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 4475 . . . . . . . 8 (𝑛 = 1 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
32adantl 482 . . . . . . 7 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
4 ovexd 7186 . . . . . . . . . 10 (𝜑 → (ℝ ↑m 𝑋) ∈ V)
5 ovnsubadd2lem.a . . . . . . . . . 10 (𝜑𝐴 ⊆ (ℝ ↑m 𝑋))
64, 5ssexd 5224 . . . . . . . . 9 (𝜑𝐴 ∈ V)
76, 5elpwd 4552 . . . . . . . 8 (𝜑𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
87adantr 481 . . . . . . 7 ((𝜑𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
93, 8eqeltrd 2917 . . . . . 6 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
109adantlr 711 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
11 simpl 483 . . . . . . . . . . 11 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1)
1211iffalsed 4480 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
13 simpr 485 . . . . . . . . . . 11 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2)
1413iftrued 4477 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
1512, 14eqtrd 2860 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
1615adantll 710 . . . . . . . 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 5224 . . . . . . . . . 10 (𝜑𝐵 ∈ V)
1918, 17elpwd 4552 . . . . . . . . 9 (𝜑𝐵 ∈ 𝒫 (ℝ ↑m 𝑋))
2019ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ ↑m 𝑋))
2116, 20eqeltrd 2917 . . . . . . 7 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
2221adantllr 715 . . . . . 6 ((((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
23 simpl 483 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1)
2423iffalsed 4480 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
25 simpr 485 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
2625iffalsed 4480 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅)
2724, 26eqtrd 2860 . . . . . . . 8 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
28 0elpw 5252 . . . . . . . . 9 ∅ ∈ 𝒫 (ℝ ↑m 𝑋)
2928a1i 11 . . . . . . . 8 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈ 𝒫 (ℝ ↑m 𝑋))
3027, 29eqeltrd 2917 . . . . . . 7 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
3130adantll 710 . . . . . 6 ((((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
3222, 31pm2.61dan 809 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑m 𝑋))
3310, 32pm2.61dan 809 . . . 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 6873 . . 3 (𝜑𝐶:ℕ⟶𝒫 (ℝ ↑m 𝑋))
361, 35ovnsubadd 42716 . 2 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
37 eldifi 4106 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ {1, 2}) → 𝑛 ∈ ℕ)
3837adantl 482 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑛 ∈ ℕ)
39 eldifn 4107 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 ∈ {1, 2})
40 vex 3502 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
4140a1i 11 . . . . . . . . . . . . . . . 16 𝑛 ∈ {1, 2} → 𝑛 ∈ V)
42 id 22 . . . . . . . . . . . . . . . 16 𝑛 ∈ {1, 2} → ¬ 𝑛 ∈ {1, 2})
4341, 42nelpr1 4589 . . . . . . . . . . . . . . 15 𝑛 ∈ {1, 2} → 𝑛 ≠ 1)
4443neneqd 3025 . . . . . . . . . . . . . 14 𝑛 ∈ {1, 2} → ¬ 𝑛 = 1)
4539, 44syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 1)
4641, 42nelpr2 4588 . . . . . . . . . . . . . . 15 𝑛 ∈ {1, 2} → 𝑛 ≠ 2)
4746neneqd 3025 . . . . . . . . . . . . . 14 𝑛 ∈ {1, 2} → ¬ 𝑛 = 2)
4839, 47syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 2)
4945, 48, 27syl2anc 584 . . . . . . . . . . . 12 (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
50 0ex 5207 . . . . . . . . . . . . 13 ∅ ∈ V
