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Theorem caratheodorylem2 39417
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 39394 . . . . . . . . . 10 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
51, 4syl 17 . . . . . . . . 9 (𝜑𝑆 ⊆ dom 𝑂)
65adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑆 ⊆ dom 𝑂)
7 caratheodorylem2.e . . . . . . . . 9 (𝜑𝐸:ℕ⟶𝑆)
87ffvelrnda 6267 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ 𝑆)
96, 8sseldd 3569 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom 𝑂)
10 elssuni 4403 . . . . . . 7 ((𝐸𝑛) ∈ dom 𝑂 → (𝐸𝑛) ⊆ dom 𝑂)
119, 10syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ dom 𝑂)
1211, 2syl6sseqr 3615 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ 𝑋)
1312ralrimiva 2949 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
14 iunss 4497 . . . 4 ( 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
1513, 14sylibr 223 . . 3 (𝜑 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
161, 2, 15omexrcl 39397 . 2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
17 nnex 10903 . . . 4 ℕ ∈ V
1817a1i 11 . . 3 (𝜑 → ℕ ∈ V)
191adantr 480 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑂 ∈ OutMeas)
2019, 2, 12omecl 39393 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
21 eqid 2610 . . . 4 (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))
2220, 21fmptd 6292 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))):ℕ⟶(0[,]+∞))
2318, 22sge0xrcl 39278 . 2 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
24 nfv 1830 . . 3 𝑛𝜑
25 nfcv 2751 . . 3 𝑛𝐸
26 nnuz 11599 . . 3 ℕ = (ℤ‘1)
271, 2, 3caragensspw 39399 . . . 4 (𝜑𝑆 ⊆ 𝒫 𝑋)
287, 27fssd 5970 . . 3 (𝜑𝐸:ℕ⟶𝒫 𝑋)
2924, 25, 1, 2, 26, 28omeiunle 39407 . 2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
30 elpwinss 38241 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ⊆ ℕ)
3130resmptd 5371 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → ((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥) = (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))))
3231fveq2d 6107 . . . . . 6 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))))
3332adantl 481 . . . . 5 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))))
34 1zzd 11285 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 1 ∈ ℤ)
3530adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ⊆ ℕ)
36 elinel2 3762 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ∈ Fin)
3736adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
3834, 26, 35, 37uzfissfz 38483 . . . . . 6 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘))
39 vex 3176 . . . . . . . . . . . . 13 𝑥 ∈ V
4039a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ (1...𝑘)) → 𝑥 ∈ V)
411ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝑂 ∈ OutMeas)
4228ad2antrr 758 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝐸:ℕ⟶𝒫 𝑋)
43 fz1ssnn 12243 . . . . . . . . . . . . . . . . . 18 (1...𝑘) ⊆ ℕ
44 ssel2 3563 . . . . . . . . . . . . . . . . . 18 ((𝑥 ⊆ (1...𝑘) ∧ 𝑛𝑥) → 𝑛 ∈ (1...𝑘))
4543, 44sseldi 3566 . . . . . . . . . . . . . . . . 17 ((𝑥 ⊆ (1...𝑘) ∧ 𝑛𝑥) → 𝑛 ∈ ℕ)
4645adantll 746 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝑛 ∈ ℕ)
4742, 46ffvelrnd 6268 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝐸𝑛) ∈ 𝒫 𝑋)
48 elpwi 4117 . . . . . . . . . . . . . . 15 ((𝐸𝑛) ∈ 𝒫 𝑋 → (𝐸𝑛) ⊆ 𝑋)
4947, 48syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝐸𝑛) ⊆ 𝑋)
5041, 2, 49omecl 39393 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
51 eqid 2610 . . . . . . . . . . . . 13 (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))) = (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))
5250, 51fmptd 6292 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ (1...𝑘)) → (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))):𝑥⟶(0[,]+∞))
5340, 52sge0xrcl 39278 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
54533adant2 1073 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
55 ovex 6577 . . . . . . . . . . . . 13 (1...𝑘) ∈ V
5655a1i 11 . . . . . . . . . . . 12 (𝜑 → (1...𝑘) ∈ V)
57 elfznn 12241 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
5857, 20sylan2 490 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
59 eqid 2610 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))
6058, 59fmptd 6292 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛))):(1...𝑘)⟶(0[,]+∞))
6156, 60sge0xrcl 39278 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
62613ad2ant1 1075 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
63163ad2ant1 1075 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
