Theorem carageniuncllem2 46938

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 4167	. . . . 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 46926	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
8		difssd 4069	. . . 4 ⊢ (𝜑 → (𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ 𝐴)
9	1, 2, 3, 4, 8	omessre 46926	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
10		rexadd 13173	. . 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 4176	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋)
14	3, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋)
15	1, 2	unidmex 45469	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ V)
16		ssexg 5253	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V)
17	3, 15, 16	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ V)
18		inex1g 5249	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
19	17, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
20		elpwg 4534	. . . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐴 ∩ (𝐹‘𝑛)) ∈ 𝒫 𝑋)
24		eqid 2735	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛)))
25	23, 24	fmptd 7055	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))):𝑍⟶𝒫 𝑋)
26		fveq2 6829	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
27	26	ineq2d 4151	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝐴 ∩ (𝐹‘𝑘)) = (𝐴 ∩ (𝐹‘𝑛)))
28	27	cbvmptv 5178	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘))) = (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛)))
29	28	feq1i 6648	. . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
33	19	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
34	28	fvmpt2 6948	. . . . . . . . . . . 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 4948	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)))
37	36	fveq2d 6833	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
38		nfv 1916	. . . . . . . . . . . . . . . 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 46876	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
43	42	eqcomd 2741	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛))
44	43	ineq2d 4151	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)))
45		iunin2 5002	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛))
46	45	eqcomi 2744	. . . . . . . . . . . . 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 2770	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)))
49	48	fveq2d 6833	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
50	49, 7	eqeltrrd 2836	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
51	37, 50	eqeltrd 2835	. . . . . . . 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 46935	. . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
55		elpwinss 45468	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ⊆ 𝑍)
56	55	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑧 ⊆ 𝑍)
57		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑧)
58	56, 57	sseldd 3918	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
59	58	adantll 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
60	19	ad2antrr 727	. . . . . . . . . . . . . 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 6833	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
63	62	sumeq2dv 15653	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
64	63	oveq1d 7371	. . . . . . . . . 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 5087	. . . . . . . . 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 229	. . . . . . . 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 3148	. . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑀 ∈ ℤ)
71	55	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ⊆ 𝑍)
72		elinel2 4133	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ∈ Fin)
73	72	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
74	70, 12, 71, 73	uzfissfz 45744	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → ∃𝑘 ∈ 𝑍 𝑧 ⊆ (𝑀...𝑘))
75	74	adantr 480	. . . . . . . 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 731	. . . . . . . . . . 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 13924	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ (𝑀...𝑘) → (𝑀...𝑘) ∈ Fin)
78		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ (𝑀...𝑘) → 𝑧 ⊆ (𝑀...𝑘))
79		ssfi 9096	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀...𝑘) ∈ Fin ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ∈ Fin)
80	77, 78, 79	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ (𝑀...𝑘) → 𝑧 ∈ Fin)
81	80	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ∈ Fin)
82	1	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → 𝑂 ∈ OutMeas)
83	3	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → 𝐴 ⊆ 𝑋)
84	4	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘𝐴) ∈ ℝ)
85		inss1 4167	. . . . . . . . . . . . . . . 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 46926	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
88	81, 87	fsumrecl 15685	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
89	52	rpred 12975	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ℝ)
90	89	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑌 ∈ ℝ)
91	88, 90	readdcld 11163	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
92	91	ad4ant14 753	. . . . . . . . . . 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 13924	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...𝑘) ∈ Fin)
94	85	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴)
95	1, 2, 3, 4, 94	omessre 46926	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
96	95	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
97	93, 96	fsumrecl 15685	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
98	97	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
99	89	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ)
100	98, 99	readdcld 11163	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
101	100	ad2antrr 727	. . . . . . . . . . 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 769	. . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
104		fzfid 13924	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (𝑀...𝑘) ∈ Fin)
105	96	adantlr 716	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
106		0xr 11181	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ*
107	106	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ∈ ℝ*)
108		pnfxr 11188	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ*
109	108	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → +∞ ∈ ℝ*)
110	1, 2, 14	omecl 46919	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞))
111	110	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞))
112		iccgelb 13344	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
113	107, 109, 111, 112	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
114	113	adantlr 716	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
115		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ⊆ (𝑀...𝑘))
116	104, 105, 114, 115	fsumless 15748	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ≤ Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
117	88, 103, 90, 116	leadd1dd 11753	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ≤ (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
