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Theorem mblfinlem3 36465
Description: The difference between two sets measurable by the criterion in ismblfin 36467 is itself measurable by the same. Corollary 0.3 of [Viaclovsky7] p. 3. (Contributed by Brendan Leahy, 25-Mar-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
Assertion
Ref Expression
mblfinlem3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ))) → sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) = (vol*‘(𝐴𝐵)))
Distinct variable groups:   𝑦,𝑏,𝐴   𝐵,𝑏,𝑦

Proof of Theorem mblfinlem3
Dummy variables 𝑓 𝑠 𝑢 𝑣 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltso 11290 . . 3 < Or ℝ
21a1i 11 . 2 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ))) → < Or ℝ)
3 difss 4130 . . . 4 (𝐴𝐵) ⊆ 𝐴
4 ovolsscl 24985 . . . 4 (((𝐴𝐵) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
53, 4mp3an1 1449 . . 3 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
653ad2ant1 1134 . 2 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ))) → (vol*‘(𝐴𝐵)) ∈ ℝ)
7 vex 3479 . . . . . 6 𝑢 ∈ V
8 eqeq1 2737 . . . . . . . 8 (𝑦 = 𝑢 → (𝑦 = (vol‘𝑏) ↔ 𝑢 = (vol‘𝑏)))
98anbi2d 630 . . . . . . 7 (𝑦 = 𝑢 → ((𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))))
109rexbidv 3179 . . . . . 6 (𝑦 = 𝑢 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))))
117, 10elab 3667 . . . . 5 (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))} ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))
12 simprl 770 . . . . . . . . 9 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))) → 𝑏 ⊆ (𝐴𝐵))
13 ssdifss 4134 . . . . . . . . 9 (𝐴 ⊆ ℝ → (𝐴𝐵) ⊆ ℝ)
14 ovolss 24984 . . . . . . . . 9 ((𝑏 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)))
1512, 13, 14syl2anr 598 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)))
16 uniretop 24261 . . . . . . . . . . . . 13 ℝ = (topGen‘ran (,))
1716cldss 22515 . . . . . . . . . . . 12 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ)
18 ovolcl 24977 . . . . . . . . . . . 12 (𝑏 ⊆ ℝ → (vol*‘𝑏) ∈ ℝ*)
1917, 18syl 17 . . . . . . . . . . 11 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (vol*‘𝑏) ∈ ℝ*)
20 ovolcl 24977 . . . . . . . . . . . 12 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
2113, 20syl 17 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
22 xrlenlt 11275 . . . . . . . . . . 11 (((vol*‘𝑏) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
2319, 21, 22syl2anr 598 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑏 ∈ (Clsd‘(topGen‘ran (,)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
2423adantrr 716 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
25 id 22 . . . . . . . . . . . . . 14 (𝑢 = (vol‘𝑏) → 𝑢 = (vol‘𝑏))
26 dfss4 4257 . . . . . . . . . . . . . . . . 17 (𝑏 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
2717, 26sylib 217 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
28 rembl 25039 . . . . . . . . . . . . . . . . 17 ℝ ∈ dom vol
2916cldopn 22517 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran (,)))
30 opnmbl 25101 . . . . . . . . . . . . . . . . . 18 ((ℝ ∖ 𝑏) ∈ (topGen‘ran (,)) → (ℝ ∖ 𝑏) ∈ dom vol)
3129, 30syl 17 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ dom vol)
32 difmbl 25042 . . . . . . . . . . . . . . . . 17 ((ℝ ∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
3328, 31, 32sylancr 588 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
3427, 33eqeltrrd 2835 . . . . . . . . . . . . . . 15 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol)
35 mblvol 25029 . . . . . . . . . . . . . . 15 (𝑏 ∈ dom vol → (vol‘𝑏) = (vol*‘𝑏))
3634, 35syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑏) = (vol*‘𝑏))
3725, 36sylan9eqr 2795 . . . . . . . . . . . . 13 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → 𝑢 = (vol*‘𝑏))
3837breq2d 5159 . . . . . . . . . . . 12 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → ((vol*‘(𝐴𝐵)) < 𝑢 ↔ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
3938notbid 318 . . . . . . . . . . 11 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → (¬ (vol*‘(𝐴𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
4039adantrl 715 . . . . . . . . . 10 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))) → (¬ (vol*‘(𝐴𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
4140adantl 483 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → (¬ (vol*‘(𝐴𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
4224, 41bitr4d 282 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < 𝑢))
4315, 42mpbid 231 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
4443rexlimdvaa 3157 . . . . . 6 (𝐴 ⊆ ℝ → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)) → ¬ (vol*‘(𝐴𝐵)) < 𝑢))
4544imp 408 . . . . 5 ((𝐴 ⊆ ℝ ∧ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
4611, 45sylan2b 595 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
4746adantlr 714 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
48473ad2antl1 1186 . 2 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ))) ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
49 simplr 768 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (vol*‘𝐴) ∈ ℝ)
50 resubcl 11520 . . . . . . . . . . . . . . . . . . 19 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ 𝑢 ∈ ℝ) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
5150adantrr 716 . . . . . . . . . . . . . . . . . 18 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
52 posdif 11703 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (𝑢 < (vol*‘(𝐴𝐵)) ↔ 0 < ((vol*‘(𝐴𝐵)) − 𝑢)))
5352ancoms 460 . . . . . . . . . . . . . . . . . . . 20 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘(𝐴𝐵)) ↔ 0 < ((vol*‘(𝐴𝐵)) − 𝑢)))
5453biimpd 228 . . . . . . . . . . . . . . . . . . 19 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘(𝐴𝐵)) → 0 < ((vol*‘(𝐴𝐵)) − 𝑢)))
5554impr 456 . . . . . . . . . . . . . . . . . 18 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → 0 < ((vol*‘(𝐴𝐵)) − 𝑢))
5651, 55elrpd 13009 . . . . . . . . . . . . . . . . 17 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ+)
57 3nn 12287 . . . . . . . . . . . . . . . . . 18 3 ∈ ℕ
58 nnrp 12981 . . . . . . . . . . . . . . . . . 18 (3 ∈ ℕ → 3 ∈ ℝ+)
5957, 58ax-mp 5 . . . . . . . . . . . . . . . . 17 3 ∈ ℝ+
60 rpdivcl 12995 . . . . . . . . . . . . . . . . 17 ((((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ+ ∧ 3 ∈ ℝ+) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
6156, 59, 60sylancl 587 . . . . . . . . . . . . . . . 16 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
625, 61sylan 581 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
6349, 62ltsubrpd 13044 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol*‘𝐴))
6463adantr 482 . . . . . . . . . . . . 13 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) → ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol*‘𝐴))
65 simpr 486 . . . . . . . . . . . . 13 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) → (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ))
6664, 65breqtrd 5173 . . . . . . . . . . . 12 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) → ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ))
67 reex 11197 . . . . . . . . . . . . . . . . . 18 ℝ ∈ V
6867ssex 5320 . . . . . . . . . . . . . . . . 17 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
6968adantr 482 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → 𝐴 ∈ V)
70 sseq1 4006 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → (𝑣 ⊆ ℝ ↔ 𝐴 ⊆ ℝ))
71 fveq2 6888 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → (vol*‘𝑣) = (vol*‘𝐴))
7271eleq1d 2819 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
7370, 72anbi12d 632 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)))
74 sseq2 4007 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = 𝐴 → (𝑏𝑣𝑏𝐴))
7574anbi1d 631 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝐴 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝐴𝑦 = (vol‘𝑏))))
7675rexbidv 3179 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝐴 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))))
