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Theorem volivth 23567
Description: The Intermediate Value Theorem for the Lebesgue volume function. For any positive 𝐵 ≤ (vol‘𝐴), there is a measurable subset of 𝐴 whose volume is 𝐵. (Contributed by Mario Carneiro, 30-Aug-2014.)
Assertion
Ref Expression
volivth ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem volivth
Dummy variables 𝑢 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 807 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ∈ dom vol)
2 mnfxr 10280 . . . . . 6 -∞ ∈ ℝ*
32a1i 11 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → -∞ ∈ ℝ*)
4 iccssxr 12441 . . . . . . 7 (0[,](vol‘𝐴)) ⊆ ℝ*
5 simpr 479 . . . . . . 7 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 𝐵 ∈ (0[,](vol‘𝐴)))
64, 5sseldi 3734 . . . . . 6 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 𝐵 ∈ ℝ*)
76adantr 472 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 𝐵 ∈ ℝ*)
8 iccssxr 12441 . . . . . . . 8 (0[,]+∞) ⊆ ℝ*
9 volf 23489 . . . . . . . . 9 vol:dom vol⟶(0[,]+∞)
109ffvelrni 6513 . . . . . . . 8 (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
118, 10sseldi 3734 . . . . . . 7 (𝐴 ∈ dom vol → (vol‘𝐴) ∈ ℝ*)
1211adantr 472 . . . . . 6 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (vol‘𝐴) ∈ ℝ*)
1312adantr 472 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → (vol‘𝐴) ∈ ℝ*)
14 0xr 10270 . . . . . . . . . 10 0 ∈ ℝ*
15 elicc1 12404 . . . . . . . . . 10 ((0 ∈ ℝ* ∧ (vol‘𝐴) ∈ ℝ*) → (𝐵 ∈ (0[,](vol‘𝐴)) ↔ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵𝐵 ≤ (vol‘𝐴))))
1614, 12, 15sylancr 698 . . . . . . . . 9 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (𝐵 ∈ (0[,](vol‘𝐴)) ↔ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵𝐵 ≤ (vol‘𝐴))))
175, 16mpbid 222 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵𝐵 ≤ (vol‘𝐴)))
1817simp2d 1137 . . . . . . 7 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 0 ≤ 𝐵)
1918adantr 472 . . . . . 6 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 0 ≤ 𝐵)
20 mnflt0 12144 . . . . . . . 8 -∞ < 0
21 xrltletr 12173 . . . . . . . 8 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*𝐵 ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ 𝐵) → -∞ < 𝐵))
2220, 21mpani 714 . . . . . . 7 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*𝐵 ∈ ℝ*) → (0 ≤ 𝐵 → -∞ < 𝐵))
232, 14, 22mp3an12 1555 . . . . . 6 (𝐵 ∈ ℝ* → (0 ≤ 𝐵 → -∞ < 𝐵))
247, 19, 23sylc 65 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → -∞ < 𝐵)
25 simpr 479 . . . . 5 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 𝐵 < (vol‘𝐴))
26 xrre2 12186 . . . . 5 (((-∞ ∈ ℝ*𝐵 ∈ ℝ* ∧ (vol‘𝐴) ∈ ℝ*) ∧ (-∞ < 𝐵𝐵 < (vol‘𝐴))) → 𝐵 ∈ ℝ)
273, 7, 13, 24, 25, 26syl32anc 1481 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 𝐵 ∈ ℝ)
28 volsup2 23565 . . . 4 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
291, 27, 25, 28syl3anc 1473 . . 3 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
30 nnre 11211 . . . . . . 7 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
3130ad2antrl 766 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝑛 ∈ ℝ)
3231renegcld 10641 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 ∈ ℝ)
3327adantr 472 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 ∈ ℝ)
34 0red 10225 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 0 ∈ ℝ)
35 nngt0 11233 . . . . . . . 8 (𝑛 ∈ ℕ → 0 < 𝑛)
3635ad2antrl 766 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 0 < 𝑛)
3731lt0neg2d 10782 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (0 < 𝑛 ↔ -𝑛 < 0))
3836, 37mpbid 222 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 < 0)
3932, 34, 31, 38, 36lttrd 10382 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 < 𝑛)
40 iccssre 12440 . . . . . 6 ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ⊆ ℝ)
4132, 31, 40syl2anc 696 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (-𝑛[,]𝑛) ⊆ ℝ)
42 ax-resscn 10177 . . . . . . 7 ℝ ⊆ ℂ
43 ssid 3757 . . . . . . 7 ℂ ⊆ ℂ
44 cncfss 22895 . . . . . . 7 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ))
4542, 43, 44mp2an 710 . . . . . 6 (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ)
461adantr 472 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐴 ∈ dom vol)
