carageniuncllem1 - Mathbox for Glauco Siliprandi

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Theorem carageniuncllem1 43949

Description: The outer measure of 𝐴 ∩ (𝐺‘𝑛) is the sum of the outer measures of 𝐴 ∩ (𝐹‘𝑚). 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	⊢ (𝜑 → 𝑂 ∈ OutMeas)
carageniuncllem1.s	⊢ 𝑆 = (CaraGen‘𝑂)
carageniuncllem1.x	⊢ 𝑋 = ∪ dom 𝑂
carageniuncllem1.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
carageniuncllem1.re	⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ)
carageniuncllem1.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
carageniuncllem1.e	⊢ (𝜑 → 𝐸:𝑍⟶𝑆)
carageniuncllem1.g	⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))
carageniuncllem1.f	⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖)))
carageniuncllem1.k	⊢ (𝜑 → 𝐾 ∈ 𝑍)

**Assertion**
Ref	Expression
carageniuncllem1	⊢ (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝐾))))

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.

Step	Hyp	Ref	Expression
1		carageniuncllem1.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑍)
2		carageniuncllem1.z	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3	1, 2	eleqtrdi 2849	. . 3 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘𝑀))
4		eluzfz2 13193	. . 3 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → 𝐾 ∈ (𝑀...𝐾))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝐾))
6		id 22	. 2 ⊢ (𝜑 → 𝜑)
7		oveq2 7263	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝑀...𝑘) = (𝑀...𝑀))
8	7	sumeq1d 15341	. . . . 5 ⊢ (𝑘 = 𝑀 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
9		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀))
10	9	ineq2d 4143	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐴 ∩ (𝐺‘𝑘)) = (𝐴 ∩ (𝐺‘𝑀)))
11	10	fveq2d 6760	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑀))))
12	8, 11	eqeq12d 2754	. . . 4 ⊢ (𝑘 = 𝑀 → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ↔ Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑀)))))
13	12	imbi2d 340	. . 3 ⊢ (𝑘 = 𝑀 → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘)))) ↔ (𝜑 → Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑀))))))
14		oveq2 7263	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑀...𝑘) = (𝑀...𝑗))
15	14	sumeq1d 15341	. . . . 5 ⊢ (𝑘 = 𝑗 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
16		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗))
17	16	ineq2d 4143	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴 ∩ (𝐺‘𝑘)) = (𝐴 ∩ (𝐺‘𝑗)))
18	17	fveq2d 6760	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗))))
19	15, 18	eqeq12d 2754	. . . 4 ⊢ (𝑘 = 𝑗 → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ↔ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))))
20	19	imbi2d 340	. . 3 ⊢ (𝑘 = 𝑗 → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘)))) ↔ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗))))))
21		oveq2 7263	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝑀...𝑘) = (𝑀...(𝑗 + 1)))
22	21	sumeq1d 15341	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
23		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = (𝑗 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑗 + 1)))
24	23	ineq2d 4143	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝐴 ∩ (𝐺‘𝑘)) = (𝐴 ∩ (𝐺‘(𝑗 + 1))))
25	24	fveq2d 6760	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
26	22, 25	eqeq12d 2754	. . . 4 ⊢ (𝑘 = (𝑗 + 1) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ↔ Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1))))))
27	26	imbi2d 340	. . 3 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘)))) ↔ (𝜑 → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))))
28		oveq2 7263	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑀...𝑘) = (𝑀...𝐾))
29	28	sumeq1d 15341	. . . . 5 ⊢ (𝑘 = 𝐾 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
30		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐺‘𝑘) = (𝐺‘𝐾))
31	30	ineq2d 4143	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐴 ∩ (𝐺‘𝑘)) = (𝐴 ∩ (𝐺‘𝐾)))
32	31	fveq2d 6760	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) = (𝑂‘(𝐴 ∩ (𝐺‘𝐾))))
33	29, 32	eqeq12d 2754	. . . 4 ⊢ (𝑘 = 𝐾 → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ↔ Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝐾)))))
34	33	imbi2d 340	. . 3 ⊢ (𝑘 = 𝐾 → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘)))) ↔ (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝐾))))))
35		eluzel2 12516	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
36	3, 35	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
37		fzsn 13227	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
39	38	sumeq1d 15341	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = Σ𝑛 ∈ {𝑀} (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
40		carageniuncllem1.o	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
41		carageniuncllem1.x	. . . . . . . 8 ⊢ 𝑋 = ∪ dom 𝑂
42		carageniuncllem1.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
43		carageniuncllem1.re	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ)
44		inss1 4159	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐹‘𝑀)) ⊆ 𝐴
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑀)) ⊆ 𝐴)
46	40, 41, 42, 43, 45	omessre 43938	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑀))) ∈ ℝ)
47	46	recnd 10934	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑀))) ∈ ℂ)
48		fveq2 6756	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (𝐹‘𝑛) = (𝐹‘𝑀))
49	48	ineq2d 4143	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (𝐴 ∩ (𝐹‘𝑛)) = (𝐴 ∩ (𝐹‘𝑀)))
50	49	fveq2d 6760	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐹‘𝑀))))
51	50	sumsn 15386	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑂‘(𝐴 ∩ (𝐹‘𝑀))) ∈ ℂ) → Σ𝑛 ∈ {𝑀} (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐹‘𝑀))))
52	36, 47, 51	syl2anc 583	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ {𝑀} (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐹‘𝑀))))
53		eqidd 2739	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐸‘𝑀))) = (𝑂‘(𝐴 ∩ (𝐸‘𝑀))))
54		carageniuncllem1.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖)))
55		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → (𝐸‘𝑛) = (𝐸‘𝑀))
56		oveq2 7263	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑀 → (𝑀..^𝑛) = (𝑀..^𝑀))
57	56	iuneq1d 4948	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖))
58	55, 57	difeq12d 4054	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖)) = ((𝐸‘𝑀) ∖ ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖)))
59		uzid 12526	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
60	36, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
61	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 = (ℤ_≥‘𝑀))
62	61	eqcomd 2744	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ_≥‘𝑀) = 𝑍)
63	60, 62	eleqtrd 2841	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
64		fvex 6769	. . . . . . . . . . . 12 ⊢ (𝐸‘𝑀) ∈ V
65		difexg 5246	. . . . . . . . . . . 12 ⊢ ((𝐸‘𝑀) ∈ V → ((𝐸‘𝑀) ∖ ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖)) ∈ V)
66	64, 65	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑀) ∖ ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖)) ∈ V
67	66	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘𝑀) ∖ ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖)) ∈ V)
68	54, 58, 63, 67	fvmptd3 6880	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑀) = ((𝐸‘𝑀) ∖ ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖)))
69		fzo0 13339	. . . . . . . . . . . . 13 ⊢ (𝑀..^𝑀) = ∅
70		iuneq1 4937	. . . . . . . . . . . . 13 ⊢ ((𝑀..^𝑀) = ∅ → ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖) = ∪ 𝑖 ∈ ∅ (𝐸‘𝑖))
71	69, 70	ax-mp 5	. . . . . . . . . . . 12 ⊢ ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖) = ∪ 𝑖 ∈ ∅ (𝐸‘𝑖)
72		0iun 4988	. . . . . . . . . . . 12 ⊢ ∪ 𝑖 ∈ ∅ (𝐸‘𝑖) = ∅
73	71, 72	eqtri 2766	. . . . . . . . . . 11 ⊢ ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖) = ∅
74	73	difeq2i 4050	. . . . . . . . . 10 ⊢ ((𝐸‘𝑀) ∖ ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖)) = ((𝐸‘𝑀) ∖ ∅)
75	74	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸‘𝑀) ∖ ∪ 𝑖 ∈ (𝑀..^𝑀)(𝐸‘𝑖)) = ((𝐸‘𝑀) ∖ ∅))
76		dif0 4303	. . . . . . . . . 10 ⊢ ((𝐸‘𝑀) ∖ ∅) = (𝐸‘𝑀)
77	76	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸‘𝑀) ∖ ∅) = (𝐸‘𝑀))
78	68, 75, 77	3eqtrd 2782	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐸‘𝑀))
79	78	ineq2d 4143	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑀)) = (𝐴 ∩ (𝐸‘𝑀)))
80	79	fveq2d 6760	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑀))) = (𝑂‘(𝐴 ∩ (𝐸‘𝑀))))
81		carageniuncllem1.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))
82		oveq2 7263	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
83	82	iuneq1d 4948	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖))
84		ovex 7288	. . . . . . . . . . . 12 ⊢ (𝑀...𝑀) ∈ V
85		fvex 6769	. . . . . . . . . . . 12 ⊢ (𝐸‘𝑖) ∈ V
86	84, 85	iunex 7784	. . . . . . . . . . 11 ⊢ ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) ∈ V
87	86	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) ∈ V)
88	81, 83, 63, 87	fvmptd3 6880	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑀) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖))
89	38	iuneq1d 4948	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) = ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖))
90		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑀 → (𝐸‘𝑖) = (𝐸‘𝑀))
91	90	iunxsng 5015	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖) = (𝐸‘𝑀))
92	36, 91	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖) = (𝐸‘𝑀))
93	88, 89, 92	3eqtrd 2782	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑀) = (𝐸‘𝑀))
94	93	ineq2d 4143	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ (𝐺‘𝑀)) = (𝐴 ∩ (𝐸‘𝑀)))
95	94	fveq2d 6760	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐺‘𝑀))) = (𝑂‘(𝐴 ∩ (𝐸‘𝑀))))
96	53, 80, 95	3eqtr4d 2788	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑀))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑀))))
97	39, 52, 96	3eqtrd 2782	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑀))))
98	97	a1i 11	. . 3 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝜑 → Σ𝑛 ∈ (𝑀...𝑀)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑀)))))
99		simp3 1136	. . . . 5 ⊢ ((𝑗 ∈ (𝑀..^𝐾) ∧ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) ∧ 𝜑) → 𝜑)
100		simp1 1134	. . . . 5 ⊢ ((𝑗 ∈ (𝑀..^𝐾) ∧ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) ∧ 𝜑) → 𝑗 ∈ (𝑀..^𝐾))
101		id 22	. . . . . . 7 ⊢ ((𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) → (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))))
102	101	imp 406	. . . . . 6 ⊢ (((𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) ∧ 𝜑) → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗))))
103	102	3adant1 1128	. . . . 5 ⊢ ((𝑗 ∈ (𝑀..^𝐾) ∧ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) ∧ 𝜑) → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗))))
104		elfzouz 13320	. . . . . . . . 9 ⊢ (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ (ℤ_≥‘𝑀))
105	104	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → 𝑗 ∈ (ℤ_≥‘𝑀))
106	40	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑗 + 1))) → 𝑂 ∈ OutMeas)
107	42	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑗 + 1))) → 𝐴 ⊆ 𝑋)
108	43	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂‘𝐴) ∈ ℝ)
109		inss1 4159	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴
110	109	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴)
111	106, 41, 107, 108, 110	omessre 43938	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
112	111	recnd 10934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℂ)
113	112	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) ∧ 𝑛 ∈ (𝑀...(𝑗 + 1))) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℂ)
114		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑛 = (𝑗 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑗 + 1)))
115	114	ineq2d 4143	. . . . . . . . 9 ⊢ (𝑛 = (𝑗 + 1) → (𝐴 ∩ (𝐹‘𝑛)) = (𝐴 ∩ (𝐹‘(𝑗 + 1))))
116	115	fveq2d 6760	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1)))))
117	105, 113, 116	fsump1 15396	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))))
118	117	3adant3 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾) ∧ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))))
119		oveq1 7262	. . . . . . 7 ⊢ (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗))) → (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))))
120	119	3ad2ant3 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾) ∧ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) → (Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))))
121		fzssp1 13228	. . . . . . . . . . . . . . . 16 ⊢ (𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1))
122		iunss1 4935	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀...𝑗) ⊆ (𝑀...(𝑗 + 1)) → ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ⊆ ∪ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸‘𝑖))
123	121, 122	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ⊆ ∪ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸‘𝑖)
124	123	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ⊆ ∪ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸‘𝑖))
125		oveq2 7263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (𝑀...𝑛) = (𝑀...𝑗))
126	125	iuneq1d 4948	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖))
127	104, 2	eleqtrrdi 2850	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ 𝑍)
128		ovex 7288	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀...𝑗) ∈ V
129	128, 85	iunex 7784	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∈ V
130	129	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∈ V)
131	81, 126, 127, 130	fvmptd3 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝐺‘𝑗) = ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖))
132		oveq2 7263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑗 + 1) → (𝑀...𝑛) = (𝑀...(𝑗 + 1)))
133	132	iuneq1d 4948	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸‘𝑖))
134		peano2uz 12570	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (𝑗 + 1) ∈ (ℤ_≥‘𝑀))
135	104, 134	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝑗 + 1) ∈ (ℤ_≥‘𝑀))
136	2	eqcomi 2747	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘𝑀) = 𝑍
137	135, 136	eleqtrdi 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝑗 + 1) ∈ 𝑍)
138		ovex 7288	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀...(𝑗 + 1)) ∈ V
139	138, 85	iunex 7784	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸‘𝑖) ∈ V
140	139	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ∪ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸‘𝑖) ∈ V)
141	81, 133, 137, 140	fvmptd3 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝐺‘(𝑗 + 1)) = ∪ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸‘𝑖))
142	131, 141	sseq12d 3950	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ((𝐺‘𝑗) ⊆ (𝐺‘(𝑗 + 1)) ↔ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ⊆ ∪ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸‘𝑖)))
143	124, 142	mpbird 256	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝐺‘𝑗) ⊆ (𝐺‘(𝑗 + 1)))
144		inabs3 42493	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑗) ⊆ (𝐺‘(𝑗 + 1)) → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗)) = (𝐴 ∩ (𝐺‘𝑗)))
145	143, 144	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗)) = (𝐴 ∩ (𝐺‘𝑗)))
146	145	fveq2d 6760	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗))))
147	146	eqcomd 2744	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝑂‘(𝐴 ∩ (𝐺‘𝑗))) = (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))))
148	147	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘(𝐴 ∩ (𝐺‘𝑗))) = (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))))
149		elfzoelz 13316	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ ℤ)
150		fzval3 13384	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℤ → (𝑀...𝑗) = (𝑀..^(𝑗 + 1)))
151	149, 150	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝑀...𝑗) = (𝑀..^(𝑗 + 1)))
152	151	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝑀..^(𝑗 + 1)) = (𝑀...𝑗))
153	152	iuneq1d 4948	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖))
154	153	difeq2d 4053	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)))
155	154	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)))
156		fveq2 6756	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → (𝐸‘𝑛) = (𝐸‘(𝑗 + 1)))
157		oveq2 7263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑗 + 1) → (𝑀..^𝑛) = (𝑀..^(𝑗 + 1)))
158	157	iuneq1d 4948	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖))
159	156, 158	difeq12d 4054	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖)))
160		fvex 6769	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸‘(𝑗 + 1)) ∈ V
161		difexg 5246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘(𝑗 + 1)) ∈ V → ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖)) ∈ V)
162	160, 161	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖)) ∈ V
163	162	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖)) ∈ V)
164	54, 159, 137, 163	fvmptd3 6880	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝐹‘(𝑗 + 1)) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖)))
165	164	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝐹‘(𝑗 + 1)) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀..^(𝑗 + 1))(𝐸‘𝑖)))
166		nfcv 2906	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝐸‘(𝑗 + 1))
167		fveq2 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑗 + 1) → (𝐸‘𝑖) = (𝐸‘(𝑗 + 1)))
168	166, 104, 167	iunp1 42503	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ∪ 𝑖 ∈ (𝑀...(𝑗 + 1))(𝐸‘𝑖) = (∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∪ (𝐸‘(𝑗 + 1))))
169	141, 168	eqtrd 2778	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (𝑀..^𝐾) → (𝐺‘(𝑗 + 1)) = (∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∪ (𝐸‘(𝑗 + 1))))
170	169, 131	difeq12d 4054	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ((𝐺‘(𝑗 + 1)) ∖ (𝐺‘𝑗)) = ((∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∪ (𝐸‘(𝑗 + 1))) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)))
171		difundir 4211	. . . . . . . . . . . . . . . . 17 ⊢ ((∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∪ (𝐸‘(𝑗 + 1))) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)) = ((∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)) ∪ ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)))
172		difid 4301	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)) = ∅
173	172	uneq1i 4089	. . . . . . . . . . . . . . . . 17 ⊢ ((∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)) ∪ ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖))) = (∅ ∪ ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)))
174		0un 4323	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∪ ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖))) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖))
175	171, 173, 174	3eqtri 2770	. . . . . . . . . . . . . . . 16 ⊢ ((∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∪ (𝐸‘(𝑗 + 1))) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖))
176	175	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ((∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∪ (𝐸‘(𝑗 + 1))) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)))
177	170, 176	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ((𝐺‘(𝑗 + 1)) ∖ (𝐺‘𝑗)) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)))
178	177	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ((𝐺‘(𝑗 + 1)) ∖ (𝐺‘𝑗)) = ((𝐸‘(𝑗 + 1)) ∖ ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖)))
179	155, 165, 178	3eqtr4d 2788	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝐹‘(𝑗 + 1)) = ((𝐺‘(𝑗 + 1)) ∖ (𝐺‘𝑗)))
180	179	ineq2d 4143	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝐴 ∩ (𝐹‘(𝑗 + 1))) = (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺‘𝑗))))
181		indif2 4201	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺‘𝑗))) = ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗))
182	181	eqcomi 2747	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)) = (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺‘𝑗)))
183	182	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)) = (𝐴 ∩ ((𝐺‘(𝑗 + 1)) ∖ (𝐺‘𝑗))))
184	180, 183	eqtr4d 2781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝐴 ∩ (𝐹‘(𝑗 + 1))) = ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)))
185	184	fveq2d 6760	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1)))) = (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗))))
186	148, 185	oveq12d 7273	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) + (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)))))
187		inss1 4159	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗)) ⊆ (𝐴 ∩ (𝐺‘(𝑗 + 1)))
188		inss1 4159	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∩ (𝐺‘(𝑗 + 1))) ⊆ 𝐴
189	187, 188	sstri 3926	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗)) ⊆ 𝐴
190	189	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗)) ⊆ 𝐴)
191	40, 41, 42, 43, 190	omessre 43938	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) ∈ ℝ)
192	191	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) ∈ ℝ)
193	40	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → 𝑂 ∈ OutMeas)
194	42	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → 𝐴 ⊆ 𝑋)
195	43	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘𝐴) ∈ ℝ)
196		difss 4062	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)) ⊆ (𝐴 ∩ (𝐺‘(𝑗 + 1)))
197	196, 188	sstri 3926	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)) ⊆ 𝐴
198	197	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)) ⊆ 𝐴)
199	193, 41, 194, 195, 198	omessre 43938	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗))) ∈ ℝ)
200		rexadd 12895	. . . . . . . . . 10 ⊢ (((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) ∈ ℝ ∧ (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗))) ∈ ℝ) → ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) +_𝑒 (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)))) = ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) + (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)))))
201	192, 199, 200	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) +_𝑒 (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)))) = ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) + (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)))))
202	201	eqcomd 2744	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) + (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)))) = ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) +_𝑒 (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)))))
203		carageniuncllem1.s	. . . . . . . . 9 ⊢ 𝑆 = (CaraGen‘𝑂)
204	131	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝐺‘𝑗) = ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖))
205		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝜑
206		fzfid 13621	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀...𝑗) ∈ Fin)
207		carageniuncllem1.e	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸:𝑍⟶𝑆)
208	207	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑗)) → 𝐸:𝑍⟶𝑆)
209		elfzuz 13181	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀...𝑗) → 𝑖 ∈ (ℤ_≥‘𝑀))
210	136	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀...𝑗) → (ℤ_≥‘𝑀) = 𝑍)
211	209, 210	eleqtrd 2841	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝑀...𝑗) → 𝑖 ∈ 𝑍)
212	211	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑗)) → 𝑖 ∈ 𝑍)
213	208, 212	ffvelrnd 6944	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑗)) → (𝐸‘𝑖) ∈ 𝑆)
214	205, 40, 203, 206, 213	caragenfiiuncl 43943	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∈ 𝑆)
215	214	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ∪ 𝑖 ∈ (𝑀...𝑗)(𝐸‘𝑖) ∈ 𝑆)
216	204, 215	eqeltrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝐺‘𝑗) ∈ 𝑆)
217	42	ssinss1d 42485	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (𝐺‘(𝑗 + 1))) ⊆ 𝑋)
218	217	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → (𝐴 ∩ (𝐺‘(𝑗 + 1))) ⊆ 𝑋)
219	193, 203, 41, 216, 218	caragensplit 43928	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∩ (𝐺‘𝑗))) +_𝑒 (𝑂‘((𝐴 ∩ (𝐺‘(𝑗 + 1))) ∖ (𝐺‘𝑗)))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
220	186, 202, 219	3eqtrd 2782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾)) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
221	220	3adant3 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾) ∧ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑗))) + (𝑂‘(𝐴 ∩ (𝐹‘(𝑗 + 1))))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
222	118, 120, 221	3eqtrd 2782	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝐾) ∧ Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
223	99, 100, 103, 222	syl3anc 1369	. . . 4 ⊢ ((𝑗 ∈ (𝑀..^𝐾) ∧ (𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) ∧ 𝜑) → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))
224	223	3exp 1117	. . 3 ⊢ (𝑗 ∈ (𝑀..^𝐾) → ((𝜑 → Σ𝑛 ∈ (𝑀...𝑗)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑗)))) → (𝜑 → Σ𝑛 ∈ (𝑀...(𝑗 + 1))(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘(𝑗 + 1)))))))
225	13, 20, 27, 34, 98, 224	fzind2 13433	. 2 ⊢ (𝐾 ∈ (𝑀...𝐾) → (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝐾)))))
226	5, 6, 225	sylc 65	1 ⊢ (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝐾))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {csn 4558 ∪ cuni 4836 ∪ ciun 4921 ↦ cmpt 5153 dom cdm 5580 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 1c1 10803 + caddc 10805 ℤcz 12249 ℤ_≥cuz 12511 +_𝑒 cxad 12775 ...cfz 13168 ..^cfzo 13311 Σcsu 15325 OutMeascome 43917 CaraGenccaragen 43919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-xadd 12778 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-ome 43918 df-caragen 43920

This theorem is referenced by: carageniuncllem2 43950

W3C validator