Theorem carageniuncllem2 46619

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 4184	. . . . 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 46607	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
8		difssd 4084	. . . 4 ⊢ (𝜑 → (𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ 𝐴)
9	1, 2, 3, 4, 8	omessre 46607	. . 3 ⊢ (𝜑 → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
10		rexadd 13131	. . 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 584	. 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 4193	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋)
14	3, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝑋)
15	1, 2	unidmex 45146	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ V)
16		ssexg 5259	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V)
17	3, 15, 16	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ V)
18		inex1g 5255	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ V → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
19	17, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
20		elpwg 4550	. . . . . . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛)))
25	23, 24	fmptd 7047	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛))):𝑍⟶𝒫 𝑋)
26		fveq2 6822	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
27	26	ineq2d 4167	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝐴 ∩ (𝐹‘𝑘)) = (𝐴 ∩ (𝐹‘𝑛)))
28	27	cbvmptv 5193	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘))) = (𝑛 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑛)))
29	28	feq1i 6642	. . . . . . . . . 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 6940	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝐴 ∩ (𝐹‘𝑛)) ∈ V) → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = (𝐴 ∩ (𝐹‘𝑛)))
35	32, 33, 34	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = (𝐴 ∩ (𝐹‘𝑛)))
36	35	iuneq2dv 4964	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)))
37	36	fveq2d 6826	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
38		nfv 1915	. . . . . . . . . . . . . . . 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 46557	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛))
44	43	ineq2d 4167	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)))
45		iunin2 5017	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛))
46	45	eqcomi 2740	. . . . . . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛)))
49	48	fveq2d 6826	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) = (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))))
50	49, 7	eqeltrrd 2832	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
51	37, 50	eqeltrd 2831	. . . . . . . 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 46616	. . . . . . 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 45145	. . . . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
59	58	adantll 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
60	19	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐴 ∩ (𝐹‘𝑛)) ∈ V)
61	59, 60, 34	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → ((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛) = (𝐴 ∩ (𝐹‘𝑛)))
62	61	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
63	62	sumeq2dv 15609	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘((𝑘 ∈ 𝑍 ↦ (𝐴 ∩ (𝐹‘𝑘)))‘𝑛)) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
64	63	oveq1d 7361	. . . . . . . . . 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 5102	. . . . . . . . 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 3145	. . . . . . 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 4149	. . . . . . . . . . 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 45424	. . . . . . . . 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 730	. . . . . . . . . . 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 13880	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ (𝑀...𝑘) → (𝑀...𝑘) ∈ Fin)
78		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ (𝑀...𝑘) → 𝑧 ⊆ (𝑀...𝑘))
79		ssfi 9082	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀...𝑘) ∈ Fin ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ∈ Fin)
80	77, 78, 79	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ (𝑀...𝑘) → 𝑧 ∈ Fin)
81	80	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑧 ∈ Fin)
82	1	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → 𝑂 ∈ OutMeas)
83	3	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → 𝐴 ⊆ 𝑋)
84	4	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘𝐴) ∈ ℝ)
85		inss1 4184	. . . . . . . . . . . . . . . 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 46607	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
88	81, 87	fsumrecl 15641	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
89	52	rpred 12934	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ℝ)
90	89	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → 𝑌 ∈ ℝ)
91	88, 90	readdcld 11141	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
92	91	ad4ant14 752	. . . . . . . . . . 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 13880	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀...𝑘) ∈ Fin)
94	85	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∩ (𝐹‘𝑛)) ⊆ 𝐴)
95	1, 2, 3, 4, 94	omessre 46607	. . . . . . . . . . . . . . . 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 15641	. . . . . . . . . . . . . 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 11141	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ∈ ℝ)
101	100	ad2antrr 726	. . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
104		fzfid 13880	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (𝑀...𝑘) ∈ Fin)
105	96	adantlr 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ ℝ)
106		0xr 11159	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ*
107	106	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ∈ ℝ*)
108		pnfxr 11166	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ*
109	108	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → +∞ ∈ ℝ*)
110	1, 2, 14	omecl 46600	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞))
111	110	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞))
112		iccgelb 13302	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
113	107, 109, 111, 112	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑘)) → 0 ≤ (𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
114	113	adantlr 715	. . . . . . . . . . . . . 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 15703	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) ≤ Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))))
117	88, 103, 90, 116	leadd1dd 11731	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ (𝑀...𝑘)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌) ≤ (Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) + 𝑌))
