caratheodorylem1 - Mathbox for Glauco Siliprandi

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Theorem caratheodorylem1 41383

Description: Lemma used to prove that Caratheodory's construction is sigma-additive. This is the proof of the statement in the middle of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
caratheodorylem1.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
caratheodorylem1.s	⊢ 𝑆 = (CaraGen‘𝑂)
caratheodorylem1.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
caratheodorylem1.e	⊢ (𝜑 → 𝐸:𝑍⟶𝑆)
caratheodorylem1.dj	⊢ (𝜑 → Disj 𝑛 ∈ 𝑍 (𝐸‘𝑛))
caratheodorylem1.g	⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))
caratheodorylem1.n	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))

**Assertion**
Ref	Expression
caratheodorylem1	⊢ (𝜑 → (𝑂‘(𝐺‘𝑁)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))))

Distinct variable groups: 𝑖,𝐸,𝑛 𝑖,𝐺,𝑛 𝑖,𝑀,𝑛 𝑖,𝑁,𝑛 𝑖,𝑂,𝑛 𝑛,𝑍 𝜑,𝑖,𝑛 Allowed substitution hints: 𝑆(𝑖,𝑛) 𝑍(𝑖)

Proof of Theorem caratheodorylem1 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caratheodorylem1.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		eluzfz2 12559	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		id 22	. 2 ⊢ (𝜑 → 𝜑)
5		2fveq3 6382	. . . . 5 ⊢ (𝑗 = 𝑀 → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘𝑀)))
6		oveq2 6852	. . . . . . 7 ⊢ (𝑗 = 𝑀 → (𝑀...𝑗) = (𝑀...𝑀))
7	6	mpteq1d 4899	. . . . . 6 ⊢ (𝑗 = 𝑀 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))))
8	7	fveq2d 6381	. . . . 5 ⊢ (𝑗 = 𝑀 → (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))))
9	5, 8	eqeq12d 2780	. . . 4 ⊢ (𝑗 = 𝑀 → ((𝑂‘(𝐺‘𝑗)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘𝑀)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))))))
10	9	imbi2d 331	. . 3 ⊢ (𝑗 = 𝑀 → ((𝜑 → (𝑂‘(𝐺‘𝑗)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺‘𝑀)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))))))
11		2fveq3 6382	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘𝑖)))
12		oveq2 6852	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝑀...𝑗) = (𝑀...𝑖))
13	12	mpteq1d 4899	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))
14	13	fveq2d 6381	. . . . 5 ⊢ (𝑗 = 𝑖 → (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))))
15	11, 14	eqeq12d 2780	. . . 4 ⊢ (𝑗 = 𝑖 → ((𝑂‘(𝐺‘𝑗)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))))
16	15	imbi2d 331	. . 3 ⊢ (𝑗 = 𝑖 → ((𝜑 → (𝑂‘(𝐺‘𝑗)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))))))
17		2fveq3 6382	. . . . 5 ⊢ (𝑗 = (𝑖 + 1) → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘(𝑖 + 1))))
18		oveq2 6852	. . . . . . 7 ⊢ (𝑗 = (𝑖 + 1) → (𝑀...𝑗) = (𝑀...(𝑖 + 1)))
19	18	mpteq1d 4899	. . . . . 6 ⊢ (𝑗 = (𝑖 + 1) → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛))))
20	19	fveq2d 6381	. . . . 5 ⊢ (𝑗 = (𝑖 + 1) → (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))))
21	17, 20	eqeq12d 2780	. . . 4 ⊢ (𝑗 = (𝑖 + 1) → ((𝑂‘(𝐺‘𝑗)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^{^}‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛))))))
22	21	imbi2d 331	. . 3 ⊢ (𝑗 = (𝑖 + 1) → ((𝜑 → (𝑂‘(𝐺‘𝑗)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^{^}‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))))))
23		2fveq3 6382	. . . . 5 ⊢ (𝑗 = 𝑁 → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘𝑁)))
24		oveq2 6852	. . . . . . 7 ⊢ (𝑗 = 𝑁 → (𝑀...𝑗) = (𝑀...𝑁))
25	24	mpteq1d 4899	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))
26	25	fveq2d 6381	. . . . 5 ⊢ (𝑗 = 𝑁 → (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))))
27	23, 26	eqeq12d 2780	. . . 4 ⊢ (𝑗 = 𝑁 → ((𝑂‘(𝐺‘𝑗)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘𝑁)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))))
28	27	imbi2d 331	. . 3 ⊢ (𝑗 = 𝑁 → ((𝜑 → (𝑂‘(𝐺‘𝑗)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺‘𝑁)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))))))
29		eluzel2 11894	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
30	1, 29	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
31		fzsn 12593	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
33	32	mpteq1d 4899	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))))
34	33	fveq2d 6381	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))))
35		caratheodorylem1.o	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
36	35	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑂 ∈ OutMeas)
37		eqid 2765	. . . . . . . 8 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂
38		caratheodorylem1.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (CaraGen‘𝑂)
39	38	caragenss 41361	. . . . . . . . . . 11 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
40	36, 39	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑆 ⊆ dom 𝑂)