7776abbidv 2802 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} = {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))})
7877sseq1d 4012 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ⊆ ℝ))
7977neeq1d 3001 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ≠ ∅))
8077raleqdv 3326 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥))
8180rexbidv 3179 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥))
8278, 79, 813anbi123d 1437 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥) ↔ ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥)))
8373, 82imbi12d 345 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐴 → (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥)) ↔ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥))))
84 simpr 486 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑣𝑦 = (vol‘𝑏)) → 𝑦 = (vol‘𝑏))
8584, 36sylan9eqr 2795 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏))) → 𝑦 = (vol*‘𝑏))
8685adantl 483 . . . . . . . . . . . . . . . . . . . . 21 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏)))) → 𝑦 = (vol*‘𝑏))
87 simprl 770 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏))) → 𝑏𝑣)
88 ovolsscl 24985 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑣𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → (vol*‘𝑏) ∈ ℝ)
89883expb 1121 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑣 ∧ (𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ)) → (vol*‘𝑏) ∈ ℝ)
9089ancoms 460 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ 𝑏𝑣) → (vol*‘𝑏) ∈ ℝ)
9187, 90sylan2 594 . . . . . . . . . . . . . . . . . . . . 21 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏)))) → (vol*‘𝑏) ∈ ℝ)
9286, 91eqeltrd 2834 . . . . . . . . . . . . . . . . . . . 20 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏)))) → 𝑦 ∈ ℝ)
9392rexlimdvaa 3157 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) → 𝑦 ∈ ℝ))
9493abssdv 4064 . . . . . . . . . . . . . . . . . 18 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ)
95 retop 24260 . . . . . . . . . . . . . . . . . . . . . 22 (topGen‘ran (,)) ∈ Top
96 0cld 22524 . . . . . . . . . . . . . . . . . . . . . 22 ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
9795, 96ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ (Clsd‘(topGen‘ran (,)))
98 0ss 4395 . . . . . . . . . . . . . . . . . . . . . 22 ∅ ⊆ 𝑣
99 0mbl 25038 . . . . . . . . . . . . . . . . . . . . . . . 24 ∅ ∈ dom vol
100 mblvol 25029 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
10199, 100ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (vol‘∅) = (vol*‘∅)
102 ovol0 24992 . . . . . . . . . . . . . . . . . . . . . . 23 (vol*‘∅) = 0
103101, 102eqtr2i 2762 . . . . . . . . . . . . . . . . . . . . . 22 0 = (vol‘∅)
10498, 103pm3.2i 472 . . . . . . . . . . . . . . . . . . . . 21 (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅))
105 sseq1 4006 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ∅ → (𝑏𝑣 ↔ ∅ ⊆ 𝑣))
106 fveq2 6888 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ∅ → (vol‘𝑏) = (vol‘∅))
107106eqeq2d 2744 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ∅ → (0 = (vol‘𝑏) ↔ 0 = (vol‘∅)))
108105, 107anbi12d 632 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = ∅ → ((𝑏𝑣 ∧ 0 = (vol‘𝑏)) ↔ (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅))))
109108rspcev 3612 . . . . . . . . . . . . . . . . . . . . 21 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏)))
11097, 104, 109mp2an 691 . . . . . . . . . . . . . . . . . . . 20 𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏))
111 c0ex 11204 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
112 eqeq1 2737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 0 → (𝑦 = (vol‘𝑏) ↔ 0 = (vol‘𝑏)))
113112anbi2d 630 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 0 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝑣 ∧ 0 = (vol‘𝑏))))
114113rexbidv 3179 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 0 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏))))
115111, 114spcev 3596 . . . . . . . . . . . . . . . . . . . 20 (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏)) → ∃𝑦𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)))
116110, 115ax-mp 5 . . . . . . . . . . . . . . . . . . 19 𝑦𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))
117 abn0 4379 . . . . . . . . . . . . . . . . . . . 20 ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ↔ ∃𝑦𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)))
118117biimpri 227 . . . . . . . . . . . . . . . . . . 19 (∃𝑦𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅)
119116, 118mp1i 13 . . . . . . . . . . . . . . . . . 18 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅)
120 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 = (vol‘𝑏))
121120, 36sylan9eqr 2795 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏))) → 𝑧 = (vol*‘𝑏))
122121adantl 483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑣 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏)))) → 𝑧 = (vol*‘𝑏))
123 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏))) → 𝑏𝑣)
124 ovolss 24984 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑣𝑣 ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘𝑣))
125124ancoms 460 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 ⊆ ℝ ∧ 𝑏𝑣) → (vol*‘𝑏) ≤ (vol*‘𝑣))
126123, 125sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑣 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘𝑣))
127122, 126eqbrtrd 5169 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏)))) → 𝑧 ≤ (vol*‘𝑣))
128127rexlimdvaa 3157 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ⊆ ℝ → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣)))
129128alrimiv 1931 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ⊆ ℝ → ∀𝑧(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣)))
130 eqeq1 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑧 → (𝑦 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑏)))
131130anbi2d 630 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑧 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝑣𝑧 = (vol‘𝑏))))
132131rexbidv 3179 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏))))
133132ralab 3686 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣) ↔ ∀𝑧(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣)))
134129, 133sylibr 233 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ⊆ ℝ → ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣))
135 brralrspcev 5207 . . . . . . . . . . . . . . . . . . . 20 (((vol*‘𝑣) ∈ ℝ ∧ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣)) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥)
136134, 135sylan2 594 . . . . . . . . . . . . . . . . . . 19 (((vol*‘𝑣) ∈ ℝ ∧ 𝑣 ⊆ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥)
137136ancoms 460 . . . . . . . . . . . . . . . . . 18 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥)
13894, 119, 1373jca 1129 . . . . . . . . . . . . . . . . 17 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥))
13983, 138vtoclg 3556 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V → ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥)))
14069, 139mpcom 38 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥))
141140adantr 482 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥))
14262rpred 13012 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ)
14349, 142resubcld 11638 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
144 suprlub 12174 . . . . . . . . . . . . . 14 ((({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥) ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣))
145141, 143, 144syl2anc 585 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣))
146145adantr 482 . . . . . . . . . . . 12 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣))
14766, 146mpbid 231 . . . . . . . . . . 11 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣)
148 eqeq1 2737 . . . . . . . . . . . . . . 15 (𝑦 = 𝑣 → (𝑦 = (vol‘𝑏) ↔ 𝑣 = (vol‘𝑏)))
