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Theorem mblfinlem3 33781
Description: The difference between two sets measurable by the criterion in ismblfin 33783 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 10320 . . 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 3888 . . . 4 (𝐴𝐵) ⊆ 𝐴
4 ovolsscl 23474 . . . 4 (((𝐴𝐵) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
53, 4mp3an1 1559 . . 3 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝐵)) ∈ ℝ)
653ad2ant1 1127 . 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 3354 . . . . . 6 𝑢 ∈ V
8 eqeq1 2775 . . . . . . . 8 (𝑦 = 𝑢 → (𝑦 = (vol‘𝑏) ↔ 𝑢 = (vol‘𝑏)))
98anbi2d 606 . . . . . . 7 (𝑦 = 𝑢 → ((𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))))
109rexbidv 3200 . . . . . 6 (𝑦 = 𝑢 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))))
117, 10elab 3501 . . . . 5 (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))} ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))
12 simprl 746 . . . . . . . . 9 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))) → 𝑏 ⊆ (𝐴𝐵))
13 ssdifss 3892 . . . . . . . . 9 (𝐴 ⊆ ℝ → (𝐴𝐵) ⊆ ℝ)
14 ovolss 23473 . . . . . . . . 9 ((𝑏 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)))
1512, 13, 14syl2anr 576 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)))
16 uniretop 22786 . . . . . . . . . . . . 13 ℝ = (topGen‘ran (,))
1716cldss 21054 . . . . . . . . . . . 12 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ)
18 ovolcl 23466 . . . . . . . . . . . 12 (𝑏 ⊆ ℝ → (vol*‘𝑏) ∈ ℝ*)
1917, 18syl 17 . . . . . . . . . . 11 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (vol*‘𝑏) ∈ ℝ*)
20 ovolcl 23466 . . . . . . . . . . . 12 ((𝐴𝐵) ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
2113, 20syl 17 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → (vol*‘(𝐴𝐵)) ∈ ℝ*)
22 xrlenlt 10305 . . . . . . . . . . 11 (((vol*‘𝑏) ∈ ℝ* ∧ (vol*‘(𝐴𝐵)) ∈ ℝ*) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
2319, 21, 22syl2anr 576 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑏 ∈ (Clsd‘(topGen‘ran (,)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
2423adantrr 688 . . . . . . . . 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 4007 . . . . . . . . . . . . . . . . 17 (𝑏 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
2717, 26sylib 208 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
28 rembl 23528 . . . . . . . . . . . . . . . . 17 ℝ ∈ dom vol
2916cldopn 21056 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran (,)))
30 opnmbl 23590 . . . . . . . . . . . . . . . . . 18 ((ℝ ∖ 𝑏) ∈ (topGen‘ran (,)) → (ℝ ∖ 𝑏) ∈ dom vol)
3129, 30syl 17 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ dom vol)
32 difmbl 23531 . . . . . . . . . . . . . . . . 17 ((ℝ ∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
3328, 31, 32sylancr 567 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
3427, 33eqeltrrd 2851 . . . . . . . . . . . . . . 15 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol)
35 mblvol 23518 . . . . . . . . . . . . . . 15 (𝑏 ∈ dom vol → (vol‘𝑏) = (vol*‘𝑏))
3634, 35syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑏) = (vol*‘𝑏))
3725, 36sylan9eqr 2827 . . . . . . . . . . . . 13 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → 𝑢 = (vol*‘𝑏))
3837breq2d 4798 . . . . . . . . . . . 12 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → ((vol*‘(𝐴𝐵)) < 𝑢 ↔ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
3938notbid 307 . . . . . . . . . . 11 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → (¬ (vol*‘(𝐴𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
4039adantrl 687 . . . . . . . . . 10 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))) → (¬ (vol*‘(𝐴𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
4140adantl 467 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → (¬ (vol*‘(𝐴𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴𝐵)) < (vol*‘𝑏)))
4224, 41bitr4d 271 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴𝐵)) ↔ ¬ (vol*‘(𝐴𝐵)) < 𝑢))
4315, 42mpbid 222 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
4443rexlimdvaa 3180 . . . . . 6 (𝐴 ⊆ ℝ → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏)) → ¬ (vol*‘(𝐴𝐵)) < 𝑢))
4544imp 393 . . . . 5 ((𝐴 ⊆ ℝ ∧ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑢 = (vol‘𝑏))) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
4611, 45sylan2b 573 . . . 4 ((𝐴 ⊆ ℝ ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
4746adantlr 686 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴𝐵)) < 