Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  caratheodorylem2 Structured version   Visualization version   GIF version

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  ℝ*cxr 10058   ≤ cle 10060  ℕcn 11005  ℤ≥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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