149148anbi2d 630 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → ((𝑏𝐴𝑦 = (vol‘𝑏)) ↔ (𝑏𝐴𝑣 = (vol‘𝑏))))
150149rexbidv 3179 . . . . . . . . . . . . 13 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏))))
151150rexab 3689 . . . . . . . . . . . 12 (∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣))
152 breq2 5151 . . . . . . . . . . . . . . . . 17 (𝑣 = (vol‘𝑏) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
153152ad2antll 728 . . . . . . . . . . . . . . . 16 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑣 = (vol‘𝑏))) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
154 sseq1 4006 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑏 → (𝑠𝐴𝑏𝐴))
155 fveq2 6888 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = 𝑏 → (vol‘𝑠) = (vol‘𝑏))
156155breq2d 5159 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑏 → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠) ↔ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
157154, 156anbi12d 632 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑏 → ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ↔ (𝑏𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏))))
158157rspcev 3612 . . . . . . . . . . . . . . . . . 18 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))
159158expr 458 . . . . . . . . . . . . . . . . 17 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑏𝐴) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠))))
160159adantrr 716 . . . . . . . . . . . . . . . 16 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑣 = (vol‘𝑏))) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠))))
161153, 160sylbid 239 . . . . . . . . . . . . . . 15 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑣 = (vol‘𝑏))) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠))))
162161rexlimiva 3148 . . . . . . . . . . . . . 14 (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠))))
163162imp 408 . . . . . . . . . . . . 13 ((∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))
164163exlimiv 1934 . . . . . . . . . . . 12 (∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))
165151, 164sylbi 216 . . . . . . . . . . 11 (∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))
166147, 165syl 17 . . . . . . . . . 10 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))
167166ex 414 . . . . . . . . 9 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠))))
168167adantlr 714 . . . . . . . 8 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠))))
169 simplrr 777 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (vol*‘𝐵) ∈ ℝ)
17062adantlr 714 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
171169, 170ltsubrpd 13044 . . . . . . . . . . . . 13 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol*‘𝐵))
172171adantr 482 . . . . . . . . . . . 12 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol*‘𝐵))
173 simpr 486 . . . . . . . . . . . 12 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ))
174172, 173breqtrd 5173 . . . . . . . . . . 11 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ))
17567ssex 5320 . . . . . . . . . . . . . . . 16 (𝐵 ⊆ ℝ → 𝐵 ∈ V)
176175adantr 482 . . . . . . . . . . . . . . 15 ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → 𝐵 ∈ V)
177 sseq1 4006 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → (𝑣 ⊆ ℝ ↔ 𝐵 ⊆ ℝ))
178 fveq2 6888 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → (vol*‘𝑣) = (vol*‘𝐵))
179178eleq1d 2819 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐵) ∈ ℝ))
180177, 179anbi12d 632 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
181 sseq2 4007 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝐵 → (𝑏𝑣𝑏𝐵))
182181anbi1d 631 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝐵 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝐵𝑦 = (vol‘𝑏))))
183182rexbidv 3179 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐵 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))))
184183abbidv 2802 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} = {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))})
185184sseq1d 4012 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ⊆ ℝ))
186184neeq1d 3001 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ≠ ∅))
187184raleqdv 3326 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥))
188187rexbidv 3179 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥))
189185, 186, 1883anbi123d 1437 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → (({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥) ↔ ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥)))
190180, 189imbi12d 345 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐵 → (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥)) ↔ ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥))))
191190, 138vtoclg 3556 . . . . . . . . . . . . . . 15 (𝐵 ∈ V → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥)))
192176, 191mpcom 38 . . . . . . . . . . . . . 14 ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥))
193192ad2antlr 726 . . . . . . . . . . . . 13 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥))
194142adantlr 714 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ)
195169, 194resubcld 11638 . . . . . . . . . . . . 13 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
196 suprlub 12174 . . . . . . . . . . . . 13 ((({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥) ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣))
197193, 195, 196syl2anc 585 . . . . . . . . . . . 12 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣))
198197adantr 482 . . . . . . . . . . 11 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣))
199174, 198mpbid 231 . . . . . . . . . 10 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣)
200148anbi2d 630 . . . . . . . . . . . . 13 (𝑦 = 𝑣 → ((𝑏𝐵𝑦 = (vol‘𝑏)) ↔ (𝑏𝐵𝑣 = (vol‘𝑏))))
201200rexbidv 3179 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑣 = (vol‘𝑏))))
202201rexab 3689 . . . . . . . . . . 11 (∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣))
203 breq2 5151 . . . . . . . . . . . . . . . 16 (𝑣 = (vol‘𝑏) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
204203ad2antll 728 . . . . . . . . . . . . . . 15 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐵𝑣 = (vol‘𝑏))) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
205 sseq1 4006 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑏 → (𝑤𝐵𝑏𝐵))
206 fveq2 6888 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑏 → (vol‘𝑤) = (vol‘𝑏))
207206breq2d 5159 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑏 → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤) ↔ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
208205, 207anbi12d 632 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑏 → ((𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)) ↔ (𝑏𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏))))
209208rspcev 3612 . . . . . . . . . . . . . . . . 17 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))
210209expr 458 . . . . . . . . . . . . . . . 16 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑏𝐵) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏) → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))))
211210adantrr 716 . . . . . . . . . . . . . . 15 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐵𝑣 = (vol‘𝑏))) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏) → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))))
212204, 211sylbid 239 . . . . . . . . . . . . . 14 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐵𝑣 = (vol‘𝑏))) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))))
213212rexlimiva 3148 . . . . . . . . . . . . 13 (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑣 = (vol‘𝑏)) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))))
214213imp 408 . . . . . . . . . . . 12 ((∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣) → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))
215214exlimiv 1934 . . . . . . . . . . 11 (∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣) → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))