5150a1i 11 . . . . . . . . . . . 12 (𝑛 ∈ (ℕ ∖ {1, 2}) → ∅ ∈ V)
5249, 51eqeltrd 2917 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
5352adantl 482 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
5434fvmpt2 6774 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V) → (𝐶𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
5538, 53, 54syl2anc 584 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
5649adantl 482 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
5755, 56eqtrd 2860 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶𝑛) = ∅)
5857ralrimiva 3186 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶𝑛) = ∅)
59 nfcv 2981 . . . . . . . 8 𝑛(ℕ ∖ {1, 2})
6059iunxdif3 5013 . . . . . . 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 2831 . . . . 5 (𝜑 𝑛 ∈ ℕ (𝐶𝑛) = 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛))
63 1nn 11641 . . . . . . . . . 10 1 ∈ ℕ
64 2nn 11702 . . . . . . . . . 10 2 ∈ ℕ
6563, 64pm3.2i 471 . . . . . . . . 9 (1 ∈ ℕ ∧ 2 ∈ ℕ)
66 prssi 4752 . . . . . . . . 9 ((1 ∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆ ℕ)
6765, 66ax-mp 5 . . . . . . . 8 {1, 2} ⊆ ℕ
68 dfss4 4238 . . . . . . . 8 ({1, 2} ⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2})
6967, 68mpbi 231 . . . . . . 7 (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2}
70 iuneq1 4931 . . . . . . 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 6666 . . . . . . . . 9 (𝑛 = 1 → (𝐶𝑛) = (𝐶‘1))
74 fveq2 6666 . . . . . . . . 9 (𝑛 = 2 → (𝐶𝑛) = (𝐶‘2))
7573, 74iunxprg 5014 . . . . . . . 8 ((1 ∈ ℕ ∧ 2 ∈ ℕ) → 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
7663, 64, 75mp2an 688 . . . . . . 7 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2))
7776a1i 11 . . . . . 6 (𝜑 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
7863a1i 11 . . . . . . . 8 (𝜑 → 1 ∈ ℕ)
7934, 2, 78, 6fvmptd3 6786 . . . . . . 7 (𝜑 → (𝐶‘1) = 𝐴)
80 id 22 . . . . . . . . . . . 12 (𝑛 = 2 → 𝑛 = 2)
81 1ne2 11837 . . . . . . . . . . . . . 14 1 ≠ 2
8281necomi 3074 . . . . . . . . . . . . 13 2 ≠ 1
8382a1i 11 . . . . . . . . . . . 12 (𝑛 = 2 → 2 ≠ 1)
8480, 83eqnetrd 3087 . . . . . . . . . . 11 (𝑛 = 2 → 𝑛 ≠ 1)
8584neneqd 3025 . . . . . . . . . 10 (𝑛 = 2 → ¬ 𝑛 = 1)
8685iffalsed 4480 . . . . . . . . 9 (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
87 iftrue 4475 . . . . . . . . 9 (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
8886, 87eqtrd 2860 . . . . . . . 8 (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
8964a1i 11 . . . . . . . 8 (𝜑 → 2 ∈ ℕ)
9034, 88, 89, 18fvmptd3 6786 . . . . . . 7 (𝜑 → (𝐶‘2) = 𝐵)
9179, 90uneq12d 4143 . . . . . 6 (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴𝐵))
92 eqidd 2826 . . . . . 6 (𝜑 → (𝐴𝐵) = (𝐴𝐵))
9377, 91, 923eqtrd 2864 . . . . 5 (𝜑 𝑛 ∈ {1, 2} (𝐶𝑛) = (𝐴𝐵))
9462, 72, 933eqtrd 2864 . . . 4 (𝜑 𝑛 ∈ ℕ (𝐶𝑛) = (𝐴𝐵))
9594fveq2d 6670 . . 3 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) = ((voln*‘𝑋)‘(𝐴𝐵)))
96 nfv 1908 . . . . . 6 𝑛𝜑
97 nnex 11636 . . . . . . 7 ℕ ∈ V
9897a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
9967a1i 11 . . . . . 6 (𝜑 → {1, 2} ⊆ ℕ)
1001adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin)
101 simpl 483 . . . . . . . 8 ((𝜑𝑛 ∈ {1, 2}) → 𝜑)
10299sselda 3970 . . . . . . . 8 ((𝜑𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ)
10335ffvelrnda 6846 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) ∈ 𝒫 (ℝ ↑m 𝑋))
104 elpwi 4553 . . . . . . . . 9 ((𝐶𝑛) ∈ 𝒫 (ℝ ↑m 𝑋) → (𝐶𝑛) ⊆ (ℝ ↑m 𝑋))