47 eqid 2752 . . . . . . . 8 (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) = (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))
4847volcn 23566 . . . . . . 7 ((𝐴 ∈ dom vol ∧ -𝑛 ∈ ℝ) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℝ))
4946, 32, 48syl2anc 696 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℝ))
5045, 49sseldi 3734 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℂ))
5141sselda 3736 . . . . . 6 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑢 ∈ (-𝑛[,]𝑛)) → 𝑢 ∈ ℝ)
52 cncff 22889 . . . . . . . 8 ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℝ) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))):ℝ⟶ℝ)
5349, 52syl 17 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))):ℝ⟶ℝ)
5453ffvelrnda 6514 . . . . . 6 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑢 ∈ ℝ) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑢) ∈ ℝ)
5551, 54syldan 488 . . . . 5 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑢 ∈ (-𝑛[,]𝑛)) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑢) ∈ ℝ)
56 oveq2 6813 . . . . . . . . . . . 12 (𝑦 = -𝑛 → (-𝑛[,]𝑦) = (-𝑛[,]-𝑛))
5756ineq2d 3949 . . . . . . . . . . 11 (𝑦 = -𝑛 → (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]-𝑛)))
5857fveq2d 6348 . . . . . . . . . 10 (𝑦 = -𝑛 → (vol‘(𝐴 ∩ (-𝑛[,]𝑦))) = (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))))
59 fvex 6354 . . . . . . . . . 10 (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))) ∈ V
6058, 47, 59fvmpt 6436 . . . . . . . . 9 (-𝑛 ∈ ℝ → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) = (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))))
6132, 60syl 17 . . . . . . . 8 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) = (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))))
62 inss2 3969 . . . . . . . . . . . 12 (𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ (-𝑛[,]-𝑛)
6332rexrd 10273 . . . . . . . . . . . . 13 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 ∈ ℝ*)
64 iccid 12405 . . . . . . . . . . . . 13 (-𝑛 ∈ ℝ* → (-𝑛[,]-𝑛) = {-𝑛})
6563, 64syl 17 . . . . . . . . . . . 12 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (-𝑛[,]-𝑛) = {-𝑛})
6662, 65syl5sseq 3786 . . . . . . . . . . 11 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ {-𝑛})
6732snssd 4477 . . . . . . . . . . 11 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → {-𝑛} ⊆ ℝ)
6866, 67sstrd 3746 . . . . . . . . . 10 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ ℝ)
69 ovolsn 23455 . . . . . . . . . . . 12 (-𝑛 ∈ ℝ → (vol*‘{-𝑛}) = 0)
7032, 69syl 17 . . . . . . . . . . 11 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (vol*‘{-𝑛}) = 0)
71 ovolssnul 23447 . . . . . . . . . . 11 (((𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ {-𝑛} ∧ {-𝑛} ⊆ ℝ ∧ (vol*‘{-𝑛}) = 0) → (vol*‘(𝐴 ∩ (-𝑛[,]-𝑛))) = 0)
7266, 67, 70, 71syl3anc 1473 . . . . . . . . . 10 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (vol*‘(𝐴 ∩ (-𝑛[,]-𝑛))) = 0)
73 nulmbl 23495 . . . . . . . . . 10 (((𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ ℝ ∧ (vol*‘(𝐴 ∩ (-𝑛[,]-𝑛))) = 0) → (𝐴 ∩ (-𝑛[,]-𝑛)) ∈ dom vol)
7468, 72, 73syl2anc 696 . . . . . . . . 9 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐴 ∩ (-𝑛[,]-𝑛)) ∈ dom vol)
75 mblvol 23490 . . . . . . . . 9 ((𝐴 ∩ (-𝑛[,]-𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))) = (vol*‘(𝐴 ∩ (-𝑛[,]-𝑛))))
7674, 75syl 17 . . . . . . . 8 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))) = (vol*‘(𝐴 ∩ (-𝑛[,]-𝑛))))
7761, 76, 723eqtrd 2790 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) = 0)
7819adantr 472 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 0 ≤ 𝐵)
7977, 78eqbrtrd 4818 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) ≤ 𝐵)
80 simprr 813 . . . . . . . 8 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
817adantr 472 . . . . . . . . 9 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 ∈ ℝ*)
82 iccmbl 23526 . . . . . . . . . . . 12 ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ∈ dom vol)
8332, 31, 82syl2anc 696 . . . . . . . . . . 11 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (-𝑛[,]𝑛) ∈ dom vol)