216202, 215sylbi 216 . . . . . . . . . 10 (∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))
217199, 216syl 17 . . . . . . . . 9 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))
218217ex 414 . . . . . . . 8 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ) → ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))))
219168, 218anim12d 610 . . . . . . 7 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → (∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))))
220 reeanv 3227 . . . . . . 7 (∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) ↔ (∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ ∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))(𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))))
221219, 220syl6ibr 252 . . . . . 6 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))))
222 eqid 2733 . . . . . . . . . . . . . 14 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
223222ovolgelb 24979 . . . . . . . . . . . . 13 ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ ∧ (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
2242233expa 1119 . . . . . . . . . . . 12 (((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
22562, 224sylan2 594 . . . . . . . . . . 11 (((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵))))) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
226225ancoms 460 . . . . . . . . . 10 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
227226an32s 651 . . . . . . . . 9 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
228 elmapi 8839 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
229 ssid 4003 . . . . . . . . . . . . . . 15 ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)
230222ovollb 24978 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
231229, 230mpan2 690 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
232231adantl 483 . . . . . . . . . . . . 13 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
233 eqid 2733 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
234233, 222ovolsf 24971 . . . . . . . . . . . . . . 15 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
235 frn 6721 . . . . . . . . . . . . . . . 16 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
236 icossxr 13405 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℝ*
237235, 236sstrdi 3993 . . . . . . . . . . . . . . 15 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
238 supxrcl 13290 . . . . . . . . . . . . . . 15 (ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
239234, 237, 2383syl 18 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
240 simpr 486 . . . . . . . . . . . . . . . . 17 ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐵) ∈ ℝ)
241 readdcl 11189 . . . . . . . . . . . . . . . . 17 (((vol*‘𝐵) ∈ ℝ ∧ (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
242240, 142, 241syl2anr 598 . . . . . . . . . . . . . . . 16 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
243242rexrd 11260 . . . . . . . . . . . . . . 15 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ*)
244243an32s 651 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ*)
245 rncoss 5969 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ⊆ ran (,)
246245unissi 4916 . . . . . . . . . . . . . . . . 17 ran ((,) ∘ 𝑓) ⊆ ran (,)
247 unirnioo 13422 . . . . . . . . . . . . . . . . 17 ℝ = ran (,)
248246, 247sseqtrri 4018 . . . . . . . . . . . . . . . 16 ran ((,) ∘ 𝑓) ⊆ ℝ
249 ovolcl 24977 . . . . . . . . . . . . . . . 16 ( ran ((,) ∘ 𝑓) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*)
250248, 249ax-mp 5 . . . . . . . . . . . . . . 15 (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*
251 xrletr 13133 . . . . . . . . . . . . . . 15 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ*) → (((vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
252250, 251mp3an1 1449 . . . . . . . . . . . . . 14 ((sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ*) → (((vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
253239, 244, 252syl2anr 598 . . . . . . . . . . . . 13 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (((vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
254232, 253mpand 694 . . . . . . . . . . . 12 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
255228, 254sylan2 594 . . . . . . . . . . 11 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
256255anim2d 613 . . . . . . . . . 10 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → ((𝐵 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))))
257256reximdva 3169 . . . . . . . . 9 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))))
258227, 257mpd 15 . . . . . . . 8 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
259 rexex 3077 . . . . . . . 8 (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → ∃𝑓(𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
260258, 259syl 17 . . . . . . 7 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ∃𝑓(𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
26116cldss 22515 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ⊆ ℝ)
262 indif2 4269 . . . . . . . . . . . . . . . . . 18 (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) = ((𝑠 ∩ ℝ) ∖ ran ((,) ∘ 𝑓))
263 df-ss 3964 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ⊆ ℝ ↔ (𝑠 ∩ ℝ) = 𝑠)
264263biimpi 215 . . . . . . . . . . . . . . . . . . 19 (𝑠 ⊆ ℝ → (𝑠 ∩ ℝ) = 𝑠)
265264difeq1d 4120 . . . . . . . . . . . . . . . . . 18 (𝑠 ⊆ ℝ → ((𝑠 ∩ ℝ) ∖ ran ((,) ∘ 𝑓)) = (𝑠 ran ((,) ∘ 𝑓)))
266262, 265eqtrid 2785 . . . . . . . . . . . . . . . . 17 (𝑠 ⊆ ℝ → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) = (𝑠 ran ((,) ∘ 𝑓)))
267261, 266syl 17 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) = (𝑠 ran ((,) ∘ 𝑓)))
268 retopbas 24259 . . . . . . . . . . . . . . . . . . . . 21 ran (,) ∈ TopBases
269 bastg 22451 . . . . . . . . . . . . . . . . . . . . 21 (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
270268, 269ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ran (,) ⊆ (topGen‘ran (,))
271245, 270sstri 3990 . . . . . . . . . . . . . . . . . . 19 ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))
272 uniopn 22381 . . . . . . . . . . . . . . . . . . 19 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))) → ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)))
27395, 271, 272mp2an 691 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))
27416opncld 22519 . . . . . . . . . . . . . . . . . 18 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))) → (ℝ ∖ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
27595, 273, 274mp2an 691 . . . . . . . . . . . . . . . . 17 (ℝ ∖ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,)))
276 incld 22529 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (ℝ ∖ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) ∈ (Clsd‘(topGen‘ran (,))))
277275, 276mpan2 690 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) ∈ (Clsd‘(topGen‘ran (,))))
278267, 277eqeltrrd 2835 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
279278adantr 482 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
280279ad2antlr 726 . . . . . . . . . . . . 13 ((((𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
281 simprll 778 . . . . . . . . . . . . . 14 ((((𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑠𝐴)
282 simplll 774 . . . . . . . . . . . . . 14 ((((𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝐵 ran ((,) ∘ 𝑓))
283281, 282ssdif2d 4142 . . . . . . . . . . . . 13 ((((𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵))
284 fveq2 6888 . . . . . . . . . . . . . . . . 17 ((𝑠 ran ((,) ∘ 𝑓)) = 𝑏 → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))
285284eqcoms 2741 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))
286285biantrud 533 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → (𝑏 ⊆ (𝐴𝐵) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))))
287 sseq1 4006 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → (𝑏 ⊆ (𝐴𝐵) ↔ (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)))
