𝐴 ∩ (𝐹𝑀)))
5049fveq2d 6673 . . . . . . 7 (𝑛 = 𝑀 → (𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐹𝑀))))
5150sumsn 15100 . . . . . 6 ((𝑀 ∈ ℤ ∧ (𝑂‘(𝐴 ∩ (𝐹𝑀))) ∈ ℂ) → Σ𝑛 ∈ {𝑀} (𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐹𝑀))))
5236, 47, 51syl2anc 586 . . . . 5 (𝜑 → Σ𝑛 ∈ {𝑀} (𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐹𝑀))))
53 eqidd 2822 . . . . . 6 (𝜑 → (𝑂‘(𝐴 ∩ (𝐸𝑀))) = (𝑂‘(𝐴 ∩ (𝐸𝑀))))
54 carageniuncllem1.f . . . . . . . . . 10 𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)))
55 fveq2 6669 . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝐸𝑛) = (𝐸𝑀))
56 oveq2 7163 . . . . . . . . . . . 12 (𝑛 = 𝑀 → (𝑀..^𝑛) = (𝑀..^𝑀))
5756iuneq1d 4945 . . . . . . . . . . 11 (𝑛 = 𝑀 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖))
5855, 57difeq12d 4099 . . . . . . . . . 10 (𝑛 = 𝑀 → ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)) = ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)))
59 uzid 12257 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
6036, 59syl 17 . . . . . . . . . . 11 (𝜑𝑀 ∈ (ℤ𝑀))
612a1i 11 . . . . . . . . . . . 12 (𝜑𝑍 = (ℤ𝑀))
6261eqcomd 2827 . . . . . . . . . . 11 (𝜑 → (ℤ𝑀) = 𝑍)
6360, 62eleqtrd 2915 . . . . . . . . . 10 (𝜑𝑀𝑍)
64 fvex 6682 . . . . . . . . . . . 12 (𝐸𝑀) ∈ V
65 difexg 5230 . . . . . . . . . . . 12 ((𝐸𝑀) ∈ V → ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)) ∈ V)
6664, 65ax-mp 5 . . . . . . . . . . 11 ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)) ∈ V
6766a1i 11 . . . . . . . . . 10 (𝜑 → ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)) ∈ V)
6854, 58, 63, 67fvmptd3 6790 . . . . . . . . 9 (𝜑 → (𝐹𝑀) = ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)))
69 fzo0 13060 . . . . . . . . . . . . 13 (𝑀..^𝑀) = ∅
70 iuneq1 4934 . . . . . . . . . . . . 13 ((𝑀..^𝑀) = ∅ → 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖) = 𝑖 ∈ ∅ (𝐸𝑖))
7169, 70ax-mp 5 . . . . . . . . . . . 12 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖) = 𝑖 ∈ ∅ (𝐸𝑖)
72 0iun 4985 . . . . . . . . . . . 12 𝑖 ∈ ∅ (𝐸𝑖) = ∅
7371, 72eqtri 2844 . . . . . . . . . . 11 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖) = ∅
7473difeq2i 4095 . . . . . . . . . 10 ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)) = ((𝐸𝑀) ∖ ∅)
7574a1i 11 . . . . . . . . 9 (𝜑 → ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)) = ((𝐸𝑀) ∖ ∅))
76 dif0 4331 . . . . . . . . . 10 ((𝐸𝑀) ∖ ∅) = (𝐸𝑀)
7776a1i 11 . . . . . . . . 9 (𝜑 → ((𝐸𝑀) ∖ ∅) = (𝐸𝑀))
7868, 75, 773eqtrd 2860 . . . . . . . 8 (𝜑 → (𝐹𝑀) = (𝐸𝑀))
7978ineq2d 4188 . . . . . . 7 (𝜑 → (𝐴 ∩ (𝐹𝑀)) = (𝐴 ∩ (𝐸𝑀)))
8079fveq2d 6673 . . . . . 6 (𝜑 → (𝑂‘(𝐴 ∩ (𝐹𝑀))) = (𝑂‘(𝐴 ∩ (𝐸𝑀))))
81 carageniuncllem1.g . . . . . . . . . 10 𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))
82 oveq2 7163 . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
8382iuneq1d 4945 . . . . . . . . . 10 (𝑛 = 𝑀 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
84 ovex 7188 . . . . . . . . . . . 12 (𝑀...𝑀) ∈ V
85 fvex 6682 . . . . . . . . . . . 12 (𝐸𝑖) ∈ V
8684, 85iunex 7668 . . . . . . . . . . 11 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V
8786a1i 11 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V)
8881, 83, 63, 87fvmptd3 6790 . . . . . . . . 9 (𝜑 → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
8938iuneq1d 4945 . . . . . . . . 9 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) = 𝑖 ∈ {𝑀} (𝐸𝑖))
90 fveq2 6669 . . . . . . . . . . 11 (𝑖 = 𝑀 → (𝐸𝑖) = (𝐸𝑀))
9190iunxsng 5011 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
9236, 91syl 17 . . . . . . . . 9 (𝜑 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
9388, 89, 923eqtrd 2860 . . . . . . . 8 (𝜑 → (𝐺𝑀) = (𝐸𝑀))
9493ineq2d 4188 . . . . . . 7 (𝜑 → (𝐴 ∩ (𝐺𝑀)) = (𝐴 ∩ (𝐸𝑀)))
9594fveq2d 6673 . . . . . 6 (𝜑 → (𝑂‘(𝐴 ∩ (𝐺𝑀))) = (𝑂‘(𝐴 ∩ (𝐸𝑀))))
9653, 80, 953eqtr4d 2866 . . . . 5 (𝜑 → (𝑂‘(𝐴 ∩ (𝐹𝑀))) = (𝑂‘(𝐴 ∩ (𝐺𝑀))))
9739, 52, 963eqtrd 2860 . . . 4 (𝜑 → Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑀))))
9897a1i 11 . . 3 (𝐾 ∈ (ℤ𝑀) → (𝜑 → Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑀)))))
