Theorem carageniuncllem2 40736

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 3833	. . . . 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 40724	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
8		difssd 3738	. . . 4 ⊢ (𝜑 → (𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ 𝐴)
9	1, 2, 3, 4, 8	omessre 40724	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
10		rexadd 12063	. . 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 693	. 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 3841	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋)
14	3, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋)
15	1, 2	unidmex 39217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ V)
16		ssexg 4804	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V)
17	3, 15, 16	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ V)
18		inex1g 4801	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
19	17, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
20		elpwg 4166	. . . . . . . . . . . . 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 247	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ∈ 𝒫 𝑋)
23	22	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐴 ∩ (𝐹‘𝑛)) ∈ 𝒫 𝑋)
24		eqid 2622	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛)))
25	23, 24	fmptd 6385	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))):𝑍⟶𝒫 𝑋)
26		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
27	26	ineq2d 3814	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝐴 ∩ (𝐹‘𝑘)) = (𝐴 ∩ (𝐹‘𝑛)))
28	27	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘))) = (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛)))
29	28	feq1i 6036	. . . . . . . . . 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 247	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘))):𝑍⟶𝒫 𝑋)
32		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
33	19	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
34	28	fvmpt2 6291	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝐴 ∩ (𝐹‘𝑛)) ∈ V) → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = (𝐴 ∩ (𝐹‘𝑛)))
35	32, 33, 34	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = (𝐴 ∩ (𝐹‘𝑛)))
36	35	iuneq2dv 4542	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)))
37	36	fveq2d 6195	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
38		nfv 1843	. . . . . . . . . . . . . . . 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 40677	. . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
43	42	eqcomd 2628	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛))
44	43	ineq2d 3814	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)))
45		iunin2 4584	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛))
46	45	eqcomi 2631	. . . . . . . . . . . . 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 2656	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)))
49	48	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
50	49, 7	eqeltrrd 2702	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
51	37, 50	eqeltrd 2701	. . . . . . . 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 40733	. . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
55		elpwinss 39216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ⊆ 𝑍)
56	55	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑧 ⊆ 𝑍)
57		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑧)
58	56, 57	sseldd 3604	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
59	58	adantll 750	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
60	19	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
61	59, 60, 34	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = (𝐴 ∩ (𝐹‘𝑛)))
62	61	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
63	62	sumeq2dv 14433	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
64	63	oveq1d 6665	. . . . . . . . . 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 4666	. . . . . . . . 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 219	. . . . . . . 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 3017	. . . . . . 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	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
70	69	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑀 ∈ ℤ)
71	55	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ⊆ 𝑍)
72		elinel2 3800	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ∈ Fin)
73	72	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
74	70, 12, 71, 73	uzfissfz 39542	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → ∃𝑘 ∈ 𝑍 𝑧 ⊆ (𝑀...𝑘))
75	74	adantr 481	. . . . . . . . 9 ⊢ (((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 766	. . . . . . . . . . . 12 ⊢ ((((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 12772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ⊆ (𝑀...𝑘) → (𝑀...𝑘) ∈ Fin)
78		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ⊆ (𝑀...𝑘) → 𝑧 ⊆ (𝑀...𝑘))
79		ssfi 8180	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀...𝑘) ∈ Fin ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ∈ Fin)
80	77, 78, 79	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ (𝑀...𝑘) → 𝑧 ∈ Fin)
81	80	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ∈ Fin)
82	1	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → 𝑂 ∈ OutMeas)
83	3	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → 𝐴 ⊆ 𝑋)
84	4	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘𝐴) ∈ ℝ)
85		inss1 3833	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴
86	85	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴)
87	82, 2, 83, 84, 86	omessre 40724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
88	81, 87	fsumrecl 14465	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
89	52	rpred 11872	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ ℝ)
90	89	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑌 ∈ ℝ)
91	88, 90	readdcld 10069	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
92	91	ad4ant14 1293	. . . . . . . . . . . 12 ⊢ ((((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 12772	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀...𝑘) ∈ Fin)
94	85	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴)
95	1, 2, 3, 4, 94	omessre 40724	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
96	95	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
97	93, 96	fsumrecl 14465	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
98	97	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
99	89	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ)
100	98, 99	readdcld 10069	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
101	100	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((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 792	. . . . . . . . . . . 12 ⊢ ((((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 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
104		fzfid 12772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (𝑀...𝑘) ∈ Fin)
105	96	adantlr 751	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
106		0xr 10086	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ*
107	106	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ∈ ℝ*)
108		pnfxr 10092	. . . . . . . . . . . . . . . . . 18 ⊢ +∞ ∈ ℝ*
109	108	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → +∞ ∈ ℝ*)
110	1, 2, 14	omecl 40717	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞))
111	110	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞))
112		iccgelb 12230	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
113	107, 109, 111, 112	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
114	113	adantlr 751	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
115		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ⊆ (𝑀...𝑘))
116	104, 105, 114, 115	fsumless 14528	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ≤ Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
117	88, 103, 90, 116	leadd1dd 10641	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ≤ (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
118	117	ad4ant14 1293	. . . . . . . . . . . 12 ⊢ ((((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 10197	. . . . . . . . . . 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 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
120	119	ex 450	. . . . . . . . . 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)))
