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.x (𝜑𝑋 ∈ Fin)
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

Proof of Theorem ovnsubadd2lem
StepHypRef Expression
1 ovnsubadd2lem.x . . 3 (𝜑𝑋 ∈ Fin)
2 iftrue 4125 . . . . . . . 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 ovexd 6720 . . . . . . . . . 10 (𝜑 → (ℝ ↑𝑚 𝑋) ∈ V)
5 ovnsubadd2lem.a . . . . . . . . . 10 (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))
64, 5ssexd 4838 . . . . . . . . 9 (𝜑𝐴 ∈ V)
76, 5elpwd 4200 . . . . . . . 8 (𝜑𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
87adantr 480 . . . . . . 7 ((𝜑𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
93, 8eqeltrd 2730 . . . . . 6 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
109adantlr 751 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
11 simpl 472 . . . . . . . . . . 11 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1)
1211iffalsed 4130 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
13 simpr 476 . . . . . . . . . . 11 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2)
1413iftrued 4127 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
1512, 14eqtrd 2685 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
1615adantll 750 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
17 ovnsubadd2lem.b . . . . . . . . . . 11 (𝜑𝐵 ⊆ (ℝ ↑𝑚 𝑋))
184, 17ssexd 4838 . . . . . . . . . 10 (𝜑𝐵 ∈ V)
1918, 17elpwd 4200 . . . . . . . . 9 (𝜑𝐵 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
2019ad2antrr 762 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
2116, 20eqeltrd 2730 . . . . . . 7 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
2221adantllr 755 . . . . . 6 ((((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
23 simpl 472 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1)
2423iffalsed 4130 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
25 simpr 476 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
2625iffalsed 4130 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅)
2724, 26eqtrd 2685 . . . . . . . 8 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
28 0elpw 4864 . . . . . . . . 9 ∅ ∈ 𝒫 (ℝ ↑𝑚 𝑋)
2928a1i 11 . . . . . . . 8 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3027, 29eqeltrd 2730 . . . . . . 7 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3130adantll 750 . . . . . 6 ((((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3222, 31pm2.61dan 849 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3310, 32pm2.61dan 849 . . . 4 ((𝜑𝑛 ∈ ℕ) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
34 ovnsubadd2lem.c . . . 4 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
3533, 34fmptd 6425 . . 3 (𝜑𝐶:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
361, 35ovnsubadd 41107 . 2 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
37 eldifi 3765 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ {1, 2}) → 𝑛 ∈ ℕ)
3837adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑛 ∈ ℕ)
39 eldifn 3766 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 ∈ {1, 2})
40 vex 3234 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
4140a1i 11 . . . . . . . . . . . . . . . 16 𝑛 ∈ {1, 2} → 𝑛 ∈ V)
42 id 22 . . . . . . . . . . . . . . . 16 𝑛 ∈ {1, 2} → ¬ 𝑛 ∈ {1, 2})
4341, 42nelpr1 39576 . . . . . . . . . . . . . . 15 𝑛 ∈ {1, 2} → 𝑛 ≠ 1)
4443neneqd 2828 . . . . . . . . . . . . . 14 𝑛 ∈ {1, 2} → ¬ 𝑛 = 1)
4539, 44syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 1)
4641, 42nelpr2 39575 . . . . . . . . . . . . . . 15 𝑛 ∈ {1, 2} → 𝑛 ≠ 2)
4746neneqd 2828 . . . . . . . . . . . . . 14 𝑛 ∈ {1, 2} → ¬ 𝑛 = 2)
4839, 47syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 2)
4945, 48, 27syl2anc 694 . . . . . . . . . . . 12 (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
50 0ex 4823 . . . . . . . . . . . . 13 ∅ ∈ V
5150a1i 11 . . . . . . . . . . . 12 (𝑛 ∈ (ℕ ∖ {1, 2}) → ∅ ∈ V)
5249, 51eqeltrd 2730 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
5352adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
5434fvmpt2 6330 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V) → (𝐶𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
5538, 53, 54syl2anc 694 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
5649adantl 481 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
5755, 56eqtrd 2685 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶𝑛) = ∅)
5857ralrimiva 2995 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶𝑛) = ∅)
59 nfcv 2793 . . . . . . . 8 𝑛(ℕ ∖ {1, 2})
6059iunxdif3 4638 . . . . . . 7 (∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶𝑛) = ∅ → 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ ℕ (𝐶𝑛))
6158, 60syl 17 . . . . . 6 (𝜑 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ ℕ (𝐶𝑛))
6261eqcomd 2657 . . . . 5 (𝜑 𝑛 ∈ ℕ (𝐶𝑛) = 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛))
63 1nn 11069 . . . . . . . . . 10 1 ∈ ℕ
64 2nn 11223 . . . . . . . . . 10 2 ∈ ℕ
6563, 64pm3.2i 470 . . . . . . . . 9 (1 ∈ ℕ ∧ 2 ∈ ℕ)
66 prssi 4385 . . . . . . . . 9 ((1 ∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆ ℕ)
6765, 66ax-mp 5 . . . . . . . 8 {1, 2} ⊆ ℕ
68 dfss4 3891 . . . . . . . 8 ({1, 2} ⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2})
6967, 68mpbi 220 . . . . . . 7 (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2}