99 simp3 1134 . . . . 5 ((𝑗 ∈ (𝑀..^𝐾) ∧ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) ∧ 𝜑) → 𝜑)
100 simp1 1132 . . . . 5 ((𝑗 ∈ (𝑀..^𝐾) ∧ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) ∧ 𝜑) → 𝑗 ∈ (𝑀..^𝐾))
101 id 22 . . . . . . 7 ((𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) → (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))))
102101imp 409 . . . . . 6 (((𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) ∧ 𝜑) → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗))))
1031023adant1 1126 . . . . 5 ((𝑗 ∈ (𝑀..^𝐾) ∧ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) ∧ 𝜑) → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗))))
104 elfzouz 13041 . . . . . . . . 9 (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ (ℤ𝑀))
105104adantl 484 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → 𝑗 ∈ (ℤ𝑀))
10640adantr 483 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → 𝑂 ∈ OutMeas)
10742adantr 483 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → 𝐴𝑋)
10843adantr 483 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂𝐴) ∈ ℝ)
109 inss1 4204 . . . . . . . . . . . 12 (𝐴 ∩ (𝐹𝑛)) ⊆ 𝐴
110109a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝐴 ∩ (𝐹𝑛)) ⊆ 𝐴)
111106, 41, 107, 108, 110omessre 42793 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂‘(𝐴 ∩ (𝐹𝑛))) ∈ ℝ)
112111recnd 10668 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂‘(𝐴 ∩ (𝐹𝑛))) ∈ ℂ)
113112adantlr 713 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝐾)) ∧ 𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂‘(𝐴 ∩ (𝐹𝑛))) ∈ ℂ)
114 fveq2 6669 . . . . . . . . . 10 (𝑛 = (𝑗 + 1) → (𝐹𝑛) = (𝐹‘(𝑗 + 1)))
115114ineq2d 4188 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → (𝐴 ∩ (𝐹𝑛)) = (𝐴 ∩ (𝐹‘(𝑗 + 1))))
116115fveq2d 6673 . . . . . . . 8 (𝑛 = (𝑗 + 1) → (𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1)))))
117105, 113, 116fsump1 15110 . . . . . . 7 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))))
1181173adant3 1128 . . . . . 6 ((𝜑𝑗 ∈ (𝑀..^𝐾) ∧ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))))
119 oveq1 7162 . . . . . . 7 𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗))) → (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = ((𝑂‘(𝐴 ∩ (𝐺𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))))
1201193ad2ant3 1131 . . . . . 6 ((𝜑𝑗 ∈ (𝑀..^𝐾) ∧ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) → (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = ((𝑂‘(𝐴 ∩ (𝐺𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))))
121 fzssp1 12949 . . . . . . . . . . . . . . . 16 (𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1))
122 iunss1 4932 . . . . . . . . . . . . . . . 16 ((𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1)) → 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ⊆ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖))
123121, 122ax-mp 5 . . . . . . . . . . . . . . 15 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ⊆ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖)
124123a1i 11 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝐾) → 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ⊆ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖))
125 oveq2 7163 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑀...𝑛) = (𝑀...𝑗))
126125iuneq1d 4945 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
127104, 2eleqtrrdi 2924 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → 𝑗𝑍)
128 ovex 7188 . . . . . . . . . . . . . . . . . 18 (𝑀...𝑗) ∈ V
129128, 85iunex 7668 . . . . . . . . . . . . . . . . 17 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∈ V
130129a1i 11 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∈ V)
13181, 126, 127, 130fvmptd3 6790 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → (𝐺𝑗) = 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
132 oveq2 7163 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑗 + 1) → (𝑀...𝑛) = (𝑀...(𝑗 + 1)))
133132iuneq1d 4945 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖))
