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Theorem ovnsubadd2lem 41341
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 4285 . . . . . . . 8 (𝑛 = 1 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
32adantl 469 . . . . . . 7 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴)
4 ovexd 6908 . . . . . . . . . 10 (𝜑 → (ℝ ↑𝑚 𝑋) ∈ V)
5 ovnsubadd2lem.a . . . . . . . . . 10 (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))
64, 5ssexd 5000 . . . . . . . . 9 (𝜑𝐴 ∈ V)
76, 5elpwd 4360 . . . . . . . 8 (𝜑𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
87adantr 468 . . . . . . 7 ((𝜑𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
93, 8eqeltrd 2885 . . . . . 6 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
109adantlr 697 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
11 simpl 470 . . . . . . . . . . 11 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1)
1211iffalsed 4290 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
13 simpr 473 . . . . . . . . . . 11 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2)
1413iftrued 4287 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
1512, 14eqtrd 2840 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
1615adantll 696 . . . . . . . 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 5000 . . . . . . . . . 10 (𝜑𝐵 ∈ V)
1918, 17elpwd 4360 . . . . . . . . 9 (𝜑𝐵 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
2019ad2antrr 708 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
2116, 20eqeltrd 2885 . . . . . . 7 (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
2221adantllr 701 . . . . . 6 ((((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
23 simpl 470 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1)
2423iffalsed 4290 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
25 simpr 473 . . . . . . . . . 10 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2)
2625iffalsed 4290 . . . . . . . . 9 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅)
2724, 26eqtrd 2840 . . . . . . . 8 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
28 0elpw 5026 . . . . . . . . 9 ∅ ∈ 𝒫 (ℝ ↑𝑚 𝑋)
2928a1i 11 . . . . . . . 8 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3027, 29eqeltrd 2885 . . . . . . 7 ((¬ 𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3130adantll 696 . . . . . 6 ((((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3222, 31pm2.61dan 838 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
3310, 32pm2.61dan 838 . . . 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 6606 . . 3 (𝜑𝐶:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
361, 35ovnsubadd 41268 . 2 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
37 eldifi 3931 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ {1, 2}) → 𝑛 ∈ ℕ)
3837adantl 469 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑛 ∈ ℕ)
39 eldifn 3932 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 ∈ {1, 2})
40 vex 3394 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
4140a1i 11 . . . . . . . . . . . . . . . 16 𝑛 ∈ {1, 2} → 𝑛 ∈ V)
42 id 22 . . . . . . . . . . . . . . . 16 𝑛 ∈ {1, 2} → ¬ 𝑛 ∈ {1, 2})
4341, 42nelpr1 39755 . . . . . . . . . . . . . . 15 𝑛 ∈ {1, 2} → 𝑛 ≠ 1)
4443neneqd 2983 . . . . . . . . . . . . . 14 𝑛 ∈ {1, 2} → ¬ 𝑛 = 1)
4539, 44syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 1)
4641, 42nelpr2 39754 . . . . . . . . . . . . . . 15 𝑛 ∈ {1, 2} → 𝑛 ≠ 2)
4746neneqd 2983 . . . . . . . . . . . . . 14 𝑛 ∈ {1, 2} → ¬ 𝑛 = 2)
4839, 47syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ {1, 2}) → ¬ 𝑛 = 2)
4945, 48, 27syl2anc 575 . . . . . . . . . . . 12 (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
50 0ex 4984 . . . . . . . . . . . . 13 ∅ ∈ V
5150a1i 11 . . . . . . . . . . . 12 (𝑛 ∈ (ℕ ∖ {1, 2}) → ∅ ∈ V)
