Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnsubadd2lem Structured version   Visualization version   GIF version

 Description: (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovnsubadd2lem.a (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))
ovnsubadd2lem.b (𝜑𝐵 ⊆ (ℝ ↑𝑚 𝑋))
ovnsubadd2lem.c 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
Assertion
Ref Expression
ovnsubadd2lem (𝜑 → ((voln*‘𝑋)‘(𝐴𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝐶,𝑛   𝑛,𝑋   𝜑,𝑛

StepHypRef Expression
1 ovnsubadd2lem.x . . 3 (𝜑𝑋 ∈ Fin)
2 iftrue 4042 . . . . . . . 8 (𝑛 = 1 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
32adantl 481 . . . . . . 7 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
4 ovex 6577 . . . . . . . . . . 11 (ℝ ↑𝑚 𝑋) ∈ V
54a1i 11 . . . . . . . . . 10 (𝜑 → (ℝ ↑𝑚 𝑋) ∈ V)
6 ovnsubadd2lem.a . . . . . . . . . 10 (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))
75, 6ssexd 4733 . . . . . . . . 9 (𝜑𝐴 ∈ V)
87, 6elpwd 38264 . . . . . . . 8 (𝜑𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
98adantr 480 . . . . . . 7 ((𝜑𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
103, 9eqeltrd 2688 . . . . . 6 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
1110adantlr 747 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
12 simpl 472 . . . . . . . . . . 11 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1)
1312iffalsed 4047 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
14 simpr 476 . . . . . . . . . . 11 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2)
1514iftrued 4044 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
1613, 15eqtrd 2644 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
1716adantll 746 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
18 ovnsubadd2lem.b . . . . . . . . . . 11 (𝜑𝐵 ⊆ (ℝ ↑𝑚 𝑋))
195, 18ssexd 4733 . . . . . . . . . 10 (𝜑𝐵 ∈ V)
2019, 18elpwd 38264 . . . . . . . . 9 (𝜑𝐵 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
2120ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
2217, 21eqeltrd 2688 . . . . . . 7 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
2322adantllr 751 . . . . . 6 ((((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
24 simpl 472 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1)
2524iffalsed 4047 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
26 simpr 476 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
2726iffalsed 4047 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅)
2825, 27eqtrd 2644 . . . . . . . 8 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
29 0elpw 4760 . . . . . . . . 9 ∅ ∈ 𝒫 (ℝ ↑𝑚 𝑋)
3029a1i 11 . . . . . . . 8 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3128, 30eqeltrd 2688 . . . . . . 7 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3231adantll 746 . . . . . 6 ((((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3323, 32pm2.61dan 828 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3411, 33pm2.61dan 828 . . . 4 ((𝜑𝑛 ∈ ℕ) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
35 ovnsubadd2lem.c . . . 4 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
3634, 35fmptd 6292 . . 3 (𝜑𝐶:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
371, 36ovnsubadd 39462 . 2 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
38 eldifi 3694 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ {1, 2}) → 𝑛 ∈ ℕ)
3938adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑛 ∈ ℕ)
40 eldifn 3695 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 ∈ {1, 2})
41 vex 3176 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
4241a1i 11 . . . . . . . . . . . . . . . 16 𝑛 ∈ {1, 2} → 𝑛 ∈ V)
43 id 22 . . . . . . . . . . . . . . . 16 𝑛 ∈ {1, 2} → ¬ 𝑛 ∈ {1, 2})