105103, 104syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) ⊆ (ℝ ↑m 𝑋))
106101, 102, 105syl2anc 584 . . . . . . 7 ((𝜑𝑛 ∈ {1, 2}) → (𝐶𝑛) ⊆ (ℝ ↑m 𝑋))
107100, 106ovncl 42711 . . . . . 6 ((𝜑𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶𝑛)) ∈ (0[,]+∞))
10857fveq2d 6670 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘∅))
1091adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑋 ∈ Fin)
110109ovn0 42710 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘∅) = 0)
111108, 110eqtrd 2860 . . . . . 6 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶𝑛)) = 0)
11296, 98, 99, 107, 111sge0ss 42556 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
113112eqcomd 2831 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
11479, 5eqsstrd 4008 . . . . . 6 (𝜑 → (𝐶‘1) ⊆ (ℝ ↑m 𝑋))
1151, 114ovncl 42711 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈ (0[,]+∞))
11690, 17eqsstrd 4008 . . . . . 6 (𝜑 → (𝐶‘2) ⊆ (ℝ ↑m 𝑋))
1171, 116ovncl 42711 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈ (0[,]+∞))
118 2fveq3 6671 . . . . 5 (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘(𝐶‘1)))
119 2fveq3 6671 . . . . 5 (𝑛 = 2 → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘(𝐶‘2)))
12081a1i 11 . . . . 5 (𝜑 → 1 ≠ 2)
12178, 89, 115, 117, 118, 119, 120sge0pr 42538 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))))
12279fveq2d 6670 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴))
12390fveq2d 6670 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵))
124122, 123oveq12d 7169 . . . 4 (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
125113, 121, 1243eqtrd 2864 . . 3 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
12695, 125breq12d 5075 . 2 (𝜑 → (((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) ↔ ((voln*‘𝑋)‘(𝐴𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))))
12736, 126mpbid 233 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 396   = wceq 1530   ∈ wcel 2107   ≠ wne 3020  ∀wral 3142  Vcvv 3499   ∖ cdif 3936   ∪ cun 3937   ⊆ wss 3939  ∅c0 4294  ifcif 4469  𝒫 cpw 4541  {cpr 4565  ∪ ciun 4916   class class class wbr 5062   ↦ cmpt 5142  ‘cfv 6351  (class class class)co 7151   ↑m cmap 8399  Fincfn 8501  ℝcr 10528  0cc0 10529  1c1 10530   ≤ cle 10668  ℕcn 11630  2c2 11684   +𝑒 cxad 12498  Σ^csumge0 42506  voln*covoln 42680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cc 9849  ax-ac2 9877  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-disj 5028  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-acn 9363  df-ac 9534  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12383  df-xneg 12500  df-xadd 12501  df-xmul 12502  df-ioo 12735  df-ico 12737  df-icc 12738  df-fz 12886  df-fzo 13027  df-fl 13155  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-clim 14838  df-rlim 14839  df-sum 15036  df-prod 15252  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-rest 16688  df-0g 16707  df-topgen 16709  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-grp 18038  df-minusg 18039  df-subg 18208  df-cmn 18830  df-abl 18831  df-mgp 19162  df-ur 19174  df-ring 19221  df-cring 19222  df-oppr 19295  df-dvdsr 19313  df-unit 19314  df-invr 19344  df-dvr 19355  df-drng 19426  df-psmet 20453  df-xmet 20454  df-met 20455  df-bl 20456  df-mopn 20457  df-cnfld 20462  df-top 21418  df-topon 21435  df-bases 21470  df-cmp 21911  df-ovol 23980  df-vol 23981  df-sumge0 42507  df-ovoln 42681 This theorem is referenced by:  ovnsubadd2  42790
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