41		caratheodorylem1.e	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸:𝑍⟶𝑆)
42	41	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝐸:𝑍⟶𝑆)
43		elsni 4353	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {𝑀} → 𝑛 = 𝑀)
44	43	adantl 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑛 = 𝑀)
45		uzid 11904	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
46	30, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
47		caratheodorylem1.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ_≥‘𝑀)
48	46, 47	syl6eleqr 2855	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
49	48	adantr 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑀 ∈ 𝑍)
50	44, 49	eqeltrd 2844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑛 ∈ 𝑍)
51	42, 50	ffvelrnd 6552	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝐸‘𝑛) ∈ 𝑆)
52	40, 51	sseldd 3764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝐸‘𝑛) ∈ dom 𝑂)
53		elssuni 4627	. . . . . . . . 9 ⊢ ((𝐸‘𝑛) ∈ dom 𝑂 → (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
54	52, 53	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
55	36, 37, 54	omecl 41360	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
56		eqid 2765	. . . . . . 7 ⊢ (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))
57	55, 56	fmptd 6576	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))):{𝑀}⟶(0[,]+∞))
58	30, 57	sge0sn 41236	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))) = ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))‘𝑀))
59		eqidd 2766	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))))
60	32	iuneq1d 4703	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) = ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖))
61		fveq2 6377	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑀 → (𝐸‘𝑖) = (𝐸‘𝑀))
62	61	iunxsng 4760	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝑍 → ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖) = (𝐸‘𝑀))
63	48, 62	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖) = (𝐸‘𝑀))
64		eqidd 2766	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘𝑀) = (𝐸‘𝑀))
65	60, 63, 64	3eqtrrd 2804	. . . . . . . . 9 ⊢ (𝜑 → (𝐸‘𝑀) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖))
66	65	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐸‘𝑀) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖))
67		fveq2 6377	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (𝐸‘𝑛) = (𝐸‘𝑀))
68	67	adantl 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐸‘𝑛) = (𝐸‘𝑀))
69		caratheodorylem1.g	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))
70	69	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)))
71		oveq2 6852	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
72	71	iuneq1d 4703	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖))
73	72	adantl 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖))
74		ovex 6876	. . . . . . . . . . . 12 ⊢ (𝑀...𝑀) ∈ V
75		fvex 6390	. . . . . . . . . . . 12 ⊢ (𝐸‘𝑖) ∈ V
76	74, 75	iunex 7347	. . . . . . . . . . 11 ⊢ ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) ∈ V
77	76	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) ∈ V)
78	70, 73, 48, 77	fvmptd 6479	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑀) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖))
79	78	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐺‘𝑀) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖))
80	66, 68, 79	3eqtr4d 2809	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐸‘𝑛) = (𝐺‘𝑀))
81	80	fveq2d 6381	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐺‘𝑀)))
82		snidg 4366	. . . . . . 7 ⊢ (𝑀 ∈ 𝑍 → 𝑀 ∈ {𝑀})
83	48, 82	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ {𝑀})
84		fvexd 6392	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝐺‘𝑀)) ∈ V)
85	59, 81, 83, 84	fvmptd 6479	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))‘𝑀) = (𝑂‘(𝐺‘𝑀)))
86	34, 58, 85	3eqtrrd 2804	. . . 4 ⊢ (𝜑 → (𝑂‘(𝐺‘𝑀)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))))
87	86	a1i 11	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → (𝑂‘(𝐺‘𝑀)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))))))
88		simp3 1168	. . . . 5 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → 𝜑)
89		simp1 1166	. . . . 5 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → 𝑖 ∈ (𝑀..^𝑁))
90		id 22	. . . . . . 7 ⊢ ((𝜑 → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (𝜑 → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))))
91	90	imp 395	. . . . . 6 ⊢ (((𝜑 → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))))
92	91	3adant1 1160	. . . . 5 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))))
93		elfzoel1 12679	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ)
94		elfzoelz 12681	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℤ)