288286, 287bitr3d 281 . . . . . . . . . . . . . 14 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → ((𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)) ↔ (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)))
289288rspcev 3612 . . . . . . . . . . . . 13 (((𝑠 ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)))
290280, 283, 289syl2anc 585 . . . . . . . . . . . 12 ((((𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)))
291290adantlll 717 . . . . . . . . . . 11 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)))
292 difss 4130 . . . . . . . . . . . . . . . 16 ((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) ⊆ (𝐴𝐵)
293292, 3sstri 3990 . . . . . . . . . . . . . . 15 ((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴
294 ovolsscl 24985 . . . . . . . . . . . . . . 15 ((((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
295293, 294mp3an1 1449 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
296295ad5antr 733 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
2975ad5antr 733 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴𝐵)) ∈ ℝ)
298 simpl 484 . . . . . . . . . . . . . 14 ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵))) → 𝑢 ∈ ℝ)
299298ad4antlr 732 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑢 ∈ ℝ)
300 difdif2 4285 . . . . . . . . . . . . . . 15 ((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) = (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))
301300fveq2i 6891 . . . . . . . . . . . . . 14 (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) = (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))))
302 difss 4130 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝐵)
303302, 3sstri 3990 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∖ 𝑠) ⊆ 𝐴
304 inss1 4227 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)
305304, 3sstri 3990 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ 𝐴
306303, 305unssi 4184 . . . . . . . . . . . . . . . . 17 (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ 𝐴
307 ovolsscl 24985 . . . . . . . . . . . . . . . . 17 (((((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ∈ ℝ)
308306, 307mp3an1 1449 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ∈ ℝ)
309308ad5antr 733 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ∈ ℝ)
310 difss 4130 . . . . . . . . . . . . . . . . . 18 (𝐴𝑠) ⊆ 𝐴
311 ovolsscl 24985 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑠) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
312310, 311mp3an1 1449 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
313312ad5antr 733 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴𝑠)) ∈ ℝ)
314169, 194readdcld 11239 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
315314, 250jctil 521 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ))
316 simpr 486 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))
317 ovolge0 24980 . . . . . . . . . . . . . . . . . . . . 21 ( ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤ (vol*‘ ran ((,) ∘ 𝑓)))
318248, 317ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 0 ≤ (vol*‘ ran ((,) ∘ 𝑓))
319316, 318jctil 521 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → (0 ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
320 xrrege0 13149 . . . . . . . . . . . . . . . . . . 19 ((((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ) ∧ (0 ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
321315, 319, 320syl2an 597 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
322 difss 4130 . . . . . . . . . . . . . . . . . . 19 ( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ran ((,) ∘ 𝑓)
323 ovolsscl 24985 . . . . . . . . . . . . . . . . . . 19 ((( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)
324322, 248, 323mp3an12 1452 . . . . . . . . . . . . . . . . . 18 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)
325321, 324syl 17 . . . . . . . . . . . . . . . . 17 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)
326325ad2antrr 725 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)
327313, 326readdcld 11239 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ)
3285, 50sylan 581 . . . . . . . . . . . . . . . . . 18 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑢 ∈ ℝ) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
329328adantrr 716 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
330329adantlr 714 . . . . . . . . . . . . . . . 16 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
331330ad3antrrr 729 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
332 ssdifss 4134 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ ℝ → (𝐴𝑠) ⊆ ℝ)
333322, 248sstri 3990 . . . . . . . . . . . . . . . . . . . . 21 ( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ
334 unss 4183 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑠) ⊆ ℝ ∧ ( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ) ↔ ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ)
335332, 333, 334sylanblc 590 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ ℝ → ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ)
336 ovolcl 24977 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ*)
337335, 336syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐴 ⊆ ℝ → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ*)
338337ad4antr 731 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ*)
339312ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘(𝐴𝑠)) ∈ ℝ)
340339, 325readdcld 11239 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ)
341 ovolge0 24980 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ → 0 ≤ (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))))
342335, 341syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))))
343342ad4antr 731 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → 0 ≤ (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))))
344332adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴𝑠) ⊆ ℝ)
345344, 312jca 513 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((𝐴𝑠) ⊆ ℝ ∧ (vol*‘(𝐴𝑠)) ∈ ℝ))
346345ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → ((𝐴𝑠) ⊆ ℝ ∧ (vol*‘(𝐴𝑠)) ∈ ℝ))
347325, 333jctil 521 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ ∧ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ))
348 ovolun 24998 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑠) ⊆ ℝ ∧ (vol*‘(𝐴𝑠)) ∈ ℝ) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ ∧ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)) → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ≤ ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))))
349346, 347, 348syl2anc 585 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ≤ ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))))
350 xrrege0 13149 . . . . . . . . . . . . . . . . . 18 ((((vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ* ∧ ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ) ∧ (0 ≤ (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∧ (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ≤ ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))))) → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ)
351338, 340, 343, 349, 350syl22anc 838 . . . . . . . . . . . . . . . . 17 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ)
352351ad2antrr 725 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ)
353 ssdif 4138 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝑠))
3543, 353ax-mp 5 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝑠)
355 incom 4200 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) = ( ran ((,) ∘ 𝑓) ∩ (𝐴𝐵))
356 indif2 4269 . . . . . . . . . . . . . . . . . . . 20 ( ran ((,) ∘ 𝑓) ∩ (𝐴𝐵)) = (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∖ 𝐵)
357355, 356eqtri 2761 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) = (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∖ 𝐵)
358 inss1 4227 . . . . . . . . . . . . . . . . . . . . 21 ( ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ran ((,) ∘ 𝑓)
