carageniuncllem2 - Mathbox for Glauco Siliprandi

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Theorem carageniuncllem2 45173

Description: The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
carageniuncllem2.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
carageniuncllem2.s	⊢ 𝑆 = (CaraGen‘𝑂)
carageniuncllem2.x	⊢ 𝑋 = ∪ dom 𝑂
carageniuncllem2.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
carageniuncllem2.re	⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ)
carageniuncllem2.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
carageniuncllem2.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
carageniuncllem2.e	⊢ (𝜑 → 𝐸:𝑍⟶𝑆)
carageniuncllem2.y	⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
carageniuncllem2.g	⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))
carageniuncllem2.f	⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖)))

**Assertion**
Ref	Expression
carageniuncllem2	⊢ (𝜑 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌))

Distinct variable groups: 𝐴,𝑛 𝑖,𝐸,𝑛 𝑛,𝐹 𝑖,𝑀,𝑛 𝑛,𝑂 𝑆,𝑖 𝑛,𝑋 𝑖,𝑍,𝑛 𝜑,𝑖,𝑛 Allowed substitution hints: 𝐴(𝑖) 𝑆(𝑛) 𝐹(𝑖) 𝐺(𝑖,𝑛) 𝑂(𝑖) 𝑋(𝑖) 𝑌(𝑖,𝑛)

