Mathbox for Brendan Leahy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mblfinlem3 Structured version   Visualization version   GIF version

Theorem mblfinlem3 34408
 Description: The difference between two sets measurable by the criterion in ismblfin 34410 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 10557 . . 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 4024 . . . 4 (𝐴𝐵) ⊆ 𝐴
4 ovolsscl 23758 . . . 4 (((𝐴𝐵) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
53, 4mp3an1 1438 . . 3 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
653ad2ant1 1124 . 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 3435 . . . . . 6 𝑢 ∈ V
8 eqeq1 2797 . . . . . . . 8 (𝑦 = 𝑢 → (𝑦 = (vol‘𝑏) ↔ 𝑢 = (vol‘𝑏)))
98anbi2d 628 . . . . . . 7 (𝑦 = 𝑢 → ((𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))))
109rexbidv 3257 . . . . . 6 (𝑦 = 𝑢 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))))
117, 10elab 3600 . . . . 5 (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))} ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))
12 simprl 767 . . . . . . . . 9 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))) → 𝑏 ⊆ (𝐴𝐵))
13 ssdifss 4028 . . . . . . . . 9 (𝐴 ⊆ ℝ → (𝐴𝐵) ⊆ ℝ)
14 ovolss 23757 . . . . . . . . 9 ((𝑏 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)))
1512, 13, 14syl2anr 596 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)))
16 uniretop 23042 . . . . . . . . . . . . 13 ℝ = (topGen‘ran (,))
1716cldss 21309 . . . . . . . . . . . 12 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ)
18 ovolcl 23750 . . . . . . . . . . . 12 (𝑏 ⊆ ℝ → (vol*‘𝑏) ∈ ℝ*)
1917, 18syl 17 . . . . . . . . . . 11 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (vol*‘𝑏) ∈ ℝ*)
20 ovolcl 23750 . . . . . . . . . . . 12 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
2113, 20syl 17 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
22 xrlenlt 10542 . . . . . . . . . . 11 (((vol*‘𝑏) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
2319, 21, 22syl2anr 596 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑏 ∈ (Clsd‘(topGen‘ran (,)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
2423adantrr 713 . . . . . . . . 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 4150 . . . . . . . . . . . . . . . . 17 (𝑏 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
2717, 26sylib 219 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
28 rembl 23812 . . . . . . . . . . . . . . . . 17 ℝ ∈ dom vol
2916cldopn 21311 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran (,)))
30 opnmbl 23874 . . . . . . . . . . . . . . . . . 18 ((ℝ ∖ 𝑏) ∈ (topGen‘ran (,)) → (ℝ ∖ 𝑏) ∈ dom vol)
3129, 30syl 17 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ dom vol)
32 difmbl 23815 . . . . . . . . . . . . . . . . 17 ((ℝ ∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
3328, 31, 32sylancr 587 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
3427, 33eqeltrrd 2882 . . . . . . . . . . . . . . 15 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol)
35 mblvol 23802 . . . . . . . . . . . . . . 15 (𝑏 ∈ dom vol → (vol‘𝑏) = (vol*‘𝑏))
3634, 35syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑏) = (vol*‘𝑏))
3725, 36sylan9eqr 2851 . . . . . . . . . . . . 13 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → 𝑢 = (vol*‘𝑏))
3837breq2d 4968 . . . . . . . . . . . 12 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → ((vol*‘(𝐴𝐵)) < 𝑢 ↔ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
3938notbid 319 . . . . . . . . . . 11 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → (¬ (vol*‘(𝐴𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
4039adantrl 712 . . . . . . . . . 10 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))) → (¬ (vol*‘(𝐴𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
4140adantl 482 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → (¬ (vol*‘(𝐴𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
4224, 41bitr4d 283 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < 𝑢))
4315, 42mpbid 233 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
4443rexlimdvaa 3245 . . . . . 6 (𝐴 ⊆ ℝ → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)) → ¬ (vol*‘(𝐴𝐵)) < 𝑢))
4544imp 407 . . . . 5 ((𝐴 ⊆ ℝ ∧ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
4611, 45sylan2b 593 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
4746adantlr 711 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
48473ad2antl1 1176 . 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 765 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (vol*‘𝐴) ∈ ℝ)
50 resubcl 10787 . . . . . . . . . . . . . . . . . . 19 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ 𝑢 ∈ ℝ) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
5150adantrr 713 . . . . . . . . . . . . . . . . . 18 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
52 posdif 10970 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (𝑢 < (vol*‘(𝐴𝐵)) ↔ 0 < ((vol*‘(𝐴𝐵)) − 𝑢)))
5352ancoms 459 . . . . . . . . . . . . . . . . . . . 20 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘(𝐴𝐵)) ↔ 0 < ((vol*‘(𝐴𝐵)) − 𝑢)))
5453biimpd 230 . . . . . . . . . . . . . . . . . . 19 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘(𝐴𝐵)) → 0 < ((vol*‘(𝐴𝐵)) − 𝑢)))
5554impr 455 . . . . . . . . . . . . . . . . . 18 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → 0 < ((vol*‘(𝐴𝐵)) − 𝑢))
5651, 55elrpd 12267 . . . . . . . . . . . . . . . . 17 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ+)
57 3nn 11553 . . . . . . . . . . . . . . . . . 18 3 ∈ ℕ
58 nnrp 12239 . . . . . . . . . . . . . . . . . 18 (3 ∈ ℕ → 3 ∈ ℝ+)
5957, 58ax-mp 5 . . . . . . . . . . . . . . . . 17 3 ∈ ℝ+
60 rpdivcl 12253 . . . . . . . . . . . . . . . . 17 ((((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ+ ∧ 3 ∈ ℝ+) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