134 peano2uz 12300 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (ℤ𝑀) → (𝑗 + 1) ∈ (ℤ𝑀))
135104, 134syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀..^𝐾) → (𝑗 + 1) ∈ (ℤ𝑀))
1362eqcomi 2830 . . . . . . . . . . . . . . . . 17 (ℤ𝑀) = 𝑍
137135, 136eleqtrdi 2923 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → (𝑗 + 1) ∈ 𝑍)
138 ovex 7188 . . . . . . . . . . . . . . . . . 18 (𝑀...(𝑗 + 1)) ∈ V
139138, 85iunex 7668 . . . . . . . . . . . . . . . . 17 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖) ∈ V
140139a1i 11 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖) ∈ V)
14181, 133, 137, 140fvmptd3 6790 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → (𝐺‘(𝑗 + 1)) = 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖))
142131, 141sseq12d 3999 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝐾) → ((𝐺𝑗) ⊆ (𝐺‘(𝑗 + 1)) ↔ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ⊆ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖)))
143124, 142mpbird 259 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑀..^𝐾) → (𝐺𝑗) ⊆ (𝐺‘(𝑗 + 1)))
144 inabs3 41318 . . . . . . . . . . . . 13 ((𝐺𝑗) ⊆ (𝐺‘(𝑗 + 1)) → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗)) = (𝐴 ∩ (𝐺𝑗)))
145143, 144syl 17 . . . . . . . . . . . 12 (𝑗 ∈ (𝑀..^𝐾) → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗)) = (𝐴 ∩ (𝐺𝑗)))
146145fveq2d 6673 . . . . . . . . . . 11 (𝑗 ∈ (𝑀..^𝐾) → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) = (𝑂‘(𝐴 ∩ (𝐺𝑗))))
147146eqcomd 2827 . . . . . . . . . 10 (𝑗 ∈ (𝑀..^𝐾) → (𝑂‘(𝐴 ∩ (𝐺𝑗))) = (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))))
148147adantl 484 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘(𝐴 ∩ (𝐺𝑗))) = (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))))
149 elfzoelz 13037 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ ℤ)
150 fzval3 13105 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℤ → (𝑀...𝑗) = (𝑀..^(𝑗 + 1)))
151149, 150syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀..^𝐾) → (𝑀...𝑗) = (𝑀..^(𝑗 + 1)))
152151eqcomd 2827 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → (𝑀..^(𝑗 + 1)) = (𝑀...𝑗))
153152iuneq1d 4945 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
154153difeq2d 4098 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝐾) → ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
155154adantl 484 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
156 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → (𝐸𝑛) = (𝐸‘(𝑗 + 1)))
157 oveq2 7163 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑗 + 1) → (𝑀..^𝑛) = (𝑀..^(𝑗 + 1)))
158157iuneq1d 4945 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖))
159156, 158difeq12d 4099 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)))
160 fvex 6682 . . . . . . . . . . . . . . . . 17 (𝐸‘(𝑗 + 1)) ∈ V
161 difexg 5230 . . . . . . . . . . . . . . . . 17 ((𝐸‘(𝑗 + 1)) ∈ V → ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)) ∈ V)
162160, 161ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)) ∈ V
163162a1i 11 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)) ∈ V)
16454, 159, 137, 163fvmptd3 6790 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝐾) → (𝐹‘(𝑗 + 1)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)))
165164adantl 484 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐹‘(𝑗 + 1)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)))
166 nfcv 2977 . . . . . . . . . . . . . . . . . 18 𝑖(𝐸‘(𝑗 + 1))
167 fveq2 6669 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑗 + 1) → (𝐸𝑖) = (𝐸‘(𝑗 + 1)))
168166, 104, 167iunp1 41328 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀..^𝐾) → 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖) = ( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∪ (𝐸‘(𝑗 + 1))))
169141, 168eqtrd 2856 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → (𝐺‘(𝑗 + 1)) = ( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∪ (𝐸‘(𝑗 + 1))))
170169, 131difeq12d 4099 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗)) = (( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∪ (𝐸‘(𝑗 + 1))) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