95	94	peano2zd 11735	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℤ)
96	93, 95, 95	3jca 1158	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ))
97	93	zred 11732	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℝ)
98	95	zred 11732	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℝ)
99	94	zred 11732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℝ)
100		elfzole1 12689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑖)
101	99	ltp1d 11210	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 < (𝑖 + 1))
102	97, 99, 98, 100, 101	lelttrd 10451	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 < (𝑖 + 1))
103	97, 98, 102	ltled 10441	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ (𝑖 + 1))
104		leid 10389	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 + 1) ∈ ℝ → (𝑖 + 1) ≤ (𝑖 + 1))
105	98, 104	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ≤ (𝑖 + 1))
106	96, 103, 105	jca32 511	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝑀..^𝑁) → ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1))))
107		elfz2 12543	. . . . . . . . . . . . . 14 ⊢ ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) ↔ ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1))))
108	106, 107	sylibr 225	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
109	108	adantl 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
110		fveq2 6377	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑖 + 1) → (𝐸‘𝑗) = (𝐸‘(𝑖 + 1)))
111	110	ssiun2s 4722	. . . . . . . . . . . 12 ⊢ ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) → (𝐸‘(𝑖 + 1)) ⊆ ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗))
112	109, 111	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗))
113		fveq2 6377	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑗 → (𝐸‘𝑖) = (𝐸‘𝑗))
114	113	cbviunv 4717	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗)
115	114	mpteq2i 4902	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) = (𝑛 ∈ 𝑍 ↦ ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗))
116	69, 115	eqtri 2787	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗))
117	116	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗)))
118		oveq2 6852	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑖 + 1) → (𝑀...𝑛) = (𝑀...(𝑖 + 1)))
119	118	iuneq1d 4703	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑖 + 1) → ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) = ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗))
120	119	adantl 473	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 = (𝑖 + 1)) → ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) = ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗))
121	30	adantr 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ)
122	94	adantl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℤ)
123	122	peano2zd 11735	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℤ)
124	121	zred 11732	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
125	123	zred 11732	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℝ)
126	122	zred 11732	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℝ)
127	100	adantl 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑖)
128	126	ltp1d 11210	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 < (𝑖 + 1))
129	124, 126, 125, 127, 128	lelttrd 10451	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 < (𝑖 + 1))
130	124, 125, 129	ltled 10441	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ (𝑖 + 1))
131	121, 123, 130	3jca 1158	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
132		eluz2 11895	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 + 1) ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
133	131, 132	sylibr 225	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (ℤ_≥‘𝑀))
134	47	eqcomi 2774	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑀) = 𝑍
135	133, 134	syl6eleq 2854	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ 𝑍)
136		ovex 6876	. . . . . . . . . . . . . . 15 ⊢ (𝑀...(𝑖 + 1)) ∈ V
137		fvex 6390	. . . . . . . . . . . . . . 15 ⊢ (𝐸‘𝑗) ∈ V
138	136, 137	iunex 7347	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ∈ V
139	138	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ∈ V)
140	117, 120, 135, 139	fvmptd 6479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗))
141	140	eqcomd 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) = (𝐺‘(𝑖 + 1)))
142	112, 141	sseqtrd 3803	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)))
143		sseqin2 3981	. . . . . . . . . . 11 ⊢ ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) ↔ ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
144	143	biimpi 207	. . . . . . . . . 10 ⊢ ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