Proof of Theorem carageniuncllem2 Dummy variables 𝑘 𝑧 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		carageniuncllem2.o	. . . 4 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		carageniuncllem2.x	. . . 4 ⊢ 𝑋 = ∪ dom 𝑂
3		carageniuncllem2.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
4		carageniuncllem2.re	. . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ)
5		inss1 4227	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ 𝐴
6	5	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ 𝐴)
7	1, 2, 3, 4, 6	omessre 45161	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
8		difssd 4131	. . . 4 ⊢ (𝜑 → (𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ 𝐴)
9	1, 2, 3, 4, 8	omessre 45161	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
10		rexadd 13207	. . 3 ⊢ (((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ ∧ (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ) → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) = ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) + (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))))
11	7, 9, 10	syl2anc 585	. 2 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) = ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) + (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))))
12		carageniuncllem2.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ_≥‘𝑀)
13		ssinss1 4236	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋)
14	3, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋)
15	1, 2	unidmex 43670	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ V)
16		ssexg 5322	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V)
17	3, 15, 16	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ V)
18		inex1g 5318	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
19	17, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
20		elpwg 4604	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∩ (𝐹‘𝑛)) ∈ V → ((𝐴 ∩ (𝐹‘𝑛)) ∈ 𝒫 𝑋 ↔ (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋))
21	19, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 ∩ (𝐹‘𝑛)) ∈ 𝒫 𝑋 ↔ (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋))
22	14, 21	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ∈ 𝒫 𝑋)
23	22	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐴 ∩ (𝐹‘𝑛)) ∈ 𝒫 𝑋)
24		eqid 2733	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛)))
25	23, 24	fmptd 7109	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))):𝑍⟶𝒫 𝑋)
26		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
27	26	ineq2d 4211	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝐴 ∩ (𝐹‘𝑘)) = (𝐴 ∩ (𝐹‘𝑛)))
28	27	cbvmptv 5260	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘))) = (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛)))
29	28	feq1i 6705	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘))):𝑍⟶𝒫 𝑋 ↔ (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))):𝑍⟶𝒫 𝑋)
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘))):𝑍⟶𝒫 𝑋 ↔ (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))):𝑍⟶𝒫 𝑋))
31	25, 30	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘))):𝑍⟶𝒫 𝑋)
32		simpr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
33	19	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
34	28	fvmpt2 7005	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝐴 ∩ (𝐹‘𝑛)) ∈ V) → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = (𝐴 ∩ (𝐹‘𝑛)))
35	32, 33, 34	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = (𝐴 ∩ (𝐹‘𝑛)))
36	35	iuneq2dv 5020	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)))
37	36	fveq2d 6892	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
38		nfv 1918	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛𝜑
39		carageniuncllem2.e	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸:𝑍⟶𝑆)
40		carageniuncllem2.f	. . . . . . . . . . . . . . . 16 ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖)))
41	38, 12, 39, 40	iundjiun 45111	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑀...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑀...𝑚)(𝐸‘𝑛) ∧ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∧ Disj 𝑛 ∈ 𝑍 (𝐹‘𝑛)))
42	41	simplrd 769	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
43	42	eqcomd 2739	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛))
44	43	ineq2d 4211	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)))
45		iunin2 5073	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛))
46	45	eqcomi 2742	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)))
48	44, 47	eqtrd 2773	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)))
49	48	fveq2d 6892	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
50	49, 7	eqeltrrd 2835	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
51	37, 50	eqeltrd 2834	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) ∈ ℝ)
52		carageniuncllem2.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
53	1, 2, 12, 31, 51, 52	omeiunltfirp 45170	. . . . . . 7 ⊢ (𝜑 → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) + 𝑌))
54	37	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
55		elpwinss 43669	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ⊆ 𝑍)
56	55	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑧 ⊆ 𝑍)
57		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑧)
58	56, 57	sseldd 3982	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
59	58	adantll 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
60	19	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
61	59, 60, 34	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = (𝐴 ∩ (𝐹‘𝑛)))
62	61	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
63	62	sumeq2dv 15645	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
64	63	oveq1d 7419	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) + 𝑌) = (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
65	54, 64	breq12d 5160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → ((𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) + 𝑌) ↔ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)))
66	65	biimpd 228	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → ((𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) + 𝑌) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)))
67	66	reximdva 3169	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) + 𝑌) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)))
68	53, 67	mpd 15	. . . . . 6 ⊢ (𝜑 → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
69		carageniuncllem2.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
70	69	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑀 ∈ ℤ)
71	55	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ⊆ 𝑍)
72		elinel2 4195	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ∈ Fin)
73	72	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
74	70, 12, 71, 73	uzfissfz 43971	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → ∃𝑘 ∈ 𝑍 𝑧 ⊆ (𝑀...𝑘))
75	74	adantr 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → ∃𝑘 ∈ 𝑍 𝑧 ⊆ (𝑀...𝑘))
76	50	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) ∧ 𝑧 ⊆ (𝑀...𝑘)) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
77		fzfid 13934	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ (𝑀...𝑘) → (𝑀...𝑘) ∈ Fin)
78		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ (𝑀...𝑘) → 𝑧 ⊆ (𝑀...𝑘))
79		ssfi 9169	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀...𝑘) ∈ Fin ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ∈ Fin)
80	77, 78, 79	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ (𝑀...𝑘) → 𝑧 ∈ Fin)
81	80	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ∈ Fin)
82	1	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → 𝑂 ∈ OutMeas)
83	3	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → 𝐴 ⊆ 𝑋)
84	4	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘𝐴) ∈ ℝ)
85		inss1 4227	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴
86	85	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴)
87	82, 2, 83, 84, 86	omessre 45161	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
88	81, 87	fsumrecl 15676	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
89	52	rpred 13012	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ℝ)
90	89	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑌 ∈ ℝ)
91	88, 90	readdcld 11239	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
92	91	ad4ant14 751	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
93		fzfid 13934	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...𝑘) ∈ Fin)
94	85	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴)
95	1, 2, 3, 4, 94	omessre 45161	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
96	95	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
97	93, 96	fsumrecl 15676	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
98	97	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
99	89	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ)
100	98, 99	readdcld 11239	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
101	100	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
102		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) ∧ 𝑧 ⊆ (𝑀...𝑘)) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
103	97	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
104		fzfid 13934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (𝑀...𝑘) ∈ Fin)
105	96	adantlr 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
106		0xr 11257	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ^*
107	106	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ∈ ℝ^*)
108		pnfxr 11264	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
109	108	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → +∞ ∈ ℝ^*)
110	1, 2, 14	omecl 45154	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞))
111	110	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞))
112		iccgelb 13376	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
113	107, 109, 111, 112	syl3anc 1372	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