6455a1i 11 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (1...𝑘) ∈ V)
65 simpl1 1057 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝜑)
6657adantl 481 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
6765, 66, 20syl2anc 691 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
68 simp3 1056 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ⊆ (1...𝑘))
6964, 67, 68sge0lessmpt 39292 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))))
701adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑂 ∈ OutMeas)
717adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐸:ℕ⟶𝑆)
72 caratheodorylem2.5 . . . . . . . . . . . . . . 15 (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))
7372adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → Disj 𝑛 ∈ ℕ (𝐸𝑛))
74 caratheodorylem2.g . . . . . . . . . . . . . . 15 𝐺 = (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛))
75 nfiu1 4486 . . . . . . . . . . . . . . . 16 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛)
76 nfcv 2751 . . . . . . . . . . . . . . . 16 𝑘 𝑚 ∈ (1...𝑛)(𝐸𝑚)
77 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝐸𝑛) = (𝐸𝑚))
7877cbviunv 4495 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑘)(𝐸𝑚)
7978a1i 11 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑘)(𝐸𝑚))
80 oveq2 6557 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (1...𝑘) = (1...𝑛))
8180iuneq1d 4481 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 𝑚 ∈ (1...𝑘)(𝐸𝑚) = 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8279, 81eqtrd 2644 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8375, 76, 82cbvmpt 4677 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛)) = (𝑛 ∈ ℕ ↦ 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8474, 83eqtri 2632 . . . . . . . . . . . . . 14 𝐺 = (𝑛 ∈ ℕ ↦ 𝑚 ∈ (1...𝑛)(𝐸𝑚))
85 id 22 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
8685, 26syl6eleq 2698 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → 𝑘 ∈ (ℤ‘1))
8786adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
8870, 3, 26, 71, 73, 84, 87caratheodorylem1 39416 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝑂‘(𝐺𝑘)) = (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))))
8988eqcomd 2616 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) = (𝑂‘(𝐺𝑘)))
9015adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
91 fvex 6113 . . . . . . . . . . . . . . . . 17 (𝐸𝑛) ∈ V
9255, 91iunex 7039 . . . . . . . . . . . . . . . 16 𝑛 ∈ (1...𝑘)(𝐸𝑛) ∈ V
9374fvmpt2 6200 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑘)(𝐸𝑛) ∈ V) → (𝐺𝑘) = 𝑛 ∈ (1...𝑘)(𝐸𝑛))
9485, 92, 93sylancl 693 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝐺𝑘) = 𝑛 ∈ (1...𝑘)(𝐸𝑛))
9543a1i 11 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (1...𝑘) ⊆ ℕ)
96 iunss1 4468 . . . . . . . . . . . . . . . 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 3602 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝐺𝑘) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
9998adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
10070, 2, 90, 99omessle 39388 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝑂‘(𝐺𝑘)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
10189, 100eqbrtrd 4605 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
1021013adant3 1074 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
10354, 62, 63, 69, 102xrletrd 11869 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
1041033exp 1256 . . . . . . . 8 (𝜑 → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))))
105104adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))))
106105rexlimdv 3012 . . . . . 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 4605 . . . 4 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
109108ralrimiva 2949 . . 3 (𝜑 → ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
11018, 22, 16sge0lefi 39291 . . 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 246 . 2 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
11216, 23, 29, 111xrletrid 11862 1 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  cin 3539  wss 3540  𝒫 cpw 4108   cuni 4372   ciun 4455  Disj wdisj 4553   class class class wbr 4583  cmpt 4643  dom cdm 5038  cres 5040  wf 5800  cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816  +∞cpnf 9950  *cxr 9952  cle 9954  cn 10897  cuz 11563  [,]cicc 12049  ...cfz 12197  Σ^csumge0 39255  OutMeascome 39379  CaraGenccaragen 39381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-sumge0 39256  df-ome 39380  df-caragen 39382
This theorem is referenced by:  caratheodory  39418
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