6156, 59, 60sylancl 586 . . . . . . . . . . . . . . . 16 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
625, 61sylan 580 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
6349, 62ltsubrpd 12302 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol*‘𝐴))
6463adantr 481 . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 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 4982 . . . . . . . . . . . 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 10463 . . . . . . . . . . . . . . . . . 18 ℝ ∈ V
6867ssex 5109 . . . . . . . . . . . . . . . . 17 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
6968adantr 481 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → 𝐴 ∈ V)
70 sseq1 3908 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → (𝑣 ⊆ ℝ ↔ 𝐴 ⊆ ℝ))
71 fveq2 6530 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → (vol*‘𝑣) = (vol*‘𝐴))
7271eleq1d 2865 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
7370, 72anbi12d 630 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)))
74 sseq2 3909 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = 𝐴 → (𝑏𝑣𝑏𝐴))
7574anbi1d 629 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝐴 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝐴𝑦 = (vol‘𝑏))))
7675rexbidv 3257 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝐴 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))))
7776abbidv 2858 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} = {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))})
7877sseq1d 3914 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ⊆ ℝ))
7977neeq1d 3041 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ≠ ∅))
8077raleqdv 3372 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥))
8180rexbidv 3257 . . . . . . . . . . . . . . . . . . 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 1426 . . . . . . . . . . . . . . . . . 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 346 . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑣𝑦 = (vol‘𝑏)) → 𝑦 = (vol‘𝑏))
8584, 36sylan9eqr 2851 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏))) → 𝑦 = (vol*‘𝑏))
8685adantl 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏)))) → 𝑦 = (vol*‘𝑏))
87 simprl 767 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏))) → 𝑏𝑣)
88 ovolsscl 23758 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑣𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → (vol*‘𝑏) ∈ ℝ)
89883expb 1111 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑣 ∧ (𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ)) → (vol*‘𝑏) ∈ ℝ)
9089ancoms 459 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ 𝑏𝑣) → (vol*‘𝑏) ∈ ℝ)
9187, 90sylan2 592 . . . . . . . . . . . . . . . . . . . . 21 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏)))) → (vol*‘𝑏) ∈ ℝ)
9286, 91eqeltrd 2881 . . . . . . . . . . . . . . . . . . . 20 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏)))) → 𝑦 ∈ ℝ)
9392rexlimdvaa 3245 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) → 𝑦 ∈ ℝ))
9493abssdv 3961 . . . . . . . . . . . . . . . . . 18 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ)
95 retop 23041 . . . . . . . . . . . . . . . . . . . . . 22 (topGen‘ran (,)) ∈ Top
96 0cld 21318 . . . . . . . . . . . . . . . . . . . . . 22 ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
9795, 96ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ (Clsd‘(topGen‘ran (,)))
98 0ss 4264 . . . . . . . . . . . . . . . . . . . . . 22 ∅ ⊆ 𝑣
99 0mbl 23811 . . . . . . . . . . . . . . . . . . . . . . . 24 ∅ ∈ dom vol
100 mblvol 23802 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
10199, 100ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (vol‘∅) = (vol*‘∅)
102 ovol0 23765 . . . . . . . . . . . . . . . . . . . . . . 23 (vol*‘∅) = 0
103101, 102eqtr2i 2818 . . . . . . . . . . . . . . . . . . . . . 22 0 = (vol‘∅)
10498, 103pm3.2i 471 . . . . . . . . . . . . . . . . . . . . 21 (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅))
105 sseq1 3908 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ∅ → (𝑏𝑣 ↔ ∅ ⊆ 𝑣))
106 fveq2 6530 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ∅ → (vol‘𝑏) = (vol‘∅))
107106eqeq2d 2803 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ∅ → (0 = (vol‘𝑏) ↔ 0 = (vol‘∅)))
108105, 107anbi12d 630 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = ∅ → ((𝑏𝑣 ∧ 0 = (vol‘𝑏)) ↔ (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅))))
109108rspcev 3554 . . . . . . . . . . . . . . . . . . . . 21 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏)))
11097, 104, 109mp2an 688 . . . . . . . . . . . . . . . . . . . 20 𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏))
111 c0ex 10470 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
112 eqeq1 2797 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 0 → (𝑦 = (vol‘𝑏) ↔ 0 = (vol‘𝑏)))
113112anbi2d 628 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 0 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝑣 ∧ 0 = (vol‘𝑏))))
114113rexbidv 3257 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 0 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏))))
115111, 114spcev 3544 . . . . . . . . . . . . . . . . . . . 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 4250 . . . . . . . . . . . . . . . . . . . 20 ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ↔ ∃𝑦𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)))
118117biimpri 229 . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 = (vol‘𝑏))
121120, 36sylan9eqr 2851 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏))) → 𝑧 = (vol*‘𝑏))
122121adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑣 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏)))) → 𝑧 = (vol*‘𝑏))
123 simprl 767 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏))) → 𝑏𝑣)
124 ovolss 23757 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑣𝑣 ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘𝑣))