70 iuneq1 4566 . . . . . . 7 ((ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2} → 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ {1, 2} (𝐶𝑛))
7169, 70ax-mp 5 . . . . . 6 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ {1, 2} (𝐶𝑛)
7271a1i 11 . . . . 5 (𝜑 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛) = 𝑛 ∈ {1, 2} (𝐶𝑛))
73 fveq2 6229 . . . . . . . . 9 (𝑛 = 1 → (𝐶𝑛) = (𝐶‘1))
74 fveq2 6229 . . . . . . . . 9 (𝑛 = 2 → (𝐶𝑛) = (𝐶‘2))
7573, 74iunxprg 4639 . . . . . . . 8 ((1 ∈ ℕ ∧ 2 ∈ ℕ) → 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
7663, 64, 75mp2an 708 . . . . . . 7 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2))
7776a1i 11 . . . . . 6 (𝜑 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
7834a1i 11 . . . . . . . 8 (𝜑𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅))))
7963a1i 11 . . . . . . . 8 (𝜑 → 1 ∈ ℕ)
8078, 3, 79, 6fvmptd 6327 . . . . . . 7 (𝜑 → (𝐶‘1) = 𝐴)
81 id 22 . . . . . . . . . . . . 13 (𝑛 = 2 → 𝑛 = 2)
82 1ne2 11278 . . . . . . . . . . . . . . 15 1 ≠ 2
8382necomi 2877 . . . . . . . . . . . . . 14 2 ≠ 1
8483a1i 11 . . . . . . . . . . . . 13 (𝑛 = 2 → 2 ≠ 1)
8581, 84eqnetrd 2890 . . . . . . . . . . . 12 (𝑛 = 2 → 𝑛 ≠ 1)
8685neneqd 2828 . . . . . . . . . . 11 (𝑛 = 2 → ¬ 𝑛 = 1)
8786iffalsed 4130 . . . . . . . . . 10 (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
88 iftrue 4125 . . . . . . . . . 10 (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
8987, 88eqtrd 2685 . . . . . . . . 9 (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
9089adantl 481 . . . . . . . 8 ((𝜑𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
9164a1i 11 . . . . . . . 8 (𝜑 → 2 ∈ ℕ)
9278, 90, 91, 18fvmptd 6327 . . . . . . 7 (𝜑 → (𝐶‘2) = 𝐵)
9380, 92uneq12d 3801 . . . . . 6 (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴𝐵))
94 eqidd 2652 . . . . . 6 (𝜑 → (𝐴𝐵) = (𝐴𝐵))
9577, 93, 943eqtrd 2689 . . . . 5 (𝜑 𝑛 ∈ {1, 2} (𝐶𝑛) = (𝐴𝐵))
9662, 72, 953eqtrd 2689 . . . 4 (𝜑 𝑛 ∈ ℕ (𝐶𝑛) = (𝐴𝐵))
9796fveq2d 6233 . . 3 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) = ((voln*‘𝑋)‘(𝐴𝐵)))
98 nfv 1883 . . . . . 6 𝑛𝜑
99 nnex 11064 . . . . . . 7 ℕ ∈ V
10099a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
10167a1i 11 . . . . . 6 (𝜑 → {1, 2} ⊆ ℕ)
1021adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin)
103 simpl 472 . . . . . . . 8 ((𝜑𝑛 ∈ {1, 2}) → 𝜑)
104101sselda 3636 . . . . . . . 8 ((𝜑𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ)
10535ffvelrnda 6399 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
106 elpwi 4201 . . . . . . . . 9 ((𝐶𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝐶𝑛) ⊆ (ℝ ↑𝑚 𝑋))
107105, 106syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) ⊆ (ℝ ↑𝑚 𝑋))
108103, 104, 107syl2anc 694 . . . . . . 7 ((𝜑𝑛 ∈ {1, 2}) → (𝐶𝑛) ⊆ (ℝ ↑𝑚 𝑋))
109102, 108ovncl 41102 . . . . . 6 ((𝜑𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶𝑛)) ∈ (0[,]+∞))
11057fveq2d 6233 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘∅))
1111adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑋 ∈ Fin)
112111ovn0 41101 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘∅) = 0)
113110, 112eqtrd 2685 . . . . . 6 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶𝑛)) = 0)
11498, 100, 101, 109, 113sge0ss 40947 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
115114eqcomd 2657 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
11680, 5eqsstrd 3672 . . . . . 6 (𝜑 → (𝐶‘1) ⊆ (ℝ ↑𝑚 𝑋))
1171, 116ovncl 41102 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈ (0[,]+∞))
11892, 17eqsstrd 3672 . . . . . 6 (𝜑 → (𝐶‘2) ⊆ (ℝ ↑𝑚 𝑋))
1191, 118ovncl 41102 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈ (0[,]+∞))
12073fveq2d 6233 . . . . 5 (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘(𝐶‘1)))
12174fveq2d 6233 . . . . 5 (𝑛 = 2 → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘(𝐶‘2)))
12282a1i 11 . . . . 5 (𝜑 → 1 ≠ 2)
12379, 91, 117, 119, 120, 121, 122sge0pr 40929 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))))
12480fveq2d 6233 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴))
12592fveq2d 6233 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵))
126124, 125oveq12d 6708 . . . 4 (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
127115, 123, 1263eqtrd 2689 . . 3 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
12897, 127breq12d 4698 . 2 (𝜑 → (((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) ↔ ((voln*‘𝑋)‘(𝐴𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))))
12936, 128mpbid 222 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 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ⊆ wss 3607  ∅c0 3948  ifcif 4119  𝒫 cpw 4191  {cpr 4212  ∪ ciun 4552   class class class wbr 4685   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  Fincfn 7997  ℝcr 9973  0cc0 9974  1c1 9975   ≤ cle 10113  ℕcn 11058  2c2 11108   +𝑒 cxad 11982  Σ^csumge0 40897  voln*covoln 41071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-ac2 9323  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-prod 14680  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-rest 16130  df-0g 16149  df-topgen 16151  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-subg 17638  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-cnfld 19795  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-sumge0 40898  df-ovoln 41072 This theorem is referenced by:  ovnsubadd2  41181
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