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Theorem caratheodorylem2 46482
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 46459 . . . . . . . . . 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)
87ffvelcdmda 7103 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ 𝑆)
96, 8sseldd 3995 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom 𝑂)
10 elssuni 4941 . . . . . . 7 ((𝐸𝑛) ∈ dom 𝑂 → (𝐸𝑛) ⊆ dom 𝑂)
119, 10syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ dom 𝑂)
1211, 2sseqtrrdi 4046 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ 𝑋)
1312ralrimiva 3143 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
14 iunss 5049 . . . 4 ( 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
1513, 14sylibr 234 . . 3 (𝜑 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
161, 2, 15omexrcl 46462 . 2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
17 nnex 12269 . . . 4 ℕ ∈ V
1817a1i 11 . . 3 (𝜑 → ℕ ∈ V)
191adantr 480 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑂 ∈ OutMeas)
2019, 2, 12omecl 46458 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
21 eqid 2734 . . . 4 (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))
2220, 21fmptd 7133 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))):ℕ⟶(0[,]+∞))
2318, 22sge0xrcl 46340 . 2 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
24 nfv 1911 . . 3 𝑛𝜑
25 nfcv 2902 . . 3 𝑛𝐸
26 nnuz 12918 . . 3 ℕ = (ℤ‘1)
271, 2, 3caragensspw 46464 . . . 4 (𝜑𝑆 ⊆ 𝒫 𝑋)
287, 27fssd 6753 . . 3 (𝜑𝐸:ℕ⟶𝒫 𝑋)
2924, 25, 1, 2, 26, 28omeiunle 46472 . 2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
30 elpwinss 44988 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ⊆ ℕ)
3130resmptd 6059 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → ((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥) = (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))))
3231fveq2d 6910 . . . . . 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 12645 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 1 ∈ ℤ)
3530adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ⊆ ℕ)
36 elinel2 4211 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ∈ Fin)
3736adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
3834, 26, 35, 37uzfissfz 45275 . . . . . 6 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘))
39 vex 3481 . . . . . . . . . . . . 13 𝑥 ∈ V
4039a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ (1...𝑘)) → 𝑥 ∈ V)
411ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝑂 ∈ OutMeas)
4228ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝐸:ℕ⟶𝒫 𝑋)
43 fz1ssnn 13591 . . . . . . . . . . . . . . . . . 18 (1...𝑘) ⊆ ℕ
44 ssel2 3989 . . . . . . . . . . . . . . . . . 18 ((𝑥 ⊆ (1...𝑘) ∧ 𝑛𝑥) → 𝑛 ∈ (1...𝑘))
4543, 44sselid 3992 . . . . . . . . . . . . . . . . 17 ((𝑥 ⊆ (1...𝑘) ∧ 𝑛𝑥) → 𝑛 ∈ ℕ)
4645adantll 714 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝑛 ∈ ℕ)
4742, 46ffvelcdmd 7104 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝐸𝑛) ∈ 𝒫 𝑋)
48 elpwi 4611 . . . . . . . . . . . . . . 15 ((𝐸𝑛) ∈ 𝒫 𝑋 → (𝐸𝑛) ⊆ 𝑋)
4947, 48syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝐸𝑛) ⊆ 𝑋)
5041, 2, 49omecl 46458 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
51 eqid 2734 . . . . . . . . . . . . 13 (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))) = (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))
5250, 51fmptd 7133 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ (1...𝑘)) → (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))):𝑥⟶(0[,]+∞))
5340, 52sge0xrcl 46340 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
54533adant2 1130 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
55 ovex 7463 . . . . . . . . . . . . 13 (1...𝑘) ∈ V
5655a1i 11 . . . . . . . . . . . 12 (𝜑 → (1...𝑘) ∈ V)
57 elfznn 13589 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
5857, 20sylan2 593 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
59 eqid 2734 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))
6058, 59fmptd 7133 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛))):(1...𝑘)⟶(0[,]+∞))
6156, 60sge0xrcl 46340 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
62613ad2ant1 1132 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
63163ad2ant1 1132 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
6455a1i 11 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (1...𝑘) ∈ V)
65 simpl1 1190 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
68 simp3 1137 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ⊆ (1...𝑘))
6964, 67, 68sge0lessmpt 46354 . . . . . . . . . 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 5031 . . . . . . . . . . . . . . . 16 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛)
76 nfcv 2902 . . . . . . . . . . . . . . . 16 𝑘 𝑚 ∈ (1...𝑛)(𝐸𝑚)
77 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝐸𝑛) = (𝐸𝑚))
7877cbviunv 5044 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑘)(𝐸𝑚)
7978a1i 11 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑘)(𝐸𝑚))
80 oveq2 7438 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (1...𝑘) = (1...𝑛))
8180iuneq1d 5023 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 𝑚 ∈ (1...𝑘)(𝐸𝑚) = 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8279, 81eqtrd 2774 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8375, 76, 82cbvmpt 5258 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛)) = (𝑛 ∈ ℕ ↦ 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8474, 83eqtri 2762 . . . . . . . . . . . . . 14 𝐺 = (𝑛 ∈ ℕ ↦ 𝑚 ∈ (1...𝑛)(𝐸𝑚))
85 id 22 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
8685, 26eleqtrdi 2848 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → 𝑘 ∈ (ℤ‘1))
8786adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
8870, 3, 26, 71, 73, 84, 87caratheodorylem1 46481 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝑂‘(𝐺𝑘)) = (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))))
8988eqcomd 2740 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) = (𝑂‘(𝐺𝑘)))
9015adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
91 fvex 6919 . . . . . . . . . . . . . . . . 17 (𝐸𝑛) ∈ V
9255, 91iunex 7991 . . . . . . . . . . . . . . . 16 𝑛 ∈ (1...𝑘)(𝐸𝑛) ∈ V
9374fvmpt2 7026 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑘)(𝐸𝑛) ∈ V) → (𝐺𝑘) = 𝑛 ∈ (1...𝑘)(𝐸𝑛))
9485, 92, 93sylancl 586 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝐺𝑘) = 𝑛 ∈ (1...𝑘)(𝐸𝑛))
9543a1i 11 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (1...𝑘) ⊆ ℕ)
96 iunss1 5010 . . . . . . . . . . . . . . . 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 4033 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝐺𝑘) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
9998adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
10070, 2, 90, 99omessle 46453 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝑂‘(𝐺𝑘)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
10189, 100eqbrtrd 5169 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
1021013adant3 1131 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
10354, 62, 63, 69, 102xrletrd 13200 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
1041033exp 1118 . . . . . . . 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 3150 . . . . . 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 5169 . . . 4 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
109108ralrimiva 3143 . . 3 (𝜑 → ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
11018, 22, 16sge0lefi 46353 . . 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 257 . 2 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
11216, 23, 29, 111xrletrid 13193 1 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  cin 3961  wss 3962  𝒫 cpw 4604   cuni 4911   ciun 4995  Disj wdisj 5114   class class class wbr 5147  cmpt 5230  dom cdm 5688  cres 5690  wf 6558  cfv 6562  (class class class)co 7430  Fincfn 8983  0cc0 11152  1c1 11153  +∞cpnf 11289  *cxr 11291  cle 11293  cn 12263  cuz 12875  [,]cicc 13386  ...cfz 13543  Σ^csumge0 46317  OutMeascome 46444  CaraGenccaragen 46446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-ac2 10500  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-oi 9547  df-card 9976  df-acn 9979  df-ac 10153  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-xadd 13152  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-sumge0 46318  df-ome 46445  df-caragen 46447
This theorem is referenced by:  caratheodory  46483
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