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Theorem caratheodorylem2 40504
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 𝑥 𝑚 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 40481 . . . . . . . . . 10 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
51, 4syl 17 . . . . . . . . 9 (𝜑𝑆 ⊆ dom 𝑂)
65adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑆 ⊆ dom 𝑂)
7 caratheodorylem2.e . . . . . . . . 9 (𝜑𝐸:ℕ⟶𝑆)
87ffvelrnda 6345 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ 𝑆)
96, 8sseldd 3596 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom 𝑂)
10 elssuni 4458 . . . . . . 7 ((𝐸𝑛) ∈ dom 𝑂 → (𝐸𝑛) ⊆ dom 𝑂)
119, 10syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ dom 𝑂)
1211, 2syl6sseqr 3644 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ 𝑋)
1312ralrimiva 2963 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
14 iunss 4552 . . . 4 ( 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
1513, 14sylibr 224 . . 3 (𝜑 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
161, 2, 15omexrcl 40484 . 2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
17 nnex 11011 . . . 4 ℕ ∈ V
1817a1i 11 . . 3 (𝜑 → ℕ ∈ V)
191adantr 481 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑂 ∈ OutMeas)
2019, 2, 12omecl 40480 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
21 eqid 2620 . . . 4 (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))
2220, 21fmptd 6371 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))):ℕ⟶(0[,]+∞))
2318, 22sge0xrcl 40365 . 2 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
24 nfv 1841 . . 3 𝑛𝜑
25 nfcv 2762 . . 3 𝑛𝐸
26 nnuz 11708 . . 3 ℕ = (ℤ‘1)
271, 2, 3caragensspw 40486 . . . 4 (𝜑𝑆 ⊆ 𝒫 𝑋)
287, 27fssd 6044 . . 3 (𝜑𝐸:ℕ⟶𝒫 𝑋)
2924, 25, 1, 2, 26, 28omeiunle 40494 . 2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
30 elpwinss 39036 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ⊆ ℕ)
3130resmptd 5440 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → ((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥) = (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))))
3231fveq2d 6182 . . . . . 6 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))))
3332adantl 482 . . . . 5 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))))
34 1zzd 11393 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 1 ∈ ℤ)
3530adantl 482 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ⊆ ℕ)
36 elinel2 3792 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ∈ Fin)
3736adantl 482 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
3834, 26, 35, 37uzfissfz 39355 . . . . . 6 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘))
39 vex 3198 . . . . . . . . . . . . 13 𝑥 ∈ V
4039a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ (1...𝑘)) → 𝑥 ∈ V)
411ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝑂 ∈ OutMeas)
4228ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝐸:ℕ⟶𝒫 𝑋)
43 fz1ssnn 12357 . . . . . . . . . . . . . . . . . 18 (1...𝑘) ⊆ ℕ
44 ssel2 3590 . . . . . . . . . . . . . . . . . 18 ((𝑥 ⊆ (1...𝑘) ∧ 𝑛𝑥) → 𝑛 ∈ (1...𝑘))
4543, 44sseldi 3593 . . . . . . . . . . . . . . . . 17 ((𝑥 ⊆ (1...𝑘) ∧ 𝑛𝑥) → 𝑛 ∈ ℕ)
4645adantll 749 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → 𝑛 ∈ ℕ)
4742, 46ffvelrnd 6346 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝐸𝑛) ∈ 𝒫 𝑋)
48 elpwi 4159 . . . . . . . . . . . . . . 15 ((𝐸𝑛) ∈ 𝒫 𝑋 → (𝐸𝑛) ⊆ 𝑋)
4947, 48syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝐸𝑛) ⊆ 𝑋)
5041, 2, 49omecl 40480 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ (1...𝑘)) ∧ 𝑛𝑥) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
51 eqid 2620 . . . . . . . . . . . . 13 (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))) = (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))
5250, 51fmptd 6371 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ (1...𝑘)) → (𝑛𝑥 ↦ (𝑂‘(𝐸𝑛))):𝑥⟶(0[,]+∞))
5340, 52sge0xrcl 40365 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
54533adant2 1078 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
55 ovex 6663 . . . . . . . . . . . . 13 (1...𝑘) ∈ V
5655a1i 11 . . . . . . . . . . . 12 (𝜑 → (1...𝑘) ∈ V)
57 elfznn 12355 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
5857, 20sylan2 491 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
59 eqid 2620 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))
6058, 59fmptd 6371 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛))):(1...𝑘)⟶(0[,]+∞))
6156, 60sge0xrcl 40365 . . . . . . . . . . 11 (𝜑 → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
62613ad2ant1 1080 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ∈ ℝ*)
63163ad2ant1 1080 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
6455a1i 11 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (1...𝑘) ∈ V)
65 simpl1 1062 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝜑)
