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Theorem caratheodorylem2 41224
Description: Caratheodory's construction is sigma-additive. Main part of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
caratheodorylem2.o (𝜑𝑂 ∈ OutMeas)
caratheodorylem2.x 𝑋 = dom 𝑂
caratheodorylem2.s 𝑆 = (CaraGen‘𝑂)
caratheodorylem2.e (𝜑𝐸:ℕ⟶𝑆)
caratheodorylem2.5 (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))
caratheodorylem2.g 𝐺 = (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛))
Assertion
Ref Expression
caratheodorylem2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
Distinct variable groups:   𝑘,𝐸,𝑛   𝑛,𝐺   𝑘,𝑂,𝑛   𝑛,𝑋   𝜑,𝑘,𝑛
Allowed substitution hints:   𝑆(𝑘,𝑛)   𝐺(𝑘)   𝑋(𝑘)

Proof of Theorem caratheodorylem2
Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caratheodorylem2.o . . 3 (𝜑𝑂 ∈ OutMeas)
2 caratheodorylem2.x . . 3 𝑋 = dom 𝑂
3 caratheodorylem2.s . . . . . . . . . . 11 𝑆 = (CaraGen‘𝑂)
43caragenss 41201 . . . . . . . . . 10 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
51, 4syl 17 . . . . . . . . 9 (𝜑𝑆 ⊆ dom 𝑂)
65adantr 468 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑆 ⊆ dom 𝑂)
7 caratheodorylem2.e . . . . . . . . 9 (𝜑𝐸:ℕ⟶𝑆)
87ffvelrnda 6584 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ 𝑆)
96, 8sseldd 3806 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom 𝑂)
10 elssuni 4668 . . . . . . 7 ((𝐸𝑛) ∈ dom 𝑂 → (𝐸𝑛) ⊆ dom 𝑂)
119, 10syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ dom 𝑂)
1211, 2syl6sseqr 3856 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ 𝑋)
1312ralrimiva 3161 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
14 iunss 4760 . . . 4 ( 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
1513, 14sylibr 225 . . 3 (𝜑 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
161, 2, 15omexrcl 41204 . 2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
17 nnex 11314 . . . 4 ℕ ∈ V
1817a1i 11 . . 3 (𝜑 → ℕ ∈ V)
191adantr 468 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑂 ∈ OutMeas)
2019, 2, 12omecl 41200 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
21 eqid 2813 . . . 4 (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))
2220, 21fmptd 6609 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))):ℕ⟶(0[,]+∞))
2318, 22sge0xrcl 41082 . 2 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
24 nfv 2005 . . 3 𝑛𝜑
25 nfcv 2955 . . 3 𝑛𝐸
26 nnuz 11944 . . 3 ℕ = (ℤ‘1)
271, 2, 3caragensspw 41206 . . . 4 (𝜑𝑆 ⊆ 𝒫 𝑋)
287, 27fssd 6273 . . 3 (𝜑𝐸:ℕ⟶𝒫 𝑋)
2924, 25, 1, 2, 26, 28omeiunle 41214 . 2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
30 elpwinss 39710 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ⊆ ℕ)
3130resmptd 5664 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → ((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥) = (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))))
3231fveq2d 6415 . . . . . 6 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))))
3332adantl 469 . . . . 5 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))))
34 1zzd 11677 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 1 ∈ ℤ)
3530adantl 469 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ⊆ ℕ)
36 elinel2 4006 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ∈ Fin)
3736adantl 469 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
3834, 26, 35, 37uzfissfz 40023 . . . . . 6 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘))
39 vex 3401 . . . . . . . . . . . . 13 𝑥 ∈ V
4039a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ (1...𝑘)) → 𝑥 ∈ V)
411ad2antrr 708 . . . . . . . . . . . . . 14 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝑂 ∈ OutMeas)
4228ad2antrr 708 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝐸:ℕ⟶𝒫 𝑋)
43 fz1ssnn 12598 . . . . . . . . . . . . . . . . . 18 (1...𝑘) ⊆ ℕ
44 ssel2 3800 . . . . . . . . . . . . . . . . . 18 ((𝑥 ⊆ (1...𝑘) ∧ 𝑛𝑥) → 𝑛 ∈ (1...𝑘))
4543, 44sseldi 3803 . . . . . . . . . . . . . . . . 17 ((𝑥 ⊆ (1...𝑘) ∧ 𝑛𝑥) → 𝑛 ∈ ℕ)
4645adantll 696 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝑛 ∈ ℕ)
4742, 46ffvelrnd 6585 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝐸𝑛) ∈ 𝒫 𝑋)
48 elpwi 4368 . . . . . . . . . . . . . . 15 ((𝐸𝑛) ∈ 𝒫 𝑋 → (𝐸𝑛) ⊆ 𝑋)
4947, 48syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝐸𝑛) ⊆ 𝑋)
5041, 2, 49omecl 41200 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
51 eqid 2813 . . . . . . . . . . . . 13 (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))) = (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))
5250, 51fmptd 6609 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ (1...𝑘)) → (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))):𝑥⟶(0[,]+∞))
5340, 52sge0xrcl 41082 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
54533adant2 1154 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
55 ovex 6909 . . . . . . . . . . . . 13 (1...𝑘) ∈ V
5655a1i 11 . . . . . . . . . . . 12 (𝜑 → (1...𝑘) ∈ V)
57 elfznn 12596 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
5857, 20sylan2 582 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
59 eqid 2813 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))
6058, 59fmptd 6609 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛))):(1...𝑘)⟶(0[,]+∞))
6156, 60sge0xrcl 41082 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
62613ad2ant1 1156 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
63163ad2ant1 1156 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
6455a1i 11 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (1...𝑘) ∈ V)
65 simpl1 1235 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝜑)
6657adantl 469 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
6765, 66, 20syl2anc 575 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
68 simp3 1161 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ⊆ (1...𝑘))
6964, 67, 68sge0lessmpt 41096 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))))
701adantr 468 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑂 ∈ OutMeas)
717adantr 468 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐸:ℕ⟶𝑆)
72 caratheodorylem2.5 . . . . . . . . . . . . . . 15 (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))
7372adantr 468 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → Disj 𝑛 ∈ ℕ (𝐸𝑛))
74 caratheodorylem2.g . . . . . . . . . . . . . . 15 𝐺 = (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛))
75 nfiu1 4749 . . . . . . . . . . . . . . . 16 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛)
76 nfcv 2955 . . . . . . . . . . . . . . . 16 𝑘 𝑚 ∈ (1...𝑛)(𝐸𝑚)
77 fveq2 6411 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝐸𝑛) = (𝐸𝑚))
7877cbviunv 4758 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑘)(𝐸𝑚)
7978a1i 11 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑘)(𝐸𝑚))
80 oveq2 6885 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (1...𝑘) = (1...𝑛))
8180iuneq1d 4744 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 𝑚 ∈ (1...𝑘)(𝐸𝑚) = 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8279, 81eqtrd 2847 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8375, 76, 82cbvmpt 4950 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛)) = (𝑛 ∈ ℕ ↦ 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8474, 83eqtri 2835 . . . . . . . . . . . . . 14 𝐺 = (𝑛 ∈ ℕ ↦ 𝑚 ∈ (1...𝑛)(𝐸𝑚))
85 id 22 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
8685, 26syl6eleq 2902 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → 𝑘 ∈ (ℤ‘1))
8786adantl 469 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
8870, 3, 26, 71, 73, 84, 87caratheodorylem1 41223 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝑂‘(𝐺𝑘)) = (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))))
8988eqcomd 2819 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) = (𝑂‘(𝐺𝑘)))
9015adantr 468 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
91 fvex 6424 . . . . . . . . . . . . . . . . 17 (𝐸𝑛) ∈ V
9255, 91iunex 7380 . . . . . . . . . . . . . . . 16 𝑛 ∈ (1...𝑘)(𝐸𝑛) ∈ V
9374fvmpt2 6515 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑘)(𝐸𝑛) ∈ V) → (𝐺𝑘) = 𝑛 ∈ (1...𝑘)(𝐸𝑛))
9485, 92, 93sylancl 576 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝐺𝑘) = 𝑛 ∈ (1...𝑘)(𝐸𝑛))
9543a1i 11 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (1...𝑘) ⊆ ℕ)
96 iunss1 4731 . . . . . . . . . . . . . . . 16 ((1...𝑘) ⊆ ℕ → 𝑛 ∈ (1...𝑘)(𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
9795, 96syl 17 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → 𝑛 ∈ (1...𝑘)(𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
9894, 97eqsstrd 3843 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝐺𝑘) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
9998adantl 469 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
10070, 2, 90, 99omessle 41195 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝑂‘(𝐺𝑘)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
10189, 100eqbrtrd 4873 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
1021013adant3 1155 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
10354, 62, 63, 69, 102xrletrd 12214 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
1041033exp 1141 . . . . . . . 8 (𝜑 → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))))
105104adantr 468 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))))
106105rexlimdv 3225 . . . . . 6 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛))))
10738, 106mpd 15 . . . . 5 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
10833, 107eqbrtrd 4873 . . . 4 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
109108ralrimiva 3161 . . 3 (𝜑 → ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
11018, 22, 16sge0lefi 41095 . . 3 (𝜑 → ((Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ↔ ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛))))
111109, 110mpbird 248 . 2 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
11216, 23, 29, 111xrletrid 12207 1 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2157  wral 3103  wrex 3104  Vcvv 3398  cin 3775  wss 3776  𝒫 cpw 4358   cuni 4637   ciun 4719  Disj wdisj 4819   class class class wbr 4851  cmpt 4930  dom cdm 5318  cres 5320  wf 6100  cfv 6104  (class class class)co 6877  Fincfn 8195  0cc0 10224  1c1 10225  +∞cpnf 10359  *cxr 10361  cle 10363  cn 11308  cuz 11907  [,]cicc 12399  ...cfz 12552  Σ^csumge0 41059  OutMeascome 41186  CaraGenccaragen 41188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-inf2 8788  ax-ac2 9573  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-disj 4820  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-omul 7804  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-sup 8590  df-oi 8657  df-card 9051  df-acn 9054  df-ac 9225  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-div 10973  df-nn 11309  df-2 11367  df-3 11368  df-n0 11563  df-z 11647  df-uz 11908  df-rp 12050  df-xadd 12166  df-ico 12402  df-icc 12403  df-fz 12553  df-fzo 12693  df-seq 13028  df-exp 13087  df-hash 13341  df-cj 14065  df-re 14066  df-im 14067  df-sqrt 14201  df-abs 14202  df-clim 14445  df-sum 14643  df-sumge0 41060  df-ome 41187  df-caragen 41189
This theorem is referenced by:  caratheodory  41225
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