121	120	reximdv 3016	. . . . . . . . 9 ⊢ (((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	. . . . . . . 8 ⊢ (((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	ex 450	. . . . . . 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)))
124	123	rexlimdva 3031	. . . . . 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)))
125	68, 124	mpd 15	. . . . 5 ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
126	49	ad2antrr 762	. . . . . . . 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))))
127		simpr 477	. . . . . . . 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))
128	126, 127	eqbrtrd 4675	. . . . . . 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))
129	128	ex 450	. . . . . 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)))
130	129	reximdva 3017	. . . . 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)))
131	125, 130	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
132		simpr 477	. . . . . . 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))
133	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑂 ∈ OutMeas)
134		carageniuncllem2.s	. . . . . . . . . 10 ⊢ 𝑆 = (CaraGen‘𝑂)
135	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ⊆ 𝑋)
136	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘𝐴) ∈ ℝ)
137	39	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐸:𝑍⟶𝑆)
138		carageniuncllem2.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))
139		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
140	133, 134, 2, 135, 136, 12, 137, 138, 40, 139	carageniuncllem1 40735	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘))))
141	140	oveq1d 6665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌))
142	141	adantr 481	. . . . . . 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))
143	132, 142	breqtrd 4679	. . . . . 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))
144	143	ex 450	. . . . 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)))
145	144	reximdva 3017	. . . 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)))
146	131, 145	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌))
147	7	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
148	9	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
149		inss1 3833	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐺‘𝑘)) ⊆ 𝐴
150	149	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∩ (𝐺‘𝑘)) ⊆ 𝐴)
151	133, 2, 135, 136, 150	omessre 40724	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ∈ ℝ)
152	89	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑌 ∈ ℝ)
153	151, 152	readdcld 10069	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) ∈ ℝ)
154	153	3adant3 1081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) ∈ ℝ)
155		difssd 3738	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ (𝐺‘𝑘)) ⊆ 𝐴)
156	133, 2, 135, 136, 155	omessre 40724	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℝ)
157	156	3adant3 1081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℝ)
158		simp3 1063	. . . . . . . 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))
159	147, 154, 158	ltled 10185	. . . . . . 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))
160	135	ssdifssd 3748	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ (𝐺‘𝑘)) ⊆ 𝑋)
161		oveq2 6658	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
162	161	iuneq1d 4545	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
163		ovex 6678	. . . . . . . . . . . . . . 15 ⊢ (𝑀...𝑘) ∈ V
164		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐸‘𝑖) ∈ V
165	163, 164	iunex 7147	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ V
166	162, 138, 165	fvmpt 6282	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
167		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → (𝐸‘𝑖) = (𝐸‘𝑛))
168	167	cbviunv 4559	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) = ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛)
169	168	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) = ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛))
170	166, 169	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) = ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛))
171		elfzuz 12338	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀...𝑘) → 𝑖 ∈ (ℤ≥‘𝑀))
172	12	eqcomi 2631	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘𝑀) = 𝑍
173	172	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀...𝑘) → (ℤ≥‘𝑀) = 𝑍)
174	171, 173	eleqtrd 2703	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀...𝑘) → 𝑖 ∈ 𝑍)
175	174	ssriv 3607	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑘) ⊆ 𝑍
176		iunss1 4532	. . . . . . . . . . . . . 14 ⊢ ((𝑀...𝑘) ⊆ 𝑍 → ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
177	175, 176	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
178	177	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
179	170, 178	eqsstrd 3639	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
180	179	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
181	180	sscond 3747	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ (𝐴 ∖ (𝐺‘𝑘)))
182	133, 2, 160, 181	omessle 40712	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ≤ (𝑂‘(𝐴 ∖ (𝐺‘𝑘))))
183	182	3adant3 1081	. . . . . . 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))))
184	147, 148, 154, 157, 159, 183	le2addd 10646	. . . . . 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)))))
185	151	recnd 10068	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ∈ ℂ)
186	89	recnd 10068	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℂ)
187	186	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑌 ∈ ℂ)
188	156	recnd 10068	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℂ)
189	185, 187, 188	add32d 10263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) + 𝑌))
190		rexadd 12063	. . . . . . . . . . . 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)))))
191	151, 156, 190	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))))
192	191	eqcomd 2628	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))))
193	138	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)))
194	162	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑛 = 𝑘) → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
195		nfv 1843	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖𝜑
196	39	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → 𝐸:𝑍⟶𝑆)
197	174	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → 𝑖 ∈ 𝑍)
198	196, 197	ffvelrnd 6360	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → (𝐸‘𝑖) ∈ 𝑆)
199	195, 1, 134, 93, 198	caragenfiiuncl 40729	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ 𝑆)
200	199	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ 𝑆)
201	193, 194, 139, 200	fvmptd 6288	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
202	201, 200	eqeltrd 2701	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝑆)
203	133, 134, 2, 202, 135	caragensplit 40714	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (𝑂‘𝐴))
204	192, 203	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (𝑂‘𝐴))
205	204	oveq1d 6665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) + 𝑌) = ((𝑂‘𝐴) + 𝑌))
206	189, 205	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘𝐴) + 𝑌))
207	206	3adant3 1081	. . . . . 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))
208	184, 207	breqtrd 4679	. . . . 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))
209	208	3exp 1264	. . . 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))))
210	209	rexlimdv 3030	. . 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)))
211	146, 210	mpd 15	. 2 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) + (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌))
212	11, 211	eqbrtrd 4675	1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +𝑒 (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ∪ ciun 4520  Disj wdisj 4620   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936   + caddc 9939  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  ℤcz 11377  ℤ_≥cuz 11687  ℝ⁺crp 11832   +_𝑒 cxad 11944  [,]cicc 12178  ...cfz 12326  ..^cfzo 12465  Σcsu 14416  OutMeascome 40703  CaraGenccaragen 40705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-sumge0 40580  df-ome 40704  df-caragen 40706

This theorem is referenced by:  carageniuncl  40737