118	117	ad4ant14 752	. . . . . . . . . . 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 11273	. . . . . . . . . 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 3147	. . . . . . . 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 3135	. . . . . 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 726	. . . . . . . 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 5111	. . . . . . 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 3145	. . . . 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 46618	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → Σ𝑛 ∈ (𝑀...𝑘)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝑘))))
140	139	oveq1d 7361	. . . . . . . 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 5115	. . . . . 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 3145	. . . 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 1133	. . . . . . 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 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ)
148		inss1 4184	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐺‘𝑘)) ⊆ 𝐴
149	148	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∩ (𝐺‘𝑘)) ⊆ 𝐴)
150	132, 2, 134, 135, 149	omessre 46607	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ∈ ℝ)
151	89	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑌 ∈ ℝ)
152	150, 151	readdcld 11141	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) ∈ ℝ)
153	152	3adant3 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) ∈ ℝ)
154		difssd 4084	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ (𝐺‘𝑘)) ⊆ 𝐴)
155	132, 2, 134, 135, 154	omessre 46607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℝ)
156	155	3adant3 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) < ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌)) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℝ)
157		simp3 1138	. . . . . . . 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 11261	. . . . . . 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 4094	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ (𝐺‘𝑘)) ⊆ 𝑋)
160		oveq2 7354	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
161	160	iuneq1d 4967	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
162		ovex 7379	. . . . . . . . . . . . . . 15 ⊢ (𝑀...𝑘) ∈ V
163		fvex 6835	. . . . . . . . . . . . . . 15 ⊢ (𝐸‘𝑖) ∈ V
164	162, 163	iunex 7900	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖) ∈ V
165	161, 137, 164	fvmpt 6929	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
166		fveq2 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → (𝐸‘𝑖) = (𝐸‘𝑛))
167	166	cbviunv 4987	. . . . . . . . . . . . . 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 2766	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → (𝐺‘𝑘) = ∪ 𝑛 ∈ (𝑀...𝑘)(𝐸‘𝑛))
170		elfzuz 13420	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀...𝑘) → 𝑖 ∈ (ℤ≥‘𝑀))
171	12	eqcomi 2740	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘𝑀) = 𝑍
172	171	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (𝑀...𝑘) → (ℤ≥‘𝑀) = 𝑍)
173	170, 172	eleqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (𝑀...𝑘) → 𝑖 ∈ 𝑍)
174	173	ssriv 3933	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑘) ⊆ 𝑍
175		iunss1 4954	. . . . . . . . . . . . . 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 3964	. . . . . . . . . . 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 4093	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ (𝐴 ∖ (𝐺‘𝑘)))
181	132, 2, 159, 180	omessle 46595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ≤ (𝑂‘(𝐴 ∖ (𝐺‘𝑘))))
182	181	3adant3 1132	. . . . . . 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 11736	. . . . . 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 11140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∩ (𝐺‘𝑘))) ∈ ℂ)
185	89	recnd 11140	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℂ)
186	185	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑌 ∈ ℂ)
187	155	recnd 11140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑂‘(𝐴 ∖ (𝐺‘𝑘))) ∈ ℂ)
188	184, 186, 187	add32d 11341	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) + 𝑌))
189		rexadd 13131	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))))
191	190	eqcomd 2737	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))))
192		nfv 1915	. . . . . . . . . . . . . . 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 7018	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑘)) → (𝐸‘𝑖) ∈ 𝑆)
196	192, 1, 133, 93, 195	caragenfiiuncl 46612	. . . . . . . . . . . . . 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 6952	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ∪ 𝑖 ∈ (𝑀...𝑘)(𝐸‘𝑖))
199	198, 197	eqeltrd 2831	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝑆)
200	132, 133, 2, 199, 134	caragensplit 46597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) +𝑒 (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (𝑂‘𝐴))
201	191, 200	eqtrd 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = (𝑂‘𝐴))
202	201	oveq1d 7361	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) + 𝑌) = ((𝑂‘𝐴) + 𝑌))
203	188, 202	eqtrd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑂‘(𝐴 ∩ (𝐺‘𝑘))) + 𝑌) + (𝑂‘(𝐴 ∖ (𝐺‘𝑘)))) = ((𝑂‘𝐴) + 𝑌))
204	203	3adant3 1132	. . . . . 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 5115	. . . . 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 1119	. . . 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 3131	. . 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 5111	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 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856  ∪ ciun 4939  Disj wdisj 5056   class class class wbr 5089   ↦ cmpt 5170  dom cdm 5614  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Fincfn 8869  ℂcc 11004  ℝcr 11005  0cc0 11006   + caddc 11009  +∞cpnf 11143  ℝ^*cxr 11145   < clt 11146   ≤ cle 11147  ℤcz 12468  ℤ_≥cuz 12732  ℝ⁺crp 12890   +_𝑒 cxad 13009  [,]cicc 13248  ...cfz 13407  ..^cfzo 13554  Σcsu 15593  OutMeascome 46586  CaraGenccaragen 46588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-ac2 10354  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-disj 5057  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-omul 8390  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-oi 9396  df-card 9832  df-acn 9835  df-ac 10007  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-rp 12891  df-xadd 13012  df-ico 13251  df-icc 13252  df-fz 13408  df-fzo 13555  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-sum 15594  df-sumge0 46460  df-ome 46587  df-caragen 46589

This theorem is referenced by:  carageniuncl  46620