145	142, 144	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
146	145	fveq2d 6381	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐸‘(𝑖 + 1))))
147		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝐸‘(𝑖 + 1))
148		elfzouz 12685	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ (ℤ_≥‘𝑀))
149	148	adantl 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ (ℤ_≥‘𝑀))
150	147, 149, 110	iunp1 39889	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) = (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))))
151	140, 150	eqtrd 2799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))))
152	151	difeq1d 3891	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))))
153		caratheodorylem1.dj	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Disj 𝑛 ∈ 𝑍 (𝐸‘𝑛))
154		fveq2 6377	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝐸‘𝑛) = (𝐸‘𝑗))
155	154	cbvdisjv 4790	. . . . . . . . . . . . . . 15 ⊢ (Disj 𝑛 ∈ 𝑍 (𝐸‘𝑛) ↔ Disj 𝑗 ∈ 𝑍 (𝐸‘𝑗))
156	153, 155	sylib 209	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Disj 𝑗 ∈ 𝑍 (𝐸‘𝑗))
157	156	adantr 472	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → Disj 𝑗 ∈ 𝑍 (𝐸‘𝑗))
158		fzssuz 12592	. . . . . . . . . . . . . . 15 ⊢ (𝑀...𝑖) ⊆ (ℤ_≥‘𝑀)
159	158, 134	sseqtri 3799	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑖) ⊆ 𝑍
160	159	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑀...𝑖) ⊆ 𝑍)
161		fzp1nel 12634	. . . . . . . . . . . . . . . 16 ⊢ ¬ (𝑖 + 1) ∈ (𝑀...𝑖)
162	161	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀..^𝑁) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
163	162	adantl 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
164	135, 163	eldifd 3745	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑍 ∖ (𝑀...𝑖)))
165	157, 160, 164, 110	disjiun2 39880	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅)
166		undif4 4197	. . . . . . . . . . . 12 ⊢ ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅ → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))))
167	165, 166	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))))
168	167	eqcomd 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))) = (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))))
169		simpl 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝜑)
170	149, 134	syl6eleq 2854	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ 𝑍)
171	116	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗)))
172		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 = 𝑖) → 𝑛 = 𝑖)
173	172	oveq2d 6860	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 = 𝑖) → (𝑀...𝑛) = (𝑀...𝑖))
174	173	iuneq1d 4703	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 = 𝑖) → ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) = ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗))
175		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
176		ovex 6876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀...𝑖) ∈ V
177	176, 137	iunex 7347	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∈ V
178	177	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∈ V)
179	171, 174, 175, 178	fvmptd 6479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗))
180	169, 170, 179	syl2anc 579	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘𝑖) = ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗))
181	180	eqcomd 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) = (𝐺‘𝑖))
182		difid 4115	. . . . . . . . . . . . 13 ⊢ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅
183	182	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅)
184	181, 183	uneq12d 3932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((𝐺‘𝑖) ∪ ∅))
185		un0 4131	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑖) ∪ ∅) = (𝐺‘𝑖)
186	185	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘𝑖) ∪ ∅) = (𝐺‘𝑖))
187	184, 186	eqtrd 2799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝐺‘𝑖))
188	152, 168, 187	3eqtrd 2803	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (𝐺‘𝑖))
189	188	fveq2d 6381	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐺‘𝑖)))
190	146, 189	oveq12d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +_𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +_𝑒 (𝑂‘(𝐺‘𝑖))))
191	190	3adant3 1162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +_𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +_𝑒 (𝑂‘(𝐺‘𝑖))))
192	35	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑂 ∈ OutMeas)
193	41	adantr 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝐸:𝑍⟶𝑆)
194	193, 135	ffvelrnd 6552	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ∈ 𝑆)
195		simpll 783	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
196	93	adantr 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ∈ ℤ)