125124ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 ⊆ ℝ ∧ 𝑏𝑣) → (vol*‘𝑏) ≤ (vol*‘𝑣))
126123, 125sylan2 592 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑣 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘𝑣))
127122, 126eqbrtrd 4978 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏)))) → 𝑧 ≤ (vol*‘𝑣))
128127rexlimdvaa 3245 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ⊆ ℝ → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣)))
129128alrimiv 1903 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ⊆ ℝ → ∀𝑧(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣)))
130 eqeq1 2797 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑧 → (𝑦 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑏)))
131130anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑧 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝑣𝑧 = (vol‘𝑏))))
132131rexbidv 3257 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏))))
133132ralab 3617 . . . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ⊆ ℝ → ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣))
135 brralrspcev 5016 . . . . . . . . . . . . . . . . . . . 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 592 . . . . . . . . . . . . . . . . . . 19 (((vol*‘𝑣) ∈ ℝ ∧ 𝑣 ⊆ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥)
137136ancoms 459 . . . . . . . . . . . . . . . . . 18 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥)
13894, 119, 1373jca 1119 . . . . . . . . . . . . . . . . 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 3505 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 12270 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ)
14349, 142resubcld 10905 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
144 suprlub 11442 . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 233 . . . . . . . . . . 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 2797 . . . . . . . . . . . . . . 15 (𝑦 = 𝑣 → (𝑦 = (vol‘𝑏) ↔ 𝑣 = (vol‘𝑏)))
149148anbi2d 628 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → ((𝑏𝐴𝑦 = (vol‘𝑏)) ↔ (𝑏𝐴𝑣 = (vol‘𝑏))))
150149rexbidv 3257 . . . . . . . . . . . . 13 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏))))
151150rexab 3619 . . . . . . . . . . . 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 4960 . . . . . . . . . . . . . . . . 17 (𝑣 = (vol‘𝑏) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
153152ad2antll 725 . . . . . . . . . . . . . . . 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 3908 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑏 → (𝑠𝐴𝑏𝐴))
155 fveq2 6530 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = 𝑏 → (vol‘𝑠) = (vol‘𝑏))
156155breq2d 4968 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑏 → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠) ↔ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
157154, 156anbi12d 630 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑏 → ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ↔ (𝑏𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏))))
158157rspcev 3554 . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . 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 3241 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 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 1906 . . . . . . . . . . . 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 218 . . . . . . . . . . 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 413 . . . . . . . . 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 711 . . . . . . . 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 774 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (vol*‘𝐵) ∈ ℝ)
17062adantlr 711 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
171169, 170ltsubrpd 12302 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 485 . . . . . . . . . . . 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 4982 . . . . . . . . . . 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 5109 . . . . . . . . . . . . . . . 16 (𝐵 ⊆ ℝ → 𝐵 ∈ V)
176175adantr 481 . . . . . . . . . . . . . . 15 ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → 𝐵 ∈ V)
177 sseq1 3908 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → (𝑣 ⊆ ℝ ↔ 𝐵 ⊆ ℝ))
178 fveq2 6530 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → (vol*‘𝑣) = (vol*‘𝐵))
179178eleq1d 2865 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐵) ∈ ℝ))
180177, 179anbi12d 630 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
181 sseq2 3909 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝐵 → (𝑏𝑣𝑏𝐵))
182181anbi1d 629 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝐵 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝐵𝑦 = (vol‘𝑏))))
183182rexbidv 3257 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐵 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))))
184183abbidv 2858 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} = {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))})
185184sseq1d 3914 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ⊆ ℝ))
186184neeq1d 3041 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ≠ ∅))
187184raleqdv 3372 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥))
188187rexbidv 3257 . . . . . . . . . . . . . . . . . 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 1426 . . . . . . . . . . . . . . . . 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 346 . . . . . . . . . . . . . . . 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 3505 . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ)
195169, 194resubcld 10905 . . . . . . . . . . . . 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 11442 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 233 . . . . . . . . . 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 628 . . . . . . . . . . . . 13 (𝑦 = 𝑣 → ((𝑏𝐵𝑦 = (vol‘𝑏)) ↔ (𝑏𝐵𝑣 = (vol‘𝑏))))
201200rexbidv 3257 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑣 = (vol‘𝑏))))