5249, 51eqeltrd 2885 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ {1, 2}) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
5352adantl 469 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V)
5434fvmpt2 6512 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V) → (𝐶𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
5538, 53, 54syl2anc 575 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))
5649adantl 469 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅)
5755, 56eqtrd 2840 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → (𝐶𝑛) = ∅)
5857ralrimiva 3154 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶𝑛) = ∅)
59 nfcv 2948 . . . . . . . 8 𝑛(ℕ ∖ {1, 2})
6059iunxdif3 4798 . . . . . . 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 2812 . . . . 5 (𝜑 𝑛 ∈ ℕ (𝐶𝑛) = 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1, 2}))(𝐶𝑛))
63 1nn 11316 . . . . . . . . . 10 1 ∈ ℕ
64 2nn 11462 . . . . . . . . . 10 2 ∈ ℕ
6563, 64pm3.2i 458 . . . . . . . . 9 (1 ∈ ℕ ∧ 2 ∈ ℕ)
66 prssi 4542 . . . . . . . . 9 ((1 ∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆ ℕ)
6765, 66ax-mp 5 . . . . . . . 8 {1, 2} ⊆ ℕ
68 dfss4 4060 . . . . . . . 8 ({1, 2} ⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2})
6967, 68mpbi 221 . . . . . . 7 (ℕ ∖ (ℕ ∖ {1, 2})) = {1, 2}
70 iuneq1 4726 . . . . . . 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 6408 . . . . . . . . 9 (𝑛 = 1 → (𝐶𝑛) = (𝐶‘1))
74 fveq2 6408 . . . . . . . . 9 (𝑛 = 2 → (𝐶𝑛) = (𝐶‘2))
7573, 74iunxprg 4799 . . . . . . . 8 ((1 ∈ ℕ ∧ 2 ∈ ℕ) → 𝑛 ∈ {1, 2} (𝐶𝑛) = ((𝐶‘1) ∪ (𝐶‘2)))
7663, 64, 75mp2an 675 . . . . . . 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 6509 . . . . . . 7 (𝜑 → (𝐶‘1) = 𝐴)
81 id 22 . . . . . . . . . . . . 13 (𝑛 = 2 → 𝑛 = 2)
82 1ne2 11507 . . . . . . . . . . . . . . 15 1 ≠ 2
8382necomi 3032 . . . . . . . . . . . . . 14 2 ≠ 1
8483a1i 11 . . . . . . . . . . . . 13 (𝑛 = 2 → 2 ≠ 1)
8581, 84eqnetrd 3045 . . . . . . . . . . . 12 (𝑛 = 2 → 𝑛 ≠ 1)
8685neneqd 2983 . . . . . . . . . . 11 (𝑛 = 2 → ¬ 𝑛 = 1)
8786iffalsed 4290 . . . . . . . . . 10 (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅))
88 iftrue 4285 . . . . . . . . . 10 (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵)
8987, 88eqtrd 2840 . . . . . . . . 9 (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
9089adantl 469 . . . . . . . 8 ((𝜑𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵)
9164a1i 11 . . . . . . . 8 (𝜑 → 2 ∈ ℕ)
9278, 90, 91, 18fvmptd 6509 . . . . . . 7 (𝜑 → (𝐶‘2) = 𝐵)
9380, 92uneq12d 3967 . . . . . 6 (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴𝐵))
94 eqidd 2807 . . . . . 6 (𝜑 → (𝐴𝐵) = (𝐴𝐵))
9577, 93, 943eqtrd 2844 . . . . 5 (𝜑 𝑛 ∈ {1, 2} (𝐶𝑛) = (𝐴𝐵))
9662, 72, 953eqtrd 2844 . . . 4 (𝜑 𝑛 ∈ ℕ (𝐶𝑛) = (𝐴𝐵))
9796fveq2d 6412 . . 3 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐶𝑛)) = ((voln*‘𝑋)‘(𝐴𝐵)))
98 nfv 2005 . . . . . 6 𝑛𝜑
99 nnex 11311 . . . . . . 7 ℕ ∈ V
10099a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
10167a1i 11 . . . . . 6 (𝜑 → {1, 2} ⊆ ℕ)
1021adantr 468 . . . . . . 7 ((𝜑𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin)
103 simpl 470 . . . . . . . 8 ((𝜑𝑛 ∈ {1, 2}) → 𝜑)
104101sselda 3798 . . . . . . . 8 ((𝜑𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ)
10535ffvelrnda 6581 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
106 elpwi 4361 . . . . . . . . 9 ((𝐶𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝐶𝑛) ⊆ (ℝ ↑𝑚 𝑋))
107105, 106syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) ⊆ (ℝ ↑𝑚 𝑋))
108103, 104, 107syl2anc 575 . . . . . . 7 ((𝜑𝑛 ∈ {1, 2}) → (𝐶𝑛) ⊆ (ℝ ↑𝑚 𝑋))
109102, 108ovncl 41263 . . . . . 6 ((𝜑𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶𝑛)) ∈ (0[,]+∞))
11057fveq2d 6412 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘∅))