171 difundir 4256 . . . . . . . . . . . . . . . . 17 (( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∪ (𝐸‘(𝑗 + 1))) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)) = (( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)) ∪ ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
172 difid 4329 . . . . . . . . . . . . . . . . . 18 ( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)) = ∅
173172uneq1i 4134 . . . . . . . . . . . . . . . . 17 (( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)) ∪ ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))) = (∅ ∪ ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
174 0un 4345 . . . . . . . . . . . . . . . . 17 (∅ ∪ ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
175171, 173, 1743eqtri 2848 . . . . . . . . . . . . . . . 16 (( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∪ (𝐸‘(𝑗 + 1))) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
176175a1i 11 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → (( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∪ (𝐸‘(𝑗 + 1))) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
177170, 176eqtrd 2856 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝐾) → ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
178177adantl 484 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
179155, 165, 1783eqtr4d 2866 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐹‘(𝑗 + 1)) = ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗)))
180179ineq2d 4188 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐴 ∩ (𝐹‘(𝑗 + 1))) = (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗))))
181 indif2 4246 . . . . . . . . . . . . 13 (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗))) = ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗))
182181eqcomi 2830 . . . . . . . . . . . 12 ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)) = (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗)))
183182a1i 11 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)) = (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗))))
184180, 183eqtr4d 2859 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐴 ∩ (𝐹‘(𝑗 + 1))) = ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))
185184fveq2d 6673 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1)))) = (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗))))
186148, 185oveq12d 7173 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘(𝐴 ∩ (𝐺𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) + (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))))
187 inss1 4204 . . . . . . . . . . . . . 14 ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗)) ⊆ (𝐴 ∩ (𝐺‘(𝑗 + 1)))
188 inss1 4204 . . . . . . . . . . . . . 14 (𝐴 ∩ (𝐺‘(𝑗 + 1))) ⊆ 𝐴
189187, 188sstri 3975 . . . . . . . . . . . . 13 ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗)) ⊆ 𝐴
190189a1i 11 . . . . . . . . . . . 12 (𝜑 → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗)) ⊆ 𝐴)
19140, 41, 42, 43, 190omessre 42793 . . . . . . . . . . 11 (𝜑 → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) ∈ ℝ)
192191adantr 483 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) ∈ ℝ)
19340adantr 483 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → 𝑂 ∈ OutMeas)
19442adantr 483 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → 𝐴𝑋)
19543adantr 483 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂𝐴) ∈ ℝ)
196 difss 4107 . . . . . . . . . . . . 13 ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)) ⊆ (𝐴 ∩ (𝐺‘(𝑗 + 1)))
197196, 188sstri 3975 . . . . . . . . . . . 12 ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)) ⊆ 𝐴
198197a1i 11 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)) ⊆ 𝐴)
199193, 41, 194, 195, 198omessre 42793 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗))) ∈ ℝ)