202201rexab 3619 . . . . . . . . . . 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 4960 . . . . . . . . . . . . . . . 16 (𝑣 = (vol‘𝑏) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
204203ad2antll 725 . . . . . . . . . . . . . . 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 3908 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑏 → (𝑤𝐵𝑏𝐵))
206 fveq2 6530 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑏 → (vol‘𝑤) = (vol‘𝑏))
207206breq2d 4968 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑏 → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤) ↔ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
208205, 207anbi12d 630 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑏 → ((𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)) ↔ (𝑏𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏))))
209208rspcev 3554 . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . 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 3241 . . . . . . . . . . . . 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 407 . . . . . . . . . . . 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 1906 . . . . . . . . . . 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 218 . . . . . . . . . 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 413 . . . . . . . 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 608 . . . . . . 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 3325 . . . . . . 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 253 . . . . . 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 2793 . . . . . . . . . . . . . 14 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
223222ovolgelb 23752 . . . . . . . . . . . . 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 1109 . . . . . . . . . . . 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 592 . . . . . . . . . . 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 459 . . . . . . . . . 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 648 . . . . . . . . 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 8269 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
229 ssid 3905 . . . . . . . . . . . . . . 15 ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)
230222ovollb 23751 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
231229, 230mpan2 687 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
232231adantl 482 . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
234233, 222ovolsf 23744 . . . . . . . . . . . . . . 15 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
235 frn 6380 . . . . . . . . . . . . . . . 16 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
236 icossxr 12660 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℝ*
237235, 236syl6ss 3896 . . . . . . . . . . . . . . 15 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
238 supxrcl 12547 . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . 17 ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐵) ∈ ℝ)
241 readdcl 10455 . . . . . . . . . . . . . . . . 17 (((vol*‘𝐵) ∈ ℝ ∧ (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
242240, 142, 241syl2anr 596 . . . . . . . . . . . . . . . 16 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
243242rexrd 10526 . . . . . . . . . . . . . . 15 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ*)
244243an32s 648 . . . . . . . . . . . . . 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 5716 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ⊆ ran (,)
246245unissi 4762 . . . . . . . . . . . . . . . . 17 ran ((,) ∘ 𝑓) ⊆ ran (,)
247 unirnioo 12676 . . . . . . . . . . . . . . . . 17 ℝ = ran (,)
248246, 247sseqtr4i 3920 . . . . . . . . . . . . . . . 16 ran ((,) ∘ 𝑓) ⊆ ℝ
249 ovolcl 23750 . . . . . . . . . . . . . . . 16 ( ran ((,) ∘ 𝑓) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*)
250248, 249ax-mp 5 . . . . . . . . . . . . . . 15 (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*
251 xrletr 12390 . . . . . . . . . . . . . . 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 1438 . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . 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 691 . . . . . . . . . . . 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 592 . . . . . . . . . . 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 611 . . . . . . . . . 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 3234 . . . . . . . . 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 3202 . . . . . . . 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 21309 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ⊆ ℝ)
262 indif2 4162 . . . . . . . . . . . . . . . . . 18 (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) = ((𝑠 ∩ ℝ) ∖ ran ((,) ∘ 𝑓))
263 df-ss 3869 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ⊆ ℝ ↔ (𝑠 ∩ ℝ) = 𝑠)
264263biimpi 217 . . . . . . . . . . . . . . . . . . 19 (𝑠 ⊆ ℝ → (𝑠 ∩ ℝ) = 𝑠)
265264difeq1d 4014 . . . . . . . . . . . . . . . . . 18 (𝑠 ⊆ ℝ → ((𝑠 ∩ ℝ) ∖ ran ((,) ∘ 𝑓)) = (𝑠 ran ((,) ∘ 𝑓)))
266262, 265syl5eq 2841 . . . . . . . . . . . . . . . . 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 23040 . . . . . . . . . . . . . . . . . . . . 21 ran (,) ∈ TopBases
269 bastg 21246 . . . . . . . . . . . . . . . . . . . . 21 (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
270268, 269ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ran (,) ⊆ (topGen‘ran (,))
271245, 270sstri 3893 . . . . . . . . . . . . . . . . . . 19 ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))
272 uniopn 21177 . . . . . . . . . . . . . . . . . . 19 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))) → ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)))
27395, 271, 272mp2an 688 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))