1111adantr 468 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → 𝑋 ∈ Fin)
112111ovn0 41262 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘∅) = 0)
113110, 112eqtrd 2840 . . . . . 6 ((𝜑𝑛 ∈ (ℕ ∖ {1, 2})) → ((voln*‘𝑋)‘(𝐶𝑛)) = 0)
11498, 100, 101, 109, 113sge0ss 41108 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
115114eqcomd 2812 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))))
11680, 5eqsstrd 3836 . . . . . 6 (𝜑 → (𝐶‘1) ⊆ (ℝ ↑𝑚 𝑋))
1171, 116ovncl 41263 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈ (0[,]+∞))
11892, 17eqsstrd 3836 . . . . . 6 (𝜑 → (𝐶‘2) ⊆ (ℝ ↑𝑚 𝑋))
1191, 118ovncl 41263 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈ (0[,]+∞))
120 2fveq3 6413 . . . . 5 (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘(𝐶‘1)))
121 2fveq3 6413 . . . . 5 (𝑛 = 2 → ((voln*‘𝑋)‘(𝐶𝑛)) = ((voln*‘𝑋)‘(𝐶‘2)))
12282a1i 11 . . . . 5 (𝜑 → 1 ≠ 2)
12379, 91, 117, 119, 120, 121, 122sge0pr 41090 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))))
12480fveq2d 6412 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴))
12592fveq2d 6412 . . . . 5 (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵))
126124, 125oveq12d 6892 . . . 4 (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒 ((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
127115, 123, 1263eqtrd 2844 . . 3 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
12897, 127breq12d 4857 . 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 223 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 384   = wceq 1637  wcel 2156  wne 2978  wral 3096  Vcvv 3391  cdif 3766  cun 3767  wss 3769  c0 4116  ifcif 4279  𝒫 cpw 4351  {cpr 4372   ciun 4712   class class class wbr 4844  cmpt 4923  cfv 6101  (class class class)co 6874  𝑚 cmap 8092  Fincfn 8192  cr 10220  0cc0 10221  1c1 10222  cle 10360  cn 11305  2c2 11356   +𝑒 cxad 12160  Σ^csumge0 41058  voln*covoln 41232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cc 9542  ax-ac2 9570  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299  ax-addf 10300  ax-mulf 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-disj 4813  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-of 7127  df-om 7296  df-1st 7398  df-2nd 7399  df-tpos 7587  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-map 8094  df-pm 8095  df-ixp 8146  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-fi 8556  df-sup 8587  df-inf 8588  df-oi 8654  df-card 9048  df-acn 9051  df-ac 9222  df-cda 9275  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-3 11365  df-4 11366  df-5 11367  df-6 11368  df-7 11369  df-8 11370  df-9 11371  df-n0 11560  df-z 11644  df-dec 11760  df-uz 11905  df-q 12008  df-rp 12047  df-xneg 12162  df-xadd 12163  df-xmul 12164  df-ioo 12397  df-ico 12399  df-icc 12400  df-fz 12550  df-fzo 12690  df-fl 12817  df-seq 13025  df-exp 13084  df-hash 13338  df-cj 14062  df-re 14063  df-im 14064  df-sqrt 14198  df-abs 14199  df-clim 14442  df-rlim 14443  df-sum 14640  df-prod 14857  df-struct 16070  df-ndx 16071  df-slot 16072  df-base 16074  df-sets 16075  df-ress 16076  df-plusg 16166  df-mulr 16167  df-starv 16168  df-tset 16172  df-ple 16173  df-ds 16175  df-unif 16176  df-rest 16288  df-0g 16307  df-topgen 16309  df-mgm 17447  df-sgrp 17489  df-mnd 17500  df-grp 17630  df-minusg 17631  df-subg 17793  df-cmn 18396  df-abl 18397  df-mgp 18692  df-ur 18704  df-ring 18751  df-cring 18752  df-oppr 18825  df-dvdsr 18843  df-unit 18844  df-invr 18874  df-dvr 18885  df-drng 18953  df-psmet 19946  df-xmet 19947  df-met 19948  df-bl 19949  df-mopn 19950  df-cnfld 19955  df-top 20912  df-topon 20929  df-bases 20964  df-cmp 21404  df-ovol 23445  df-vol 23446  df-sumge0 41059  df-ovoln 41233
This theorem is referenced by:  ovnsubadd2  41342
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