118	117	ad4ant14 753	. . . . . . . . . . 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 11295	. . . . . . . . . 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 412	. . . . . . . . 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 3150	. . . . . . . 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 3138	. . . . . 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 727	. . . . . . . 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 484	. . . . . . . 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 5096	. . . . . . 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 412	. . . . . 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 3148	. . . . 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 484	. . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑂 ∈ OutMeas)
133		carageniuncllem2.s	. . . . . . . . . 10 ⊢ 𝑆 = (CaraGen‘𝑂)
134	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ⊆ 𝑋)
135	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘𝐴) ∈ ℝ)
136	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐸:𝑍⟶𝑆)
137		carageniuncllem2.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))
138		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
139	132, 133, 2, 134, 135, 12, 136, 137, 40, 138	carageniuncllem1 46937	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘))))
140	139	oveq1d 7371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌))
141	140	adantr 480	. . . . . . 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 5100	. . . . . 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 412	. . . . 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 3148	. . . 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 4167	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐺‘𝑘)) ⊆ 𝐴
149	148	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∩ (𝐺‘𝑘)) ⊆ 𝐴)
150	132, 2, 134, 135, 149	omessre 46926	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ∈ ℝ)
151	89	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑌 ∈ ℝ)
152	150, 151	readdcld 11163	. . . . . . . 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 4069	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ (𝐺‘𝑘)) ⊆ 𝐴)
155	132, 2, 134, 135, 154	omessre 46926	. . . . . . . 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 11283	. . . . . . 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 4079	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ (𝐺‘𝑘)) ⊆ 𝑋)
160		oveq2 7364	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
161	160	iuneq1d 4951	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
162		ovex 7389	. . . . . . . . . . . . . . 15 ⊢ (𝑀...𝑘) ∈ V
163		fvex 6842	. . . . . . . . . . . . . . 15 ⊢ (𝐸‘𝑖) ∈ V
164	162, 163	iunex 7910	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ V
165	161, 137, 164	fvmpt 6936	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
166		fveq2 6829	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → (𝐸‘𝑖) = (𝐸‘𝑛))
167	166	cbviunv 4970	. . . . . . . . . . . . . 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 2770	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) = ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛))
170		elfzuz 13463	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀...𝑘) → 𝑖 ∈ (ℤ≥‘𝑀))
171	12	eqcomi 2744	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘𝑀) = 𝑍
172	171	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀...𝑘) → (ℤ≥‘𝑀) = 𝑍)
173	170, 172	eleqtrd 2837	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀...𝑘) → 𝑖 ∈ 𝑍)
174	173	ssriv 3921	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑘) ⊆ 𝑍
175		iunss1 4938	. . . . . . . . . . . . . 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 3951	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
179	178	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
180	179	sscond 4078	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ (𝐴 ∖ (𝐺‘𝑘)))
181	132, 2, 159, 180	omessle 46914	. . . . . . . 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 11758	. . . . . 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 11162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ∈ ℂ)
185	89	recnd 11162	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℂ)
186	185	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑌 ∈ ℂ)
187	155	recnd 11162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℂ)
188	184, 186, 187	add32d 11363	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) + 𝑌))
189		rexadd 13173	. . . . . . . . . . . 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 2741	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))))
192		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖𝜑
193	39	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → 𝐸:𝑍⟶𝑆)
194	173	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → 𝑖 ∈ 𝑍)
195	193, 194	ffvelcdmd 7026	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → (𝐸‘𝑖) ∈ 𝑆)
196	192, 1, 133, 93, 195	caragenfiiuncl 46931	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ 𝑆)
197	196	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ 𝑆)
198	137, 161, 138, 197	fvmptd3 6960	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
199	198, 197	eqeltrd 2835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝑆)
200	132, 133, 2, 199, 134	caragensplit 46916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (𝑂‘𝐴))
201	191, 200	eqtrd 2770	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (𝑂‘𝐴))
202	201	oveq1d 7371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) + 𝑌) = ((𝑂‘𝐴) + 𝑌))
203	188, 202	eqtrd 2770	. . . . . . 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 5100	. . . . 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 3134	. . 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 5096	1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +𝑒 (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3049  ∃wrex 3059  Vcvv 3427   ∖ cdif 3882   ∩ cin 3884   ⊆ wss 3885  𝒫 cpw 4531  ∪ cuni 4840  ∪ ciun 4923  Disj wdisj 5041   class class class wbr 5074   ↦ cmpt 5155  dom cdm 5620  ⟶wf 6483  ‘cfv 6487  (class class class)co 7356  Fincfn 8882  ℂcc 11025  ℝcr 11026  0cc0 11027   + caddc 11030  +∞cpnf 11165  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169  ℤcz 12513  ℤ_≥cuz 12777  ℝ⁺crp 12931   +_𝑒 cxad 13050  [,]cicc 13290  ...cfz 13450  ..^cfzo 13597  Σcsu 15637  OutMeascome 46905  CaraGenccaragen 46907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-inf2 9551  ax-ac2 10374  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-pre-sup 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-disj 5042  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-oadd 8398  df-omul 8399  df-er 8632  df-map 8764  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-sup 9344  df-oi 9414  df-card 9852  df-acn 9855  df-ac 10027  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12164  df-2 12233  df-3 12234  df-n0 12427  df-z 12514  df-uz 12778  df-rp 12932  df-xadd 13053  df-ico 13293  df-icc 13294  df-fz 13451  df-fzo 13598  df-seq 13953  df-exp 14013  df-hash 14282  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15439  df-sum 15638  df-sumge0 46779  df-ome 46906  df-caragen 46908

This theorem is referenced by:  carageniuncl  46939