27416opncld 21313 . . . . . . . . . . . . . . . . . 18 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))) → (ℝ ∖ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
27595, 273, 274mp2an 688 . . . . . . . . . . . . . . . . 17 (ℝ ∖ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,)))
276 incld 21323 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (ℝ ∖ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) ∈ (Clsd‘(topGen‘ran (,))))
277275, 276mpan2 687 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) ∈ (Clsd‘(topGen‘ran (,))))
278267, 277eqeltrrd 2882 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
279278adantr 481 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
280279ad2antlr 723 . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . 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 4036 . . . . . . . . . . . . 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 6530 . . . . . . . . . . . . . . . . 17 ((𝑠 ran ((,) ∘ 𝑓)) = 𝑏 → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))
285284eqcoms 2801 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))
286285biantrud 532 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → (𝑏 ⊆ (𝐴𝐵) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))))
287 sseq1 3908 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → (𝑏 ⊆ (𝐴𝐵) ↔ (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)))
288286, 287bitr3d 282 . . . . . . . . . . . . . 14 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → ((𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)) ↔ (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)))
289288rspcev 3554 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 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 714 . . . . . . . . . . 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 4024 . . . . . . . . . . . . . . . 16 ((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) ⊆ (𝐴𝐵)
293292, 3sstri 3893 . . . . . . . . . . . . . . 15 ((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴
294 ovolsscl 23758 . . . . . . . . . . . . . . 15 ((((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
295293, 294mp3an1 1438 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
296295ad5antr 730 . . . . . . . . . . . . 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 730 . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵))) → 𝑢 ∈ ℝ)
299298ad4antlr 729 . . . . . . . . . . . . 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 4176 . . . . . . . . . . . . . . 15 ((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) = (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))
301300fveq2i 6533 . . . . . . . . . . . . . 14 (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) = (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))))
302 difss 4024 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝐵)
303302, 3sstri 3893 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∖ 𝑠) ⊆ 𝐴
304 inss1 4120 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)
305304, 3sstri 3893 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ 𝐴
306303, 305unssi 4077 . . . . . . . . . . . . . . . . 17 (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ 𝐴
307 ovolsscl 23758 . . . . . . . . . . . . . . . . 17 (((((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ∈ ℝ)
308306, 307mp3an1 1438 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ∈ ℝ)
309308ad5antr 730 . . . . . . . . . . . . . . 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 4024 . . . . . . . . . . . . . . . . . 18 (𝐴𝑠) ⊆ 𝐴
311 ovolsscl 23758 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑠) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
312310, 311mp3an1 1438 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
313312ad5antr 730 . . . . . . . . . . . . . . . 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 10505 . . . . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))
317 ovolge0 23753 . . . . . . . . . . . . . . . . . . . . 21 ( ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤ (vol*‘ ran ((,) ∘ 𝑓)))
318248, 317ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 0 ≤ (vol*‘ ran ((,) ∘ 𝑓))
319316, 318jctil 520 . . . . . . . . . . . . . . . . . . 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 12406 . . . . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . . . . 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 4024 . . . . . . . . . . . . . . . . . . 19 ( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ran ((,) ∘ 𝑓)
323 ovolsscl 23758 . . . . . . . . . . . . . . . . . . 19 ((( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)
324322, 248, 323mp3an12 1441 . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . 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 10505 . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . 18 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑢 ∈ ℝ) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
329328adantrr 713 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
330329adantlr 711 . . . . . . . . . . . . . . . 16 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
331330ad3antrrr 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)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
332 ssdifss 4028 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ ℝ → (𝐴𝑠) ⊆ ℝ)
333322, 248sstri 3893 . . . . . . . . . . . . . . . . . . . . 21 ( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ
334 unss 4076 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑠) ⊆ ℝ ∧ ( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ) ↔ ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ)
335332, 333, 334sylanblc 589 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ ℝ → ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ)
336 ovolcl 23750 . . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . 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 10505 . . . . . . . . . . . . . . . . . 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 23753 . . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴𝑠) ⊆ ℝ)