359358a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ( ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ran ((,) ∘ 𝑓))
360 simprrl 780 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑤𝐵)
361359, 360ssdif2d 4142 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∖ 𝐵) ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝑤))
362357, 361eqsstrid 4029 . . . . . . . . . . . . . . . . . 18 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝑤))
363 unss12 4181 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝑠) ∧ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) → (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)))
364354, 362, 363sylancr 588 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)))
365335ad6antr 735 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ)
366 ovolss 24984 . . . . . . . . . . . . . . . . 17 (((((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∧ ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ≤ (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))))
367364, 365, 366syl2anc 585 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ≤ (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))))
368332ad6antr 735 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝐴𝑠) ⊆ ℝ)
369326, 333jctil 521 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ ∧ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ))
370368, 313, 369, 348syl21anc 837 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ≤ ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))))
371309, 352, 327, 367, 370letrd 11367 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ≤ ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))))
372194ad3antrrr 729 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ)
373194, 194readdcld 11239 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
374373ad3antrrr 729 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
375 eleq1w 2817 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = 𝑠 → (𝑏 ∈ dom vol ↔ 𝑠 ∈ dom vol))
376375, 34vtoclga 3565 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ∈ dom vol)
377 mblvol 25029 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ dom vol → (vol‘𝑠) = (vol*‘𝑠))
378376, 377syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑠) = (vol*‘𝑠))
379378adantr 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑠) = (vol*‘𝑠))
380 sseqin2 4214 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠𝐴 ↔ (𝐴𝑠) = 𝑠)
381380biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑠𝐴 → (𝐴𝑠) = 𝑠)
382381eqcomd 2739 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠𝐴𝑠 = (𝐴𝑠))
383382fveq2d 6892 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠𝐴 → (vol*‘𝑠) = (vol*‘(𝐴𝑠)))
384383ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → (vol*‘𝑠) = (vol*‘(𝐴𝑠)))
385379, 384sylan9eq 2793 . . . . . . . . . . . . . . . . . . . . 21 (((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑠) = (vol*‘(𝐴𝑠)))
386385oveq2d 7420 . . . . . . . . . . . . . . . . . . . 20 (((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol‘𝑠)) = ((vol*‘𝐴) − (vol*‘(𝐴𝑠))))
387386adantll 713 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol‘𝑠)) = ((vol*‘𝐴) − (vol*‘(𝐴𝑠))))
388376adantr 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑠 ∈ dom vol)
389 simplll 774 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
390 mblsplit 25031 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝐴) = ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))))
391390eqcomd 2739 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))) = (vol*‘𝐴))
3923913expb 1121 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ dom vol ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))) = (vol*‘𝐴))
393388, 389, 392syl2anr 598 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))) = (vol*‘𝐴))
394393adantr 482 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))) = (vol*‘𝐴))
395 simp-6r 787 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝐴) ∈ ℝ)
396395recnd 11238 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝐴) ∈ ℂ)
397 inss1 4227 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑠) ⊆ 𝐴
398 ovolsscl 24985 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴𝑠) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
399397, 398mp3an1 1449 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
400399recnd 11238 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℂ)
401400ad5antr 733 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴𝑠)) ∈ ℂ)
402312recnd 11238 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℂ)
403402ad5antr 733 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴𝑠)) ∈ ℂ)
404396, 401, 403subaddd 11585 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘𝐴) − (vol*‘(𝐴𝑠))) = (vol*‘(𝐴𝑠)) ↔ ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))) = (vol*‘𝐴)))
405394, 404mpbird 257 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol*‘(𝐴𝑠))) = (vol*‘(𝐴𝑠)))
406387, 405eqtrd 2773 . . . . . . . . . . . . . . . . . 18 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol‘𝑠)) = (vol*‘(𝐴𝑠)))
407379ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑠) = (vol*‘𝑠))
408 simpll 766 . . . . . . . . . . . . . . . . . . . . 21 (((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → 𝑠𝐴)
409 simp-4l 782 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
410 ovolsscl 24985 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝑠) ∈ ℝ)
4114103expb 1121 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠𝐴 ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → (vol*‘𝑠) ∈ ℝ)
412408, 409, 411syl2anr 598 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝑠) ∈ ℝ)
413407, 412eqeltrd 2834 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑠) ∈ ℝ)
414 simprlr 779 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠))
415395, 372, 413, 414ltsub23d 11815 . . . . . . . . . . . . . . . . . 18 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol‘𝑠)) < (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
416406, 415eqbrtrrd 5171 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴𝑠)) < (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
417321recnd 11238 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℂ)
418417ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℂ)
419240ad5antlr 734 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝐵) ∈ ℝ)
420419recnd 11238 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝐵) ∈ ℂ)
421 eleq1w 2817 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 = 𝑤 → (𝑏 ∈ dom vol ↔ 𝑤 ∈ dom vol))
422421, 34vtoclga 3565 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ∈ (Clsd‘(topGen‘ran (,))) → 𝑤 ∈ dom vol)
423 mblvol 25029 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ∈ dom vol → (vol‘𝑤) = (vol*‘𝑤))
424422, 423syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑤) = (vol*‘𝑤))
425424adantl 483 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑤) = (vol*‘𝑤))
426425ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑤) = (vol*‘𝑤))
427 simprl 770 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → 𝑤𝐵)
428 simp-4r 783 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
429 ovolsscl 24985 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝑤) ∈ ℝ)
4304293expb 1121 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑤𝐵 ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝑤) ∈ ℝ)
431427, 428, 430syl2anr 598 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝑤) ∈ ℝ)
432426, 431eqeltrd 2834 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑤) ∈ ℝ)
433432recnd 11238 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑤) ∈ ℂ)
434418, 420, 433npncand 11591 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) + ((vol*‘𝐵) − (vol‘𝑤))) = ((vol*‘ ran ((,) ∘ 𝑓)) − (vol‘𝑤)))
435 simplrl 776 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) → 𝐵 ran ((,) ∘ 𝑓))
436427, 435sylan9ssr 3995 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑤 ran ((,) ∘ 𝑓))
437 sseqin2 4214 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ran ((,) ∘ 𝑓) ↔ ( ran ((,) ∘ 𝑓) ∩ 𝑤) = 𝑤)
438436, 437sylib 217 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ( ran ((,) ∘ 𝑓) ∩ 𝑤) = 𝑤)
439438fveq2d 6892 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) = (vol*‘𝑤))
440426, 439eqtr4d 2776 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑤) = (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)))
441440oveq2d 7420 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol‘𝑤)) = ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤))))
442422adantl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑤 ∈ dom vol)
443321, 248jctil 521 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → ( ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ))
444 mblsplit 25031 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘ ran ((,) ∘ 𝑓)) = ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))))
445444eqcomd 2739 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑤 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘ ran ((,) ∘ 𝑓)))
4464453expb 1121 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑤 ∈ dom vol ∧ ( ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)) → ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘ ran ((,) ∘ 𝑓)))
447442, 443, 446syl2anr 598 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) → ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘ ran ((,) ∘ 𝑓)))
448447adantr 482 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘ ran ((,) ∘ 𝑓)))
449 inss1 4227 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ran ((,) ∘ 𝑓)
450 ovolsscl 24985 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ)
451449, 248, 450mp3an12 1452 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ)
452321, 451syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ)
453452recnd 11238 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℂ)
454325recnd 11238 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℂ)
455417, 453, 454subaddd 11585 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤))) = (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ↔ ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘ ran ((,) ∘ 𝑓))))
456455ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤))) = (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ↔ ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘ ran ((,) ∘ 𝑓))))
457448, 456mpbird 257 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤))) = (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)))
458434, 441, 4573eqtrd 2777 . . . . . . . . . . . . . . . . . 18 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) + ((vol*‘𝐵) − (vol‘𝑤))) = (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)))
459240ad3antlr 730 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘𝐵) ∈ ℝ)
460321, 459resubcld 11638 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ∈ ℝ)
461460ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ∈ ℝ)
462419, 432resubcld 11638 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐵) − (vol‘𝑤)) ∈ ℝ)
463 simprr 772 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))
464194adantr 482 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ)
465321, 459, 464lesubadd2d 11809 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ≤ (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ↔ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
466463, 465mpbird 257 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ≤ (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
467466ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ≤ (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
468 simprrr 781 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))
469419, 372, 432, 468ltsub23d 11815 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐵) − (vol‘𝑤)) < (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
470461, 462, 372, 372, 467, 469leltaddd 11832 . . . . . . . . . . . . . . . . . 18 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) + ((vol*‘𝐵) − (vol‘𝑤))) < ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))
471458, 470eqbrtrrd 5171 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) < ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))
472313, 326, 372, 374, 416, 471lt2addd 11833 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) < ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
473 df-3 12272 . . . . . . . . . . . . . . . . . . . . . 22 3 = (2 + 1)
474 2cn 12283 . . . . . . . . . . . . . . . . . . . . . . 23 2 ∈ ℂ
475 ax-1cn 11164 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℂ
476474, 475addcomi 11401 . . . . . . . . . . . . . . . . . . . . . 22 (2 + 1) = (1 + 2)
477473, 476eqtri 2761 . . . . . . . . . . . . . . . . . . . . 21 3 = (1 + 2)
478477oveq1i 7414 . . . . . . . . . . . . . . . . . . . 20 (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((1 + 2) · (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
47962rpcnd 13014 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℂ)
480 adddir 11201 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℂ ∧ 2 ∈ ℂ ∧ (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℂ) → ((1 + 2) · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((1 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) + (2 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
481475, 474, 480mp3an12 1452 . . . . . . . . . . . . . . . . . . . . . 22 ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℂ → ((1 + 2) · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((1 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) + (2 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
482479, 481syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((1 + 2) · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((1 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) + (2 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
483479mullidd 11228 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (1 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
4844792timesd 12451 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (2 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))
485483, 484oveq12d 7422 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((1 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) + (2 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) = ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
486482, 485eqtrd 2773 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((1 + 2) · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
487478, 486eqtrid 2785 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
488329recnd 11238 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℂ)
489 3cn 12289 . . . . . . . . . . . . . . . . . . . . 21 3 ∈ ℂ
490 3ne0 12314 . . . . . . . . . . . . . . . . . . . . 21 3 ≠ 0
491 divcan2 11876 . . . . . . . . . . . . . . . . . . . . 21 ((((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴𝐵)) − 𝑢))
492489, 490, 491mp3an23 1454 . . . . . . . . . . . . . . . . . . . 20 (((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℂ → (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴𝐵)) − 𝑢))
493488, 492syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴𝐵)) − 𝑢))
494487, 493eqtr3d 2775 . . . . . . . . . . . . . . . . . 18 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) = ((vol*‘(𝐴𝐵)) − 𝑢))
495494adantlr 714 . . . . . . . . . . . . . . . . 17 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) = ((vol*‘(𝐴𝐵)) − 𝑢))
496495ad3antrrr 729 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) = ((vol*‘(𝐴𝐵)) − 𝑢))
497472, 496breqtrd 5173 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) < ((vol*‘(𝐴𝐵)) − 𝑢))
498309, 327, 331, 371, 497lelttrd 11368 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) < ((vol*‘(𝐴𝐵)) − 𝑢))
499301, 498eqbrtrid 5182 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) < ((vol*‘(𝐴𝐵)) − 𝑢))
500296, 297, 299, 499ltsub13d 11816 . . . . . . . . . . . 12 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑢 < ((vol*‘(𝐴𝐵)) − (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))))
501283adantlll 717 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵))
502 sseqin2 4214 . . . . . . . . . . . . . . 15 ((𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵) ↔ ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) = (𝑠 ran ((,) ∘ 𝑓)))
503501, 502sylib 217 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) = (𝑠 ran ((,) ∘ 𝑓)))
504503fveq2d 6892 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
505 opnmbl 25101 . . . . . . . . . . . . . . . . . . 19 ( ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) → ran ((,) ∘ 𝑓) ∈ dom vol)
506273, 505ax-mp 5 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ∈ dom vol
507 difmbl 25042 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ∈ dom vol) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
508376, 506, 507sylancl 587 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
509508adantr 482 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
510509ad2antlr 726 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
51113adantr 482 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴𝐵) ⊆ ℝ)
512511, 5jca 513 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ))
513512ad5antr 733 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ))
514 mblsplit 25031 . . . . . . . . . . . . . . . . 17 (((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol ∧ (𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))))
5155143expb 1121 . . . . . . . . . . . . . . . 16 (((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol ∧ ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ)) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))))
516515eqcomd 2739 . . . . . . . . . . . . . . 15 (((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol ∧ ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ)) → ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))) = (vol*‘(𝐴𝐵)))
517510, 513, 516syl2anc 585 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))) = (vol*‘(𝐴𝐵)))
518297recnd 11238 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴𝐵)) ∈ ℂ)
519296recnd 11238 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℂ)
520 inss1 4227 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) ⊆ (𝐴𝐵)
521520, 3sstri 3990 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴
522 ovolsscl 24985 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
523521, 522mp3an1 1449 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
524523ad5antr 733 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
525524recnd 11238 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℂ)
526518, 519, 525subadd2d 11586 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘(𝐴𝐵)) − (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))) = (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) ↔ ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))) = (vol*‘(𝐴𝐵))))
527517, 526mpbird 257 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴𝐵)) − (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))) = (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))))
528 mblvol 25029 . . . . . . . . . . . . . . . . 17 ((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
529507, 528syl 17 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ∈ dom vol) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
530376, 506, 529sylancl 587 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
531530adantr 482 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
532531ad2antlr 726 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
533504, 527, 5323eqtr4rd 2784 . . . . . . . . . . . 12 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = ((vol*‘(𝐴𝐵)) − (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))))
534500, 533breqtrrd 5175 . . . . . . . . . . 11 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑓))))
535 fvex 6901 . . . . . . . . . . . 12 (vol‘(𝑠 ran ((,) ∘ 𝑓))) ∈ V
536 eqeq1 2737 . . . . . . . . . . . . . . 15 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → (𝑣 = (vol‘𝑏) ↔ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)))
537536anbi2d 630 . . . . . . . . . . . . . 14 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → ((𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))))
538537rexbidv 3179 . . . . . . . . . . . . 13 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))))
539 breq2 5151 . . . . . . . . . . . . 13 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → (𝑢 < 𝑣𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑓)))))
540538, 539anbi12d 632 . . . . . . . . . . . 12 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → ((∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣) ↔ (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)) ∧ 𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑓))))))
541535, 540spcev 3596 . . . . . . . . . . 11 ((∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)) ∧ 𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑓)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
542291, 534, 541syl2anc 585 . . . . . . . . . 10 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
543148anbi2d 630 . . . . . . . . . . . 12 (𝑦 = 𝑣 → ((𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏))))
544543rexbidv 3179 . . . . . . . . . . 11 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏))))
545544rexab 3689 . . . . . . . . . 10 (∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣 ↔ ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
546542, 545sylibr 233 . . . . . . . . 9 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) ∧ ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣)
547546ex 414 . . . . . . . 8 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))) → (((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣))
548547rexlimdvva 3212 . . . . . . 7 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))) → (∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣))
549260, 548exlimddv 1939 . . . . . 6 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))∃𝑤 ∈ (Clsd‘(topGen‘ran (,)))((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣))
550221, 549syld 47 . . . . 5 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣))
551550exp31 421 . . . 4 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵))) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣))))
552551com34 91 . . 3 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < )) → ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣))))
5535523imp1 1348 . 2 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ))) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣)
5542, 6, 48, 553eqsupd 9448 1 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}, ℝ, < ))) → sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) = (vol*‘(𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wne 2941  wral 3062  wrex 3071  Vcvv 3475  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321   cuni 4907   class class class wbr 5147   Or wor 5586   × cxp 5673  dom cdm 5675  ran crn 5676  ccom 5679  wf 6536  cfv 6540  (class class class)co 7404  m cmap 8816  supcsup 9431  cc 11104  cr 11105  0cc0 11106  1c1 11107   + caddc 11109   · cmul 11111  +∞cpnf 11241  *cxr 11243   < clt 11244  cle 11245  cmin 11440   / cdiv 11867  cn 12208  2c2 12263  3c3 12264  +crp 12970  (,)cioo 13320  [,)cico 13322  seqcseq 13962  abscabs 15177  topGenctg 17379  Topctop 22377  TopBasesctb 22430  Clsdccld 22502  vol*covol 24961  volcvol 24962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7665  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-omul 8466  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-acn 9933  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-rlim 15429  df-sum 15629  df-rest 17364  df-topgen 17385  df-psmet 20921  df-xmet 20922  df-met 20923  df-bl 20924  df-mopn 20925  df-top 22378  df-topon 22395  df-bases 22431  df-cld 22505  df-cmp 22873  df-ovol 24963  df-vol 24964
This theorem is referenced by:  ismblfin  36467
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