197		elfzelz 12552	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑗 ∈ ℤ)
198	197	adantl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ ℤ)
199		elfzle1 12554	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑀 ≤ 𝑗)
200	199	adantl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ≤ 𝑗)
201	196, 198, 200	3jca 1158	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗))
202		eluz2 11895	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗))
203	201, 202	sylibr 225	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ (ℤ_≥‘𝑀))
204	203, 134	syl6eleq 2854	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ 𝑍)
205	204	adantll 705	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ 𝑍)
206	35, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
207	206	adantr 472	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑆 ⊆ dom 𝑂)
208	41	ffvelrnda 6551	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ 𝑆)
209	207, 208	sseldd 3764	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ dom 𝑂)
210		elssuni 4627	. . . . . . . . . . . . . 14 ⊢ ((𝐸‘𝑗) ∈ dom 𝑂 → (𝐸‘𝑗) ⊆ ∪ dom 𝑂)
211	209, 210	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ⊆ ∪ dom 𝑂)
212	195, 205, 211	syl2anc 579	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝐸‘𝑗) ⊆ ∪ dom 𝑂)
213	212	ralrimiva 3113	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom 𝑂)
214		iunss 4719	. . . . . . . . . . 11 ⊢ (∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom 𝑂)
215	213, 214	sylibr 225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom 𝑂)
216	140, 215	eqsstrd 3801	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) ⊆ ∪ dom 𝑂)
217	192, 38, 37, 194, 216	caragensplit 41357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +_𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = (𝑂‘(𝐺‘(𝑖 + 1))))
218	217	eqcomd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +_𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))))
219	218	3adant3 1162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +_𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))))
220	192	adantr 472	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑂 ∈ OutMeas)
221	169	adantr 472	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
222		elfzuz 12548	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ (ℤ_≥‘𝑀))
223	222, 134	syl6eleq 2854	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ 𝑍)
224	223	adantl 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑛 ∈ 𝑍)
225	41, 206	fssd 6239	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑂)
226	225	ffvelrnda 6551	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑂)
227	226, 53	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
228	221, 224, 227	syl2anc 579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
229	220, 37, 228	omecl 41360	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
230		2fveq3 6382	. . . . . . . . 9 ⊢ (𝑛 = (𝑖 + 1) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘(𝑖 + 1))))
231	149, 229, 230	sge0p1 41271	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (Σ^{^}‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))) = ((Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) +_𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
232	231	3adant3 1162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (Σ^{^}‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))) = ((Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) +_𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
233		id 22	. . . . . . . . . 10 ⊢ ((𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))))
234	233	eqcomd 2771	. . . . . . . . 9 ⊢ ((𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) → (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) = (𝑂‘(𝐺‘𝑖)))
235	234	oveq1d 6859	. . . . . . . 8 ⊢ ((𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) → ((Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) +_𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐺‘𝑖)) +_𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
236	235	3ad2ant3 1165	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → ((Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) +_𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐺‘𝑖)) +_𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
237		simpl 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑖)) → 𝜑)
238	159	sseli 3759	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑀...𝑖) → 𝑗 ∈ 𝑍)
239	238	adantl 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑖)) → 𝑗 ∈ 𝑍)
240	237, 239, 211	syl2anc 579	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸‘𝑗) ⊆ ∪ dom 𝑂)
241	240	adantlr 706	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸‘𝑗) ⊆ ∪ dom 𝑂)