345344, 312jca 512 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((𝐴𝑠) ⊆ ℝ ∧ (vol*‘(𝐴𝑠)) ∈ ℝ))
346345ad3antrrr 726 . . . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . . . 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 23771 . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . 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 12406 . . . . . . . . . . . . . . . . . 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 835 . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . 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 4032 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝑠))
3543, 353ax-mp 5 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝑠)
355 incom 4094 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) = ( ran ((,) ∘ 𝑓) ∩ (𝐴𝐵))
356 indif2 4162 . . . . . . . . . . . . . . . . . . . 20 ( ran ((,) ∘ 𝑓) ∩ (𝐴𝐵)) = (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∖ 𝐵)
357355, 356eqtri 2817 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) = (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∖ 𝐵)
358 inss1 4120 . . . . . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . . . 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 4036 . . . . . . . . . . . . . . . . . . 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, 361syl5eqss 3931 . . . . . . . . . . . . . . . . . 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 4074 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝑠) ∧ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) → (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)))
364354, 362, 363sylancr 587 . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . 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 23757 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . 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 834 . . . . . . . . . . . . . . . 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 10633 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . 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 10505 . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . 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 2863 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = 𝑠 → (𝑏 ∈ dom vol ↔ 𝑠 ∈ dom vol))
376375, 34vtoclga 3512 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ∈ dom vol)
377 mblvol 23802 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ dom vol → (vol‘𝑠) = (vol*‘𝑠))
378376, 377syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑠) = (vol*‘𝑠))
379378adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑠) = (vol*‘𝑠))
380 sseqin2 4107 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠𝐴 ↔ (𝐴𝑠) = 𝑠)
381380biimpi 217 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑠𝐴 → (𝐴𝑠) = 𝑠)
382381eqcomd 2799 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠𝐴𝑠 = (𝐴𝑠))
383382fveq2d 6534 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠𝐴 → (vol*‘𝑠) = (vol*‘(𝐴𝑠)))
384383ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 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 2849 . . . . . . . . . . . . . . . . . . . . 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 7023 . . . . . . . . . . . . . . . . . . . 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 710 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑠 ∈ dom vol)
389 simplll 771 . . . . . . . . . . . . . . . . . . . . . 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 23804 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝐴) = ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))))
391390eqcomd 2799 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))) = (vol*‘𝐴))
3923913expb 1111 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ dom vol ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))) = (vol*‘𝐴))
393388, 389, 392syl2anr 596 . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . . . . . . . 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 4120 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑠) ⊆ 𝐴
398 ovolsscl 23758 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴𝑠) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
399397, 398mp3an1 1438 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
400399recnd 10504 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℂ)
401400ad5antr 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)))) ∧ (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 10504 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℂ)
403402ad5antr 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)))) ∧ (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 10852 . . . . . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . . . 21 (((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → 𝑠𝐴)
409 simp-4l 779 . . . . . . . . . . . . . . . . . . . . 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 23758 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝑠) ∈ ℝ)
4114103expb 1111 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠𝐴 ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → (vol*‘𝑠) ∈ ℝ)
412408, 409, 411syl2anr 596 . . . . . . . . . . . . . . . . . . . 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 2881 . . . . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . 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 11082 . . . . . . . . . . . . . . . . . 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 4980 . . . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . . . . . . 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 2863 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 = 𝑤 → (𝑏 ∈ dom vol ↔ 𝑤 ∈ dom vol))
422421, 34vtoclga 3512 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ∈ (Clsd‘(topGen‘ran (,))) → 𝑤 ∈ dom vol)