84 inmbl 23502 . . . . . . . . . . 11 ((𝐴 ∈ dom vol ∧ (-𝑛[,]𝑛) ∈ dom vol) → (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
8546, 83, 84syl2anc 696 . . . . . . . . . 10 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
869ffvelrni 6513 . . . . . . . . . . 11 ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ (0[,]+∞))
878, 86sseldi 3734 . . . . . . . . . 10 ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ*)
8885, 87syl 17 . . . . . . . . 9 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ*)
89 xrltle 12167 . . . . . . . . 9 ((𝐵 ∈ ℝ* ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ*) → (𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) → 𝐵 ≤ (vol‘(𝐴 ∩ (-𝑛[,]𝑛)))))
9081, 88, 89syl2anc 696 . . . . . . . 8 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) → 𝐵 ≤ (vol‘(𝐴 ∩ (-𝑛[,]𝑛)))))
9180, 90mpd 15 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 ≤ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
92 oveq2 6813 . . . . . . . . . . 11 (𝑦 = 𝑛 → (-𝑛[,]𝑦) = (-𝑛[,]𝑛))
9392ineq2d 3949 . . . . . . . . . 10 (𝑦 = 𝑛 → (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]𝑛)))
9493fveq2d 6348 . . . . . . . . 9 (𝑦 = 𝑛 → (vol‘(𝐴 ∩ (-𝑛[,]𝑦))) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
95 fvex 6354 . . . . . . . . 9 (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ V
9694, 47, 95fvmpt 6436 . . . . . . . 8 (𝑛 ∈ ℝ → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑛) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
9731, 96syl 17 . . . . . . 7 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑛) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
9891, 97breqtrrd 4824 . . . . . 6 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 ≤ ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑛))
9979, 98jca 555 . . . . 5 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) ≤ 𝐵𝐵 ≤ ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑛)))
10032, 31, 33, 39, 41, 50, 55, 99ivthle 23417 . . . 4 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ∃𝑧 ∈ (-𝑛[,]𝑛)((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = 𝐵)
10141sselda 3736 . . . . . . . 8 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → 𝑧 ∈ ℝ)
102 oveq2 6813 . . . . . . . . . . 11 (𝑦 = 𝑧 → (-𝑛[,]𝑦) = (-𝑛[,]𝑧))
103102ineq2d 3949 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]𝑧)))
104103fveq2d 6348 . . . . . . . . 9 (𝑦 = 𝑧 → (vol‘(𝐴 ∩ (-𝑛[,]𝑦))) = (vol‘(𝐴 ∩ (-𝑛[,]𝑧))))
105 fvex 6354 . . . . . . . . 9 (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) ∈ V
106104, 47, 105fvmpt 6436 . . . . . . . 8 (𝑧 ∈ ℝ → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = (vol‘(𝐴 ∩ (-𝑛[,]𝑧))))
107101, 106syl 17 . . . . . . 7 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = (vol‘(𝐴 ∩ (-𝑛[,]𝑧))))
108107eqeq1d 2754 . . . . . 6 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → (((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = 𝐵 ↔ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵))
10946adantr 472 . . . . . . . . 9 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → 𝐴 ∈ dom vol)
11032adantr 472 . . . . . . . . . 10 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → -𝑛 ∈ ℝ)
111101adantrr 755 . . . . . . . . . 10 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → 𝑧 ∈ ℝ)
112 iccmbl 23526 . . . . . . . . . 10 ((-𝑛 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝑛[,]𝑧) ∈ dom vol)
113110, 111, 112syl2anc 696 . . . . . . . . 9 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → (-𝑛[,]𝑧) ∈ dom vol)
114 inmbl 23502 . . . . . . . . 9 ((𝐴 ∈ dom vol ∧ (-𝑛[,]𝑧) ∈ dom vol) → (𝐴 ∩ (-𝑛[,]𝑧)) ∈ dom vol)
115109, 113, 114syl2anc 696 . . . . . . . 8 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → (𝐴 ∩ (-𝑛[,]𝑧)) ∈ dom vol)
116 inss1 3968 . . . . . . . . 9 (𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴
117116a1i 11 . . . . . . . 8 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → (𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴)
118 simprr 813 . . . . . . . 8 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)
119 sseq1 3759 . . . . . . . . . 10 (𝑥 = (𝐴 ∩ (-𝑛[,]𝑧)) → (𝑥𝐴 ↔ (𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴))