114	113	adantlr 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
115		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ⊆ (𝑀...𝑘))
116	104, 105, 114, 115	fsumless 15738	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ≤ Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
117	88, 103, 90, 116	leadd1dd 11824	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ≤ (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
118	117	ad4ant14 751	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ≤ (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
119	76, 92, 101, 102, 118	ltletrd 11370	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) ∧ 𝑧 ⊆ (𝑀...𝑘)) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
120	119	ex 414	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → (𝑧 ⊆ (𝑀...𝑘) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)))
121	120	reximdv 3171	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → (∃𝑘 ∈ 𝑍 𝑧 ⊆ (𝑀...𝑘) → ∃𝑘 ∈ 𝑍 (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)))
122	75, 121	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → ∃𝑘 ∈ 𝑍 (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
123	122	rexlimdva2 3158	. . . . . 6 ⊢ (𝜑 → (∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) → ∃𝑘 ∈ 𝑍 (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)))
124	68, 123	mpd 15	. . . . 5 ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
125	49	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
126		simpr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
127	125, 126	eqbrtrd 5169	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
128	127	ex 414	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)))
129	128	reximdva 3169	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝑍 (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) → ∃𝑘 ∈ 𝑍 (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)))
130	124, 129	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
131		simpr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
132	1	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑂 ∈ OutMeas)
133		carageniuncllem2.s	. . . . . . . . . 10 ⊢ 𝑆 = (CaraGen‘𝑂)
134	3	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ⊆ 𝑋)
135	4	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘𝐴) ∈ ℝ)
136	39	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐸:𝑍⟶𝑆)
137		carageniuncllem2.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))
138		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
139	132, 133, 2, 134, 135, 12, 136, 137, 40, 138	carageniuncllem1 45172	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘))))
140	139	oveq1d 7419	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌))
141	140	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌))
142	131, 141	breqtrd 5173	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌)) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌))
143	142	ex 414	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)))
144	143	reximdva 3169	. . . 4 ⊢ (𝜑 → (∃𝑘 ∈ 𝑍 (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) → ∃𝑘 ∈ 𝑍 (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)))
145	130, 144	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌))
146	7	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
147	9	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
148		inss1 4227	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐺‘𝑘)) ⊆ 𝐴
149	148	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∩ (𝐺‘𝑘)) ⊆ 𝐴)
150	132, 2, 134, 135, 149	omessre 45161	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ∈ ℝ)
151	89	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑌 ∈ ℝ)
152	150, 151	readdcld 11239	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) ∈ ℝ)
153	152	3adant3 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) ∈ ℝ)
154		difssd 4131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ (𝐺‘𝑘)) ⊆ 𝐴)
155	132, 2, 134, 135, 154	omessre 45161	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℝ)
156	155	3adant3 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℝ)
157		simp3 1139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌))
158	146, 153, 157	ltled 11358	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ≤ ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌))
159	134	ssdifssd 4141	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ (𝐺‘𝑘)) ⊆ 𝑋)
160		oveq2 7412	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
161	160	iuneq1d 5023	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
162		ovex 7437	. . . . . . . . . . . . . . 15 ⊢ (𝑀...𝑘) ∈ V
163		fvex 6901	. . . . . . . . . . . . . . 15 ⊢ (𝐸‘𝑖) ∈ V
164	162, 163	iunex 7950	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ V
165	161, 137, 164	fvmpt 6994	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
166		fveq2 6888	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → (𝐸‘𝑖) = (𝐸‘𝑛))
167	166	cbviunv 5042	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) = ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛)
168	167	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) = ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛))
169	165, 168	eqtrd 2773	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) = ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛))
170		elfzuz 13493	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀...𝑘) → 𝑖 ∈ (ℤ_≥‘𝑀))
171	12	eqcomi 2742	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘𝑀) = 𝑍
172	171	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀...𝑘) → (ℤ_≥‘𝑀) = 𝑍)
173	170, 172	eleqtrd 2836	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀...𝑘) → 𝑖 ∈ 𝑍)
174	173	ssriv 3985	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑘) ⊆ 𝑍
175		iunss1 5010	. . . . . . . . . . . . . 14 ⊢ ((𝑀...𝑘) ⊆ 𝑍 → ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
176	174, 175	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
177	176	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
178	169, 177	eqsstrd 4019	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
179	178	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
180	179	sscond 4140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ (𝐴 ∖ (𝐺‘𝑘)))
181	132, 2, 159, 180	omessle 45149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ≤ (𝑂‘(𝐴 ∖ (𝐺‘𝑘))))
182	181	3adant3 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ≤ (𝑂‘(𝐴 ∖ (𝐺‘𝑘))))
183	146, 147, 153, 156, 158, 182	le2addd 11829	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) + (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))))
184	150	recnd 11238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ∈ ℂ)
185	89	recnd 11238	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℂ)
186	185	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑌 ∈ ℂ)
187	155	recnd 11238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℂ)
188	184, 186, 187	add32d 11437	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) + 𝑌))
189		rexadd 13207	. . . . . . . . . . . 12 ⊢ (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ∈ ℝ ∧ (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℝ) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +_𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))))
190	150, 155, 189	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +_𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))))
191	190	eqcomd 2739	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +_𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))))
192		nfv 1918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖𝜑
193	39	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → 𝐸:𝑍⟶𝑆)
194	173	adantl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → 𝑖 ∈ 𝑍)
195	193, 194	ffvelcdmd 7083	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → (𝐸‘𝑖) ∈ 𝑆)
196	192, 1, 133, 93, 195	caragenfiiuncl 45166	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ 𝑆)
197	196	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ 𝑆)
198	137, 161, 138, 197	fvmptd3 7017	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
199	198, 197	eqeltrd 2834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝑆)
200	132, 133, 2, 199, 134	caragensplit 45151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +_𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (𝑂‘𝐴))
201	191, 200	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (𝑂‘𝐴))
202	201	oveq1d 7419	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) + 𝑌) = ((𝑂‘𝐴) + 𝑌))
203	188, 202	eqtrd 2773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘𝐴) + 𝑌))
204	203	3adant3 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘𝐴) + 𝑌))
205	183, 204	breqtrd 5173	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) + (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌))
206	205	3exp 1120	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) + (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌))))
207	206	rexlimdv 3154	. . 3 ⊢ (𝜑 → (∃𝑘 ∈ 𝑍 (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) + (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌)))
208	145, 207	mpd 15	. 2 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) + (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌))
209	11, 208	eqbrtrd 5169	1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 ∪ ciun 4996 Disj wdisj 5112 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 Fincfn 8935 ℂcc 11104 ℝcr 11105 0cc0 11106 + caddc 11109 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 +_𝑒 cxad 13086 [,]cicc 13323 ...cfz 13480 ..^cfzo 13623 Σcsu 15628 OutMeascome 45140 CaraGenccaragen 45142

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-ac2 10454 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-xadd 13089 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-sumge0 45014 df-ome 45141 df-caragen 45143

This theorem is referenced by: carageniuncl 45174

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