423 mblvol 23802 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ∈ dom vol → (vol‘𝑤) = (vol*‘𝑤))
424422, 423syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑤) = (vol*‘𝑤))
425424adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑤) = (vol*‘𝑤))
426425ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → 𝑤𝐵)
428 simp-4r 780 . . . . . . . . . . . . . . . . . . . . . . 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 23758 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝑤) ∈ ℝ)
4304293expb 1111 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑤𝐵 ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝑤) ∈ ℝ)
431427, 428, 430syl2anr 596 . . . . . . . . . . . . . . . . . . . . . 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 2881 . . . . . . . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . . . . . . 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 10858 . . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . . . . . 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 3898 . . . . . . . . . . . . . . . . . . . . . . 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 4107 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ran ((,) ∘ 𝑓) ↔ ( ran ((,) ∘ 𝑓) ∩ 𝑤) = 𝑤)
438436, 437sylib 219 . . . . . . . . . . . . . . . . . . . . . 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 6534 . . . . . . . . . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . . . . . . 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 7023 . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑤 ∈ dom vol)
443321, 248jctil 520 . . . . . . . . . . . . . . . . . . . . . 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 23804 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘ ran ((,) ∘ 𝑓)) = ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))))
445444eqcomd 2799 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑤 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘ ran ((,) ∘ 𝑓)))
4464453expb 1111 . . . . . . . . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . 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 4120 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ran ((,) ∘ 𝑓)
450 ovolsscl 23758 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ)
451449, 248, 450mp3an12 1441 . . . . . . . . . . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . . . . . . . . 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 10852 . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . 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 10905 . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . 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 10905 . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 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 11076 . . . . . . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . . . . 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 11082 . . . . . . . . . . . . . . . . . . 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 11099 . . . . . . . . . . . . . . . . . 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 4980 . . . . . . . . . . . . . . . . 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 11100 . . . . . . . . . . . . . . . 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 11538 . . . . . . . . . . . . . . . . . . . . . 22 3 = (2 + 1)
474 2cn 11549 . . . . . . . . . . . . . . . . . . . . . . 23 2 ∈ ℂ
475 ax-1cn 10430 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℂ
476474, 475addcomi 10667 . . . . . . . . . . . . . . . . . . . . . 22 (2 + 1) = (1 + 2)
477473, 476eqtri 2817 . . . . . . . . . . . . . . . . . . . . 21 3 = (1 + 2)
478477oveq1i 7017 . . . . . . . . . . . . . . . . . . . 20 (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((1 + 2) · (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
47962rpcnd 12272 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℂ)
480 adddir 10467 . . . . . . . . . . . . . . . . . . . . . . 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 1441 . . . . . . . . . . . . . . . . . . . . . 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))))
483479mulid2d 10494 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (1 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
4844792timesd 11717 . . . . . . . . . . . . . . . . . . . . . 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 7025 . . . . . . . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . . . . . . 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, 486syl5eq 2841 . . . . . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℂ)
489 3cn 11555 . . . . . . . . . . . . . . . . . . . . 21 3 ∈ ℂ
490 3ne0 11580 . . . . . . . . . . . . . . . . . . . . 21 3 ≠ 0
491 divcan2 11143 . . . . . . . . . . . . . . . . . . . . 21 ((((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴𝐵)) − 𝑢))
492489, 490, 491mp3an23 1443 . . . . . . . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 4982 . . . . . . . . . . . . . . 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 10634 . . . . . . . . . . . . . 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 4991 . . . . . . . . . . . . 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 11083 . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . 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 4107 . . . . . . . . . . . . . . 15 ((𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵) ↔ ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) = (𝑠 ran ((,) ∘ 𝑓)))
503501, 502sylib 219 . . . . . . . . . . . . . 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 6534 . . . . . . . . . . . . 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 23874 . . . . . . . . . . . . . . . . . . 19 ( ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) → ran ((,) ∘ 𝑓) ∈ dom vol)
506273, 505ax-mp 5 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ∈ dom vol