200 rexadd 12624 . . . . . . . . . 10 (((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) ∈ ℝ ∧ (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗))) ∈ ℝ) → ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) +𝑒 (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))) = ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) + (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))))
201192, 199, 200syl2anc 586 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) +𝑒 (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))) = ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) + (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))))
202201eqcomd 2827 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) + (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))) = ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) +𝑒 (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))))
203 carageniuncllem1.s . . . . . . . . 9 𝑆 = (CaraGen‘𝑂)
204131adantl 484 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐺𝑗) = 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
205 nfv 1911 . . . . . . . . . . . 12 𝑖𝜑
206 fzfid 13340 . . . . . . . . . . . 12 (𝜑 → (𝑀...𝑗) ∈ Fin)
207 carageniuncllem1.e . . . . . . . . . . . . . 14 (𝜑𝐸:𝑍𝑆)
208207adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀...𝑗)) → 𝐸:𝑍𝑆)
209 elfzuz 12903 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀...𝑗) → 𝑖 ∈ (ℤ𝑀))
210136a1i 11 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀...𝑗) → (ℤ𝑀) = 𝑍)
211209, 210eleqtrd 2915 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀...𝑗) → 𝑖𝑍)
212211adantl 484 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀...𝑗)) → 𝑖𝑍)
213208, 212ffvelrnd 6851 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀...𝑗)) → (𝐸𝑖) ∈ 𝑆)
214205, 40, 203, 206, 213caragenfiiuncl 42798 . . . . . . . . . . 11 (𝜑 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∈ 𝑆)
215214adantr 483 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∈ 𝑆)
216204, 215eqeltrd 2913 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐺𝑗) ∈ 𝑆)
21742ssinss1d 41310 . . . . . . . . . 10 (𝜑 → (𝐴 ∩ (𝐺‘(𝑗 + 1))) ⊆ 𝑋)
218217adantr 483 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐴 ∩ (𝐺‘(𝑗 + 1))) ⊆ 𝑋)
219193, 203, 41, 216, 218caragensplit 42783 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) +𝑒 (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
220186, 202, 2193eqtrd 2860 . . . . . . 7 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘(𝐴 ∩ (𝐺𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
2212203adant3 1128 . . . . . 6 ((𝜑𝑗 ∈ (𝑀..^𝐾) ∧ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) → ((𝑂‘(𝐴 ∩ (𝐺𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
222118, 120, 2213eqtrd 2860 . . . . 5 ((𝜑𝑗 ∈ (𝑀..^𝐾) ∧ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
22399, 100, 103, 222syl3anc 1367 . . . 4 ((𝑗 ∈ (𝑀..^𝐾) ∧ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) ∧ 𝜑) → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
2242233exp 1115 . . 3 (𝑗 ∈ (𝑀..^𝐾) → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) → (𝜑 → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))))
22513, 20, 27, 34, 98, 224fzind2 13154 . 2 (𝐾 ∈ (𝑀...𝐾) → (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝐾)))))
2265, 6, 225sylc 65 1 (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝐾))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  Vcvv 3494  cdif 3932  cun 3933  cin 3934  wss 3935  c0 4290  {csn 4566   cuni 4837   ciun 4918  cmpt 5145  dom cdm 5554  wf 6350  cfv 6354  (class class class)co 7155  cc 10534  cr 10535  1c1 10537   + caddc 10539  cz 11980  cuz 12242   +𝑒 cxad 12504  ...cfz 12891  ..^cfzo 13032  Σcsu 15041  OutMeascome 42772  CaraGenccaragen 42774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-oi 8973  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-xadd 12507  df-ico 12743  df-icc 12744  df-fz 12892  df-fzo 13033  df-seq 13369  df-exp 13429  df-hash 13690  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844  df-sum 15042  df-ome 42773  df-caragen 42775
This theorem is referenced by:  carageniuncllem2  42805
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