4442, 43nelpr1 38289 . . . . . . . . . . . . . . 15 𝑛 ∈ {1, 2} → 𝑛 ≠ 1)
4544neneqd 2787 . . . . . . . . . . . . . 14 𝑛 ∈ {1, 2} → ¬ 𝑛 = 1)
4640, 45syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 1)
4742, 43nelpr2 38288 . . . . . . . . . . . . . . 15 𝑛 ∈ {1, 2} → 𝑛 ≠ 2)
4847neneqd 2787 . . . . . . . . . . . . . 14 𝑛 ∈ {1, 2} → ¬ 𝑛 = 2)
4940, 48syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 2)
5046, 49, 28syl2anc 691 . . . . . . . . . . . 12 (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
51 0ex 4718 . . . . . . . . . . . . 13 ∅ ∈ V
5251a1i 11 . . . . . . . . . . . 12 (𝑛 ∈ (ℕ ∖ {1, 2}) → ∅ ∈ V)
5350, 52eqeltrd 2688 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
5453adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
5535fvmpt2 6200 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V) → (𝐶𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
5639, 54, 55syl2anc 691 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
5750adantl 481 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
5856, 57eqtrd 2644 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶𝑛) = ∅)
5958ralrimiva 2949 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶𝑛) = ∅)
60 nfcv 2751 . . . . . . . 8 𝑛(ℕ ∖ {1, 2})
6160iunxdif3 4542 . . . . . . 7 (∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶𝑛) = ∅ → 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ ℕ (𝐶𝑛))
6259, 61syl 17 . . . . . 6 (𝜑 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ ℕ (𝐶𝑛))
6362eqcomd 2616 . . . . 5 (𝜑 𝑛 ∈ ℕ (𝐶𝑛) = 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛))
64 1nn 10908 . . . . . . . . . 10 1 ∈ ℕ
65 2nn 11062 . . . . . . . . . 10 2 ∈ ℕ
6664, 65pm3.2i 470 . . . . . . . . 9 (1 ∈ ℕ ∧ 2 ∈ ℕ)
67 prssi 4293 . . . . . . . . 9 ((1 ∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆ ℕ)
6866, 67ax-mp 5 . . . . . . . 8 {1, 2} ⊆ ℕ
69 dfss4 3820 . . . . . . . 8 ({1, 2} ⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2})
7068, 69mpbi 219 . . . . . . 7 (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2}
71 iuneq1 4470 . . . . . . 7 ((ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2} → 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ {1, 2} (𝐶𝑛))
7270, 71ax-mp 5 . . . . . 6 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ {1, 2} (𝐶𝑛)
7372a1i 11 . . . . 5 (𝜑 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ {1, 2} (𝐶𝑛))
74 fveq2 6103 . . . . . . . . 9 (𝑛 = 1 → (𝐶𝑛) = (𝐶‘1))
75 fveq2 6103 . . . . . . . . 9 (𝑛 = 2 → (𝐶𝑛) = (𝐶‘2))
7674, 75iunxprg 4543 . . . . . . . 8 ((1 ∈ ℕ ∧ 2 ∈ ℕ) → 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
7764, 65, 76mp2an 704 . . . . . . 7 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2))
7877a1i 11 . . . . . 6 (𝜑 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
7935a1i 11 . . . . . . . 8 (𝜑𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅))))
8064a1i 11 . . . . . . . 8 (𝜑 → 1 ∈ ℕ)
8179, 3, 80, 7fvmptd 6197 . . . . . . 7 (𝜑 → (𝐶‘1) = 𝐴)
82 id 22 . . . . . . . . . . . . 13 (𝑛 = 2 → 𝑛 = 2)
83 1ne2 11117 . . . . . . . . . . . . . . 15 1 ≠ 2
8483necomi 2836 . . . . . . . . . . . . . 14 2 ≠ 1
8584a1i 11 . . . . . . . . . . . . 13 (𝑛 = 2 → 2 ≠ 1)
8682, 85eqnetrd 2849 . . . . . . . . . . . 12 (𝑛 = 2 → 𝑛 ≠ 1)
8786neneqd 2787 . . . . . . . . . . 11 (𝑛 = 2 → ¬ 𝑛 = 1)
8887iffalsed 4047 . . . . . . . . . 10 (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
89 iftrue 4042 . . . . . . . . . 10 (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
9088, 89eqtrd 2644 . . . . . . . . 9 (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
9190adantl 481 . . . . . . . 8 ((𝜑𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
9265a1i 11 . . . . . . . 8 (𝜑 → 2 ∈ ℕ)
9379, 91, 92, 19fvmptd 6197 . . . . . . 7 (𝜑 → (𝐶‘2) = 𝐵)