507 difmbl 23815 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ∈ dom vol) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
508376, 506, 507sylancl 586 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
509508adantr 481 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
510509ad2antlr 723 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴𝐵) ⊆ ℝ)
512511, 5jca 512 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ))
513512ad5antr 730 . . . . . . . . . . . . . . 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 23804 . . . . . . . . . . . . . . . . 17 (((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol ∧ (𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))))
5155143expb 1111 . . . . . . . . . . . . . . . 16 (((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol ∧ ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ)) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))))
516515eqcomd 2799 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . 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 4120 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) ⊆ (𝐴𝐵)
521520, 3sstri 3893 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴
522 ovolsscl 23758 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
523521, 522mp3an1 1438 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
524523ad5antr 730 . . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . 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 10853 . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . 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 23802 . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
531530adantr 481 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
532531ad2antlr 723 . . . . . . . . . . . . 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 2840 . . . . . . . . . . . 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 4984 . . . . . . . . . . 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 6543 . . . . . . . . . . . 12 (vol‘(𝑠 ran ((,) ∘ 𝑓))) ∈ V
536 eqeq1 2797 . . . . . . . . . . . . . . 15 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → (𝑣 = (vol‘𝑏) ↔ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)))
537536anbi2d 628 . . . . . . . . . . . . . 14 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → ((𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))))
538537rexbidv 3257 . . . . . . . . . . . . 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 4960 . . . . . . . . . . . . 13 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → (𝑢 < 𝑣𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑓)))))
540538, 539anbi12d 630 . . . . . . . . . . . 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 3544 . . . . . . . . . . 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 584 . . . . . . . . . 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 628 . . . . . . . . . . . 12 (𝑦 = 𝑣 → ((𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏))))
544543rexbidv 3257 . . . . . . . . . . 11 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏))))
545544rexab 3619 . . . . . . . . . 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 235 . . . . . . . . 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 413 . . . . . . . 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 3254 . . . . . . 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 1911 . . . . . 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 420 . . . 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 1338 . 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 8757 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 207   ∧ wa 396   ∧ w3a 1078  ∀wal 1518   = wceq 1520  ∃wex 1759   ∈ wcel 2079  {cab 2773   ≠ wne 2982  ∀wral 3103  ∃wrex 3104  Vcvv 3432   ∖ cdif 3851   ∪ cun 3852   ∩ cin 3853   ⊆ wss 3854  ∅c0 4206  ∪ cuni 4739   class class class wbr 4956   Or wor 5353   × cxp 5433  dom cdm 5435  ran crn 5436   ∘ ccom 5439  ⟶wf 6213  ‘cfv 6217  (class class class)co 7007   ↑𝑚 cmap 8247  supcsup 8740  ℂcc 10370  ℝcr 10371  0cc0 10372  1c1 10373   + caddc 10375   · cmul 10377  +∞cpnf 10507  ℝ*cxr 10509   < clt 10510   ≤ cle 10511   − cmin 10706   / cdiv 11134  ℕcn 11475  2c2 11529  3c3 11530  ℝ+crp 12228  (,)cioo 12577  [,)cico 12579  seqcseq 13207  abscabs 14415  topGenctg 16528  Topctop 21173  TopBasesctb 21225  Clsdccld 21296  vol*covol 23734  volcvol 23735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-inf2 8939  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449  ax-pre-sup 10450 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-fal 1533  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-iin 4822  df-disj 4925  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-se 5395  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-isom 6226  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-of 7258  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-2o 7945  df-oadd 7948  df-omul 7949  df-er 8130  df-map 8249  df-pm 8250  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-fi 8711  df-sup 8742  df-inf 8743  df-oi 8810  df-dju 9165  df-card 9203  df-acn 9206  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-div 11135  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-n0 11735  df-z 11819  df-uz 12083  df-q 12187  df-rp 12229  df-xneg 12346  df-xadd 12347  df-xmul 12348  df-ioo 12581  df-ico 12583  df-icc 12584  df-fz 12732  df-fzo 12873  df-fl 13000  df-seq 13208  df-exp 13268  df-hash 13529  df-cj 14280  df-re 14281  df-im 14282  df-sqrt 14416  df-abs 14417  df-clim 14667  df-rlim 14668  df-sum 14865  df-rest 16513  df-topgen 16534  df-psmet 20207  df-xmet 20208  df-met 20209  df-bl 20210  df-mopn 20211  df-top 21174  df-topon 21191  df-bases 21226  df-cld 21299  df-cmp 21667  df-ovol 23736  df-vol 23737 This theorem is referenced by:  ismblfin  34410
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