120 fveq2 6344 . . . . . . . . . . 11 (𝑥 = (𝐴 ∩ (-𝑛[,]𝑧)) → (vol‘𝑥) = (vol‘(𝐴 ∩ (-𝑛[,]𝑧))))
121120eqeq1d 2754 . . . . . . . . . 10 (𝑥 = (𝐴 ∩ (-𝑛[,]𝑧)) → ((vol‘𝑥) = 𝐵 ↔ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵))
122119, 121anbi12d 749 . . . . . . . . 9 (𝑥 = (𝐴 ∩ (-𝑛[,]𝑧)) → ((𝑥𝐴 ∧ (vol‘𝑥) = 𝐵) ↔ ((𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴 ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)))
123122rspcev 3441 . . . . . . . 8 (((𝐴 ∩ (-𝑛[,]𝑧)) ∈ dom vol ∧ ((𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴 ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))
124115, 117, 118, 123syl12anc 1471 . . . . . . 7 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))
125124expr 644 . . . . . 6 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → ((vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵 → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵)))
126108, 125sylbid 230 . . . . 5 (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → (((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = 𝐵 → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵)))
127126rexlimdva 3161 . . . 4 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (∃𝑧 ∈ (-𝑛[,]𝑛)((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = 𝐵 → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵)))
128100, 127mpd 15 . . 3 ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))
12929, 128rexlimddv 3165 . 2 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))
130 simpll 807 . . 3 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → 𝐴 ∈ dom vol)
131 ssid 3757 . . . 4 𝐴𝐴
132131a1i 11 . . 3 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → 𝐴𝐴)
133 simpr 479 . . . 4 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → 𝐵 = (vol‘𝐴))
134133eqcomd 2758 . . 3 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → (vol‘𝐴) = 𝐵)
135 sseq1 3759 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
136 fveq2 6344 . . . . . 6 (𝑥 = 𝐴 → (vol‘𝑥) = (vol‘𝐴))
137136eqeq1d 2754 . . . . 5 (𝑥 = 𝐴 → ((vol‘𝑥) = 𝐵 ↔ (vol‘𝐴) = 𝐵))
138135, 137anbi12d 749 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐴 ∧ (vol‘𝑥) = 𝐵) ↔ (𝐴𝐴 ∧ (vol‘𝐴) = 𝐵)))
139138rspcev 3441 . . 3 ((𝐴 ∈ dom vol ∧ (𝐴𝐴 ∧ (vol‘𝐴) = 𝐵)) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))
140130, 132, 134, 139syl12anc 1471 . 2 (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))
14117simp3d 1138 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 𝐵 ≤ (vol‘𝐴))
142 xrleloe 12162 . . . 4 ((𝐵 ∈ ℝ* ∧ (vol‘𝐴) ∈ ℝ*) → (𝐵 ≤ (vol‘𝐴) ↔ (𝐵 < (vol‘𝐴) ∨ 𝐵 = (vol‘𝐴))))
1436, 12, 142syl2anc 696 . . 3 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (𝐵 ≤ (vol‘𝐴) ↔ (𝐵 < (vol‘𝐴) ∨ 𝐵 = (vol‘𝐴))))
144141, 143mpbid 222 . 2 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (𝐵 < (vol‘𝐴) ∨ 𝐵 = (vol‘𝐴)))
145129, 140, 144mpjaodan 862 1 ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1624  wcel 2131  wrex 3043  cin 3706  wss 3707  {csn 4313   class class class wbr 4796  cmpt 4873  dom cdm 5258  wf 6037  cfv 6041  (class class class)co 6805  cc 10118  cr 10119  0cc0 10120  +∞cpnf 10255  -∞cmnf 10256  *cxr 10257   < clt 10258  cle 10259  -cneg 10451  cn 11204  [,]cicc 12363  cnccncf 22872  vol*covol 23423  volcvol 23424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cc 9441  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-disj 4765  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fi 8474  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8947  df-cda 9174  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-q 11974  df-rp 12018  df-xneg 12131  df-xadd 12132  df-xmul 12133  df-ioo 12364  df-ico 12366  df-icc 12367  df-fz 12512  df-fzo 12652  df-fl 12779  df-seq 12988  df-exp 13047  df-hash 13304  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-clim 14410  df-rlim 14411  df-sum 14608  df-rest 16277  df-topgen 16298  df-psmet 19932  df-xmet 19933  df-met 19934  df-bl 19935  df-mopn 19936  df-top 20893  df-topon 20910  df-bases 20944  df-cmp 21384  df-cncf 22874  df-ovol 23425  df-vol 23426
This theorem is referenced by:  itg2const2  23699
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