9481, 93uneq12d 3730 . . . . . 6 (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴𝐵))
95 eqidd 2611 . . . . . 6 (𝜑 → (𝐴𝐵) = (𝐴𝐵))
9678, 94, 953eqtrd 2648 . . . . 5 (𝜑 𝑛 ∈ {1, 2} (𝐶𝑛) = (𝐴𝐵))
9763, 73, 963eqtrd 2648 . . . 4 (𝜑 𝑛 ∈ ℕ (𝐶𝑛) = (𝐴𝐵))
9897fveq2d 6107 . . 3 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) = ((voln*‘𝑋)‘(𝐴𝐵)))
99 nfv 1830 . . . . . 6 𝑛𝜑
100 nnex 10903 . . . . . . 7 ℕ ∈ V
101100a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
10268a1i 11 . . . . . 6 (𝜑 → {1, 2} ⊆ ℕ)
1031adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin)
104 simpl 472 . . . . . . . 8 ((𝜑𝑛 ∈ {1, 2}) → 𝜑)
105102sselda 3568 . . . . . . . 8 ((𝜑𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ)
10636ffvelrnda 6267 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
107 elpwi 4117 . . . . . . . . 9 ((𝐶𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝐶𝑛) ⊆ (ℝ ↑𝑚 𝑋))
108106, 107syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) ⊆ (ℝ ↑𝑚 𝑋))
109104, 105, 108syl2anc 691 . . . . . . 7 ((𝜑𝑛 ∈ {1, 2}) → (𝐶𝑛) ⊆ (ℝ ↑𝑚 𝑋))
110103, 109ovncl 39457 . . . . . 6 ((𝜑𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶𝑛)) ∈ (0[,]+∞))
11158fveq2d 6107 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘∅))
1121adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑋 ∈ Fin)
113112ovn0 39456 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘∅) = 0)
114111, 113eqtrd 2644 . . . . . 6 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶𝑛)) = 0)
11599, 101, 102, 110, 114sge0ss 39305 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
116115eqcomd 2616 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
11781, 6eqsstrd 3602 . . . . . 6 (𝜑 → (𝐶‘1) ⊆ (ℝ ↑𝑚 𝑋))
1181, 117ovncl 39457 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈ (0[,]+∞))
11993, 18eqsstrd 3602 . . . . . 6 (𝜑 → (𝐶‘2) ⊆ (ℝ ↑𝑚 𝑋))
1201, 119ovncl 39457 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈ (0[,]+∞))
12174fveq2d 6107 . . . . 5 (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘(𝐶‘1)))
12275fveq2d 6107 . . . . 5 (𝑛 = 2 → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘(𝐶‘2)))
12383a1i 11 . . . . 5 (𝜑 → 1 ≠ 2)
12480, 92, 118, 120, 121, 122, 123sge0pr 39287 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))))
12581fveq2d 6107 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴))
12693fveq2d 6107 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵))
127125, 126oveq12d 6567 . . . 4 (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
128116, 124, 1273eqtrd 2648 . . 3 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
12998, 128breq12d 4596 . 2 (𝜑 → (((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) ↔ ((voln*‘𝑋)‘(𝐴𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))))
13037, 129mpbid 221 1 (𝜑 → ((voln*‘𝑋)‘(𝐴𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {cpr 4127  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841  ℝcr 9814  0cc0 9815  1c1 9816   ≤ cle 9954  ℕcn 10897  2c2 10947   +𝑒 cxad 11820  Σ^csumge0 39255  voln*covoln 39426 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-cc 9140  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  ax-addf 9894  ax-mulf 9895 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-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  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-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  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-rlim 14068  df-sum 14265  df-prod 14475  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-0g 15925  df-topgen 15927  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256  df-ovoln 39427 This theorem is referenced by:  ovnsubadd2  39536
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