242	241	ralrimiva 3113	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ∀𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom 𝑂)
243		iunss 4719	. . . . . . . . . . . . 13 ⊢ (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom 𝑂)
244	242, 243	sylibr 225	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom 𝑂)
245	179, 244	eqsstrd 3801	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) ⊆ ∪ dom 𝑂)
246	169, 170, 245	syl2anc 579	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘𝑖) ⊆ ∪ dom 𝑂)
247	192, 37, 246	omexrcl 41364	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘𝑖)) ∈ ℝ^*)
248	112, 215	sstrd 3773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ ∪ dom 𝑂)
249	192, 37, 248	omexrcl 41364	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐸‘(𝑖 + 1))) ∈ ℝ^*)
250	247, 249	xaddcomd 40181	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘(𝐺‘𝑖)) +_𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +_𝑒 (𝑂‘(𝐺‘𝑖))))
251	250	3adant3 1162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → ((𝑂‘(𝐺‘𝑖)) +_𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +_𝑒 (𝑂‘(𝐺‘𝑖))))
252	232, 236, 251	3eqtrd 2803	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (Σ^{^}‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +_𝑒 (𝑂‘(𝐺‘𝑖))))
253	191, 219, 252	3eqtr4d 2809	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^{^}‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))))
254	88, 89, 92, 253	syl3anc 1490	. . . 4 ⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^{^}‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))))
255	254	3exp 1148	. . 3 ⊢ (𝑖 ∈ (𝑀..^𝑁) → ((𝜑 → (𝑂‘(𝐺‘𝑖)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (𝜑 → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^{^}‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))))))
256	10, 16, 22, 28, 87, 255	fzind2 12797	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (𝑂‘(𝐺‘𝑁)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))))
257	3, 4, 256	sylc 65	1 ⊢ (𝜑 → (𝑂‘(𝐺‘𝑁)) = (Σ^{^}‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∧ w3a 1107 = wceq 1652 ∈ wcel 2155 ∀wral 3055 Vcvv 3350 ∖ cdif 3731 ∪ cun 3732 ∩ cin 3733 ⊆ wss 3734 ∅c0 4081 {csn 4336 ∪ cuni 4596 ∪ ciun 4678 Disj wdisj 4779 class class class wbr 4811 ↦ cmpt 4890 dom cdm 5279 ⟶wf 6066 ‘cfv 6070 (class class class)co 6844 ℝcr 10190 0cc0 10191 1c1 10192 + caddc 10194 +∞cpnf 10327 ≤ cle 10331 ℤcz 11626 ℤ_≥cuz 11889 +_𝑒 cxad 12147 [,]cicc 12383 ...cfz 12536 ..^cfzo 12676 Σ^{^}csumge0 41219 OutMeascome 41346 CaraGenccaragen 41348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2070 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2352 ax-ext 2743 ax-rep 4932 ax-sep 4943 ax-nul 4951 ax-pow 5003 ax-pr 5064 ax-un 7149 ax-inf2 8755 ax-cnex 10247 ax-resscn 10248 ax-1cn 10249 ax-icn 10250 ax-addcl 10251 ax-addrcl 10252 ax-mulcl 10253 ax-mulrcl 10254 ax-mulcom 10255 ax-addass 10256 ax-mulass 10257 ax-distr 10258 ax-i2m1 10259 ax-1ne0 10260 ax-1rid 10261 ax-rnegex 10262 ax-rrecex 10263 ax-cnre 10264 ax-pre-lttri 10265 ax-pre-lttrn 10266 ax-pre-ltadd 10267 ax-pre-mulgt0 10268 ax-pre-sup 10269

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2063 df-mo 2565 df-eu 2582 df-clab 2752 df-cleq 2758 df-clel 2761 df-nfc 2896 df-ne 2938 df-nel 3041 df-ral 3060 df-rex 3061 df-reu 3062 df-rmo 3063 df-rab 3064 df-v 3352 df-sbc 3599 df-csb 3694 df-dif 3737 df-un 3739 df-in 3741 df-ss 3748 df-pss 3750 df-nul 4082 df-if 4246 df-pw 4319 df-sn 4337 df-pr 4339 df-tp 4341 df-op 4343 df-uni 4597 df-int 4636 df-iun 4680 df-disj 4780 df-br 4812 df-opab 4874 df-mpt 4891 df-tr 4914 df-id 5187 df-eprel 5192 df-po 5200 df-so 5201 df-fr 5238 df-se 5239 df-we 5240 df-xp 5285 df-rel 5286 df-cnv 5287 df-co 5288 df-dm 5289 df-rn 5290 df-res 5291 df-ima 5292 df-pred 5867 df-ord 5913 df-on 5914 df-lim 5915 df-suc 5916 df-iota 6033 df-fun 6072 df-fn 6073 df-f 6074 df-f1 6075 df-fo 6076 df-f1o 6077 df-fv 6078 df-isom 6079 df-riota 6805 df-ov 6847 df-oprab 6848 df-mpt2 6849 df-om 7266 df-1st 7368 df-2nd 7369 df-wrecs 7612 df-recs 7674 df-rdg 7712 df-1o 7766 df-oadd 7770 df-er 7949 df-en 8163 df-dom 8164 df-sdom 8165 df-fin 8166 df-sup 8557 df-oi 8624 df-card 9018 df-pnf 10332 df-mnf 10333 df-xr 10334 df-ltxr 10335 df-le 10336 df-sub 10524 df-neg 10525 df-div 10941 df-nn 11277 df-2 11337 df-3 11338 df-n0 11541 df-z 11627 df-uz 11890 df-rp 12032 df-xadd 12150 df-ico 12386 df-icc 12387 df-fz 12537 df-fzo 12677 df-seq 13012 df-exp 13071 df-hash 13325 df-cj 14127 df-re 14128 df-im 14129 df-sqrt 14263 df-abs 14264 df-clim 14507 df-sum 14705 df-sumge0 41220 df-ome 41347 df-caragen 41349

This theorem is referenced by: caratheodorylem2 41384

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