6657adantl 482 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
6765, 66, 20syl2anc 692 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
68 simp3 1061 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ⊆ (1...𝑘))
6964, 67, 68sge0lessmpt 40379 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))))
701adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑂 ∈ OutMeas)
717adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐸:ℕ⟶𝑆)
72 caratheodorylem2.5 . . . . . . . . . . . . . . 15 (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))
7372adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → Disj 𝑛 ∈ ℕ (𝐸𝑛))
74 caratheodorylem2.g . . . . . . . . . . . . . . 15 𝐺 = (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛))
75 nfiu1 4541 . . . . . . . . . . . . . . . 16 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛)
76 nfcv 2762 . . . . . . . . . . . . . . . 16 𝑘 𝑚 ∈ (1...𝑛)(𝐸𝑚)
77 fveq2 6178 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝐸𝑛) = (𝐸𝑚))
7877cbviunv 4550 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑘)(𝐸𝑚)
7978a1i 11 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑘)(𝐸𝑚))
80 oveq2 6643 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (1...𝑘) = (1...𝑛))
8180iuneq1d 4536 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 𝑚 ∈ (1...𝑘)(𝐸𝑚) = 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8279, 81eqtrd 2654 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 𝑛 ∈ (1...𝑘)(𝐸𝑛) = 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8375, 76, 82cbvmpt 4740 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛)) = (𝑛 ∈ ℕ ↦ 𝑚 ∈ (1...𝑛)(𝐸𝑚))
8474, 83eqtri 2642 . . . . . . . . . . . . . 14 𝐺 = (𝑛 ∈ ℕ ↦ 𝑚 ∈ (1...𝑛)(𝐸𝑚))
85 id 22 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
8685, 26syl6eleq 2709 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → 𝑘 ∈ (ℤ‘1))
8786adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
8870, 3, 26, 71, 73, 84, 87caratheodorylem1 40503 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝑂‘(𝐺𝑘)) = (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))))
8988eqcomd 2626 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) = (𝑂‘(𝐺𝑘)))
9015adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ ℕ (𝐸𝑛) ⊆ 𝑋)
91 fvex 6188 . . . . . . . . . . . . . . . . 17 (𝐸𝑛) ∈ V
9255, 91iunex 7132 . . . . . . . . . . . . . . . 16 𝑛 ∈ (1...𝑘)(𝐸𝑛) ∈ V
9374fvmpt2 6278 . . . . . . . . . . . . . . . 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 4523 . . . . . . . . . . . . . . . 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 3631 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝐺𝑘) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
9998adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
10070, 2, 90, 99omessle 40475 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝑂‘(𝐺𝑘)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
10189, 100eqbrtrd 4666 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
1021013adant3 1079 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
10354, 62, 63, 69, 102xrletrd 11978 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
1041033exp 1262 . . . . . . . 8 (𝜑 → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))))
105104adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛𝑥 ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))))
106105rexlimdv 3026 . . . . . 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 4666 . . . 4 ((𝜑𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
109108ralrimiva 2963 . . 3 (𝜑 → ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛))) ↾ 𝑥)) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
11018, 22, 16sge0lefi 40378 . . 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 247 . 2 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))) ≤ (𝑂 𝑛 ∈ ℕ (𝐸𝑛)))
11216, 23, 29, 111xrletrid 11971 1 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  cin 3566  wss 3567  𝒫 cpw 4149   cuni 4427   ciun 4511  Disj wdisj 4611   class class class wbr 4644  cmpt 4720  dom cdm 5104  cres 5106  wf 5872  cfv 5876  (class class class)co 6635  Fincfn 7940  0cc0 9921  1c1 9922  +∞cpnf 10056  *cxr 10058  cle 10060  cn 11005  cuz 11672  [,]cicc 12163  ...cfz 12311  Σ^csumge0 40342  OutMeascome 40466  CaraGenccaragen 40468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-disj 4612  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-omul 7550  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-oi 8400  df-card 8750  df-acn 8753  df-ac 8924  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-rp 11818  df-xadd 11932  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398  df-sumge0 40343  df-ome 40467  df-caragen 40469
This theorem is referenced by:  caratheodory  40505
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