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Theorem carageniuncllem1 46142
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 2836 . . 3 (𝜑𝐾 ∈ (ℤ𝑀))
4 eluzfz2 13563 . . 3 (𝐾 ∈ (ℤ𝑀) → 𝐾 ∈ (𝑀...𝐾))
53, 4syl 17 . 2 (𝜑𝐾 ∈ (𝑀...𝐾))
6 id 22 . 2 (𝜑𝜑)
7 oveq2 7432 . . . . . 6 (𝑘 = 𝑀 → (𝑀...𝑘) = (𝑀...𝑀))
87sumeq1d 15705 . . . . 5 (𝑘 = 𝑀 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹𝑛))))
9 fveq2 6901 . . . . . . 7 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
109ineq2d 4213 . . . . . 6 (𝑘 = 𝑀 → (𝐴 ∩ (𝐺𝑘)) = (𝐴 ∩ (𝐺𝑀)))
1110fveq2d 6905 . . . . 5 (𝑘 = 𝑀 → (𝑂‘(𝐴 ∩ (𝐺𝑘))) = (𝑂‘(𝐴 ∩ (𝐺𝑀))))
128, 11eqeq12d 2742 . . . 4 (𝑘 = 𝑀 → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘))) ↔ Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑀)))))
1312imbi2d 339 . . 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 7432 . . . . . 6 (𝑘 = 𝑗 → (𝑀...𝑘) = (𝑀...𝑗))
1514sumeq1d 15705 . . . . 5 (𝑘 = 𝑗 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))))
16 fveq2 6901 . . . . . . 7 (𝑘 = 𝑗 → (𝐺𝑘) = (𝐺𝑗))
1716ineq2d 4213 . . . . . 6 (𝑘 = 𝑗 → (𝐴 ∩ (𝐺𝑘)) = (𝐴 ∩ (𝐺𝑗)))
1817fveq2d 6905 . . . . 5 (𝑘 = 𝑗 → (𝑂‘(𝐴 ∩ (𝐺𝑘))) = (𝑂‘(𝐴 ∩ (𝐺𝑗))))
1915, 18eqeq12d 2742 . . . 4 (𝑘 = 𝑗 → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘))) ↔ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))))
2019imbi2d 339 . . 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 7432 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝑀...𝑘) = (𝑀...(𝑗 + 1)))
2221sumeq1d 15705 . . . . 5 (𝑘 = (𝑗 + 1) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))))
23 fveq2 6901 . . . . . . 7 (𝑘 = (𝑗 + 1) → (𝐺𝑘) = (𝐺‘(𝑗 + 1)))
2423ineq2d 4213 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝐴 ∩ (𝐺𝑘)) = (𝐴 ∩ (𝐺‘(𝑗 + 1))))
2524fveq2d 6905 . . . . 5 (𝑘 = (𝑗 + 1) → (𝑂‘(𝐴 ∩ (𝐺𝑘))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
2622, 25eqeq12d 2742 . . . 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 339 . . 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 7432 . . . . . 6 (𝑘 = 𝐾 → (𝑀...𝑘) = (𝑀...𝐾))
2928sumeq1d 15705 . . . . 5 (𝑘 = 𝐾 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))))
30 fveq2 6901 . . . . . . 7 (𝑘 = 𝐾 → (𝐺𝑘) = (𝐺𝐾))
3130ineq2d 4213 . . . . . 6 (𝑘 = 𝐾 → (𝐴 ∩ (𝐺𝑘)) = (𝐴 ∩ (𝐺𝐾)))
3231fveq2d 6905 . . . . 5 (𝑘 = 𝐾 → (𝑂‘(𝐴 ∩ (𝐺𝑘))) = (𝑂‘(𝐴 ∩ (𝐺𝐾))))
3329, 32eqeq12d 2742 . . . 4 (𝑘 = 𝐾 → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑘))) ↔ Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝐾)))))
3433imbi2d 339 . . 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 12879 . . . . . . . 8 (𝐾 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
363, 35syl 17 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
37 fzsn 13597 . . . . . . 7 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3836, 37syl 17 . . . . . 6 (𝜑 → (𝑀...𝑀) = {𝑀})
3938sumeq1d 15705 . . . . 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 4230 . . . . . . . . 9 (𝐴 ∩ (𝐹𝑀)) ⊆ 𝐴
4544a1i 11 . . . . . . . 8 (𝜑 → (𝐴 ∩ (𝐹𝑀)) ⊆ 𝐴)
4640, 41, 42, 43, 45omessre 46131 . . . . . . 7 (𝜑 → (𝑂‘(𝐴 ∩ (𝐹𝑀))) ∈ ℝ)
4746recnd 11292 . . . . . 6 (𝜑 → (𝑂‘(𝐴 ∩ (𝐹𝑀))) ∈ ℂ)
48 fveq2 6901 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐹𝑛) = (𝐹𝑀))
4948ineq2d 4213 . . . . . . . 8 (𝑛 = 𝑀 → (𝐴 ∩ (𝐹𝑛)) = (𝐴 ∩ (𝐹𝑀)))
5049fveq2d 6905 . . . . . . 7 (𝑛 = 𝑀 → (𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐹𝑀))))
5150sumsn 15750 . . . . . 6 ((𝑀 ∈ ℤ ∧ (𝑂‘(𝐴 ∩ (𝐹𝑀))) ∈ ℂ) → Σ𝑛 ∈ {𝑀} (𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐹𝑀))))
5236, 47, 51syl2anc 582 . . . . 5 (𝜑 → Σ𝑛 ∈ {𝑀} (𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐹𝑀))))
53 eqidd 2727 . . . . . 6 (𝜑 → (𝑂‘(𝐴 ∩ (𝐸𝑀))) = (𝑂‘(𝐴 ∩ (𝐸𝑀))))
54 carageniuncllem1.f . . . . . . . . . 10 𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)))
55 fveq2 6901 . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝐸𝑛) = (𝐸𝑀))
56 oveq2 7432 . . . . . . . . . . . 12 (𝑛 = 𝑀 → (𝑀..^𝑛) = (𝑀..^𝑀))
5756iuneq1d 5028 . . . . . . . . . . 11 (𝑛 = 𝑀 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖))
5855, 57difeq12d 4122 . . . . . . . . . 10 (𝑛 = 𝑀 → ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)) = ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)))
59 uzid 12889 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
6036, 59syl 17 . . . . . . . . . . 11 (𝜑𝑀 ∈ (ℤ𝑀))
612a1i 11 . . . . . . . . . . . 12 (𝜑𝑍 = (ℤ𝑀))
6261eqcomd 2732 . . . . . . . . . . 11 (𝜑 → (ℤ𝑀) = 𝑍)
6360, 62eleqtrd 2828 . . . . . . . . . 10 (𝜑𝑀𝑍)
64 fvex 6914 . . . . . . . . . . . 12 (𝐸𝑀) ∈ V
65 difexg 5334 . . . . . . . . . . . 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 7032 . . . . . . . . 9 (𝜑 → (𝐹𝑀) = ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)))
69 fzo0 13710 . . . . . . . . . . . . 13 (𝑀..^𝑀) = ∅
70 iuneq1 5017 . . . . . . . . . . . . 13 ((𝑀..^𝑀) = ∅ → 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖) = 𝑖 ∈ ∅ (𝐸𝑖))
7169, 70ax-mp 5 . . . . . . . . . . . 12 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖) = 𝑖 ∈ ∅ (𝐸𝑖)
72 0iun 5071 . . . . . . . . . . . 12 𝑖 ∈ ∅ (𝐸𝑖) = ∅
7371, 72eqtri 2754 . . . . . . . . . . 11 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖) = ∅
7473difeq2i 4118 . . . . . . . . . 10 ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)) = ((𝐸𝑀) ∖ ∅)
7574a1i 11 . . . . . . . . 9 (𝜑 → ((𝐸𝑀) ∖ 𝑖 ∈ (𝑀..^𝑀)(𝐸𝑖)) = ((𝐸𝑀) ∖ ∅))
76 dif0 4377 . . . . . . . . . 10 ((𝐸𝑀) ∖ ∅) = (𝐸𝑀)
7776a1i 11 . . . . . . . . 9 (𝜑 → ((𝐸𝑀) ∖ ∅) = (𝐸𝑀))
7868, 75, 773eqtrd 2770 . . . . . . . 8 (𝜑 → (𝐹𝑀) = (𝐸𝑀))
7978ineq2d 4213 . . . . . . 7 (𝜑 → (𝐴 ∩ (𝐹𝑀)) = (𝐴 ∩ (𝐸𝑀)))
8079fveq2d 6905 . . . . . 6 (𝜑 → (𝑂‘(𝐴 ∩ (𝐹𝑀))) = (𝑂‘(𝐴 ∩ (𝐸𝑀))))
81 carageniuncllem1.g . . . . . . . . . 10 𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))
82 oveq2 7432 . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
8382iuneq1d 5028 . . . . . . . . . 10 (𝑛 = 𝑀 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
84 ovex 7457 . . . . . . . . . . . 12 (𝑀...𝑀) ∈ V
85 fvex 6914 . . . . . . . . . . . 12 (𝐸𝑖) ∈ V
8684, 85iunex 7982 . . . . . . . . . . 11 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V
8786a1i 11 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V)
8881, 83, 63, 87fvmptd3 7032 . . . . . . . . 9 (𝜑 → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
8938iuneq1d 5028 . . . . . . . . 9 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) = 𝑖 ∈ {𝑀} (𝐸𝑖))
90 fveq2 6901 . . . . . . . . . . 11 (𝑖 = 𝑀 → (𝐸𝑖) = (𝐸𝑀))
9190iunxsng 5098 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
9236, 91syl 17 . . . . . . . . 9 (𝜑 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
9388, 89, 923eqtrd 2770 . . . . . . . 8 (𝜑 → (𝐺𝑀) = (𝐸𝑀))
9493ineq2d 4213 . . . . . . 7 (𝜑 → (𝐴 ∩ (𝐺𝑀)) = (𝐴 ∩ (𝐸𝑀)))
9594fveq2d 6905 . . . . . 6 (𝜑 → (𝑂‘(𝐴 ∩ (𝐺𝑀))) = (𝑂‘(𝐴 ∩ (𝐸𝑀))))
9653, 80, 953eqtr4d 2776 . . . . 5 (𝜑 → (𝑂‘(𝐴 ∩ (𝐹𝑀))) = (𝑂‘(𝐴 ∩ (𝐺𝑀))))
9739, 52, 963eqtrd 2770 . . . 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 1135 . . . . 5 ((𝑗 ∈ (𝑀..^𝐾) ∧ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) ∧ 𝜑) → 𝜑)
100 simp1 1133 . . . . 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 405 . . . . . 6 (((𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗)))) ∧ 𝜑) → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝑗))))
1031023adant1 1127 . . . . 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 13690 . . . . . . . . 9 (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ (ℤ𝑀))
105104adantl 480 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → 𝑗 ∈ (ℤ𝑀))
10640adantr 479 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → 𝑂 ∈ OutMeas)
10742adantr 479 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → 𝐴𝑋)
10843adantr 479 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂𝐴) ∈ ℝ)
109 inss1 4230 . . . . . . . . . . . 12 (𝐴 ∩ (𝐹𝑛)) ⊆ 𝐴
110109a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝐴 ∩ (𝐹𝑛)) ⊆ 𝐴)
111106, 41, 107, 108, 110omessre 46131 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂‘(𝐴 ∩ (𝐹𝑛))) ∈ ℝ)
112111recnd 11292 . . . . . . . . 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 6901 . . . . . . . . . 10 (𝑛 = (𝑗 + 1) → (𝐹𝑛) = (𝐹‘(𝑗 + 1)))
115114ineq2d 4213 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → (𝐴 ∩ (𝐹𝑛)) = (𝐴 ∩ (𝐹‘(𝑗 + 1))))
116115fveq2d 6905 . . . . . . . 8 (𝑛 = (𝑗 + 1) → (𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1)))))
117105, 113, 116fsump1 15760 . . . . . . 7 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹𝑛))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))))
1181173adant3 1129 . . . . . 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 7431 . . . . . . 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 1132 . . . . . 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 13598 . . . . . . . . . . . . . . . 16 (𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1))
122 iunss1 5015 . . . . . . . . . . . . . . . 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 7432 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑀...𝑛) = (𝑀...𝑗))
126125iuneq1d 5028 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
127104, 2eleqtrrdi 2837 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → 𝑗𝑍)
128 ovex 7457 . . . . . . . . . . . . . . . . . 18 (𝑀...𝑗) ∈ V
129128, 85iunex 7982 . . . . . . . . . . . . . . . . 17 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∈ V
130129a1i 11 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∈ V)
13181, 126, 127, 130fvmptd3 7032 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → (𝐺𝑗) = 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
132 oveq2 7432 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑗 + 1) → (𝑀...𝑛) = (𝑀...(𝑗 + 1)))
133132iuneq1d 5028 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖))
134 peano2uz 12937 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (ℤ𝑀) → (𝑗 + 1) ∈ (ℤ𝑀))
135104, 134syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀..^𝐾) → (𝑗 + 1) ∈ (ℤ𝑀))
1362eqcomi 2735 . . . . . . . . . . . . . . . . 17 (ℤ𝑀) = 𝑍
137135, 136eleqtrdi 2836 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → (𝑗 + 1) ∈ 𝑍)
138 ovex 7457 . . . . . . . . . . . . . . . . . 18 (𝑀...(𝑗 + 1)) ∈ V
139138, 85iunex 7982 . . . . . . . . . . . . . . . . 17 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖) ∈ V
140139a1i 11 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖) ∈ V)
14181, 133, 137, 140fvmptd3 7032 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → (𝐺‘(𝑗 + 1)) = 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖))
142131, 141sseq12d 4013 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝐾) → ((𝐺𝑗) ⊆ (𝐺‘(𝑗 + 1)) ↔ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ⊆ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖)))
143124, 142mpbird 256 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑀..^𝐾) → (𝐺𝑗) ⊆ (𝐺‘(𝑗 + 1)))
144 inabs3 44657 . . . . . . . . . . . . 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 6905 . . . . . . . . . . 11 (𝑗 ∈ (𝑀..^𝐾) → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) = (𝑂‘(𝐴 ∩ (𝐺𝑗))))
147146eqcomd 2732 . . . . . . . . . 10 (𝑗 ∈ (𝑀..^𝐾) → (𝑂‘(𝐴 ∩ (𝐺𝑗))) = (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))))
148147adantl 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘(𝐴 ∩ (𝐺𝑗))) = (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))))
149 elfzoelz 13686 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ ℤ)
150 fzval3 13755 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℤ → (𝑀...𝑗) = (𝑀..^(𝑗 + 1)))
151149, 150syl 17 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀..^𝐾) → (𝑀...𝑗) = (𝑀..^(𝑗 + 1)))
152151eqcomd 2732 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → (𝑀..^(𝑗 + 1)) = (𝑀...𝑗))
153152iuneq1d 5028 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
154153difeq2d 4121 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝐾) → ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
155154adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
156 fveq2 6901 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → (𝐸𝑛) = (𝐸‘(𝑗 + 1)))
157 oveq2 7432 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑗 + 1) → (𝑀..^𝑛) = (𝑀..^(𝑗 + 1)))
158157iuneq1d 5028 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖))
159156, 158difeq12d 4122 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)))
160 fvex 6914 . . . . . . . . . . . . . . . . 17 (𝐸‘(𝑗 + 1)) ∈ V
161 difexg 5334 . . . . . . . . . . . . . . . . 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 7032 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝐾) → (𝐹‘(𝑗 + 1)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)))
165164adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐹‘(𝑗 + 1)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸𝑖)))
166 nfcv 2892 . . . . . . . . . . . . . . . . . 18 𝑖(𝐸‘(𝑗 + 1))
167 fveq2 6901 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑗 + 1) → (𝐸𝑖) = (𝐸‘(𝑗 + 1)))
168166, 104, 167iunp1 44667 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀..^𝐾) → 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸𝑖) = ( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∪ (𝐸‘(𝑗 + 1))))
169141, 168eqtrd 2766 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (𝑀..^𝐾) → (𝐺‘(𝑗 + 1)) = ( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∪ (𝐸‘(𝑗 + 1))))
170169, 131difeq12d 4122 . . . . . . . . . . . . . . 15 (𝑗 ∈ (𝑀..^𝐾) → ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗)) = (( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∪ (𝐸‘(𝑗 + 1))) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
171 difundir 4282 . . . . . . . . . . . . . . . . 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 4375 . . . . . . . . . . . . . . . . . 18 ( 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)) = ∅
173172uneq1i 4159 . . . . . . . . . . . . . . . . 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 4397 . . . . . . . . . . . . . . . . 17 (∅ ∪ ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
175171, 173, 1743eqtri 2758 . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝐾) → ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
178177adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗)) = ((𝐸‘(𝑗 + 1)) ∖ 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖)))
179155, 165, 1783eqtr4d 2776 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐹‘(𝑗 + 1)) = ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗)))
180179ineq2d 4213 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐴 ∩ (𝐹‘(𝑗 + 1))) = (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗))))
181 indif2 4272 . . . . . . . . . . . . 13 (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺𝑗))) = ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗))
182181eqcomi 2735 . . . . . . . . . . . 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 2769 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐴 ∩ (𝐹‘(𝑗 + 1))) = ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)))
185184fveq2d 6905 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1)))) = (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗))))
186148, 185oveq12d 7442 . . . . . . . 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 4230 . . . . . . . . . . . . . 14 ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗)) ⊆ (𝐴 ∩ (𝐺‘(𝑗 + 1)))
188 inss1 4230 . . . . . . . . . . . . . 14 (𝐴 ∩ (𝐺‘(𝑗 + 1))) ⊆ 𝐴
189187, 188sstri 3989 . . . . . . . . . . . . 13 ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗)) ⊆ 𝐴
190189a1i 11 . . . . . . . . . . . 12 (𝜑 → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗)) ⊆ 𝐴)
19140, 41, 42, 43, 190omessre 46131 . . . . . . . . . . 11 (𝜑 → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) ∈ ℝ)
192191adantr 479 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺𝑗))) ∈ ℝ)
19340adantr 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → 𝑂 ∈ OutMeas)
19442adantr 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → 𝐴𝑋)
19543adantr 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂𝐴) ∈ ℝ)
196 difss 4131 . . . . . . . . . . . . 13 ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)) ⊆ (𝐴 ∩ (𝐺‘(𝑗 + 1)))
197196, 188sstri 3989 . . . . . . . . . . . 12 ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)) ⊆ 𝐴
198197a1i 11 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗)) ⊆ 𝐴)
199193, 41, 194, 195, 198omessre 46131 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺𝑗))) ∈ ℝ)
200 rexadd 13265 . . . . . . . . . 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 582 . . . . . . . . 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 2732 . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐺𝑗) = 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖))
205 nfv 1910 . . . . . . . . . . . 12 𝑖𝜑
206 fzfid 13993 . . . . . . . . . . . 12 (𝜑 → (𝑀...𝑗) ∈ Fin)
207 carageniuncllem1.e . . . . . . . . . . . . . 14 (𝜑𝐸:𝑍𝑆)
208207adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀...𝑗)) → 𝐸:𝑍𝑆)
209 elfzuz 13551 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀...𝑗) → 𝑖 ∈ (ℤ𝑀))
210136a1i 11 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀...𝑗) → (ℤ𝑀) = 𝑍)
211209, 210eleqtrd 2828 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀...𝑗) → 𝑖𝑍)
212211adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀...𝑗)) → 𝑖𝑍)
213208, 212ffvelcdmd 7099 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀...𝑗)) → (𝐸𝑖) ∈ 𝑆)
214205, 40, 203, 206, 213caragenfiiuncl 46136 . . . . . . . . . . 11 (𝜑 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∈ 𝑆)
215214adantr 479 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → 𝑖 ∈ (𝑀...𝑗)(𝐸𝑖) ∈ 𝑆)
216204, 215eqeltrd 2826 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐺𝑗) ∈ 𝑆)
21742ssinss1d 44649 . . . . . . . . . 10 (𝜑 → (𝐴 ∩ (𝐺‘(𝑗 + 1))) ⊆ 𝑋)
218217adantr 479 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → (𝐴 ∩ (𝐺‘(𝑗 + 1))) ⊆ 𝑋)
219193, 203, 41, 216, 218caragensplit 46121 . . . . . . . 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 2770 . . . . . . 7 ((𝜑𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘(𝐴 ∩ (𝐺𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
2212203adant3 1129 . . . . . 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 2770 . . . . 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 1368 . . . 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 1116 . . 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 13805 . 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 394  w3a 1084   = wceq 1534  wcel 2099  Vcvv 3462  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4325  {csn 4633   cuni 4913   ciun 5001  cmpt 5236  dom cdm 5682  wf 6550  cfv 6554  (class class class)co 7424  cc 11156  cr 11157  1c1 11159   + caddc 11161  cz 12610  cuz 12874   +𝑒 cxad 13144  ...cfz 13538  ..^cfzo 13681  Σcsu 15690  OutMeascome 46110  CaraGenccaragen 46112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-rp 13029  df-xadd 13147  df-ico 13384  df-icc 13385  df-fz 13539  df-fzo 13682  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-sum 15691  df-ome 46111  df-caragen 46113
This theorem is referenced by:  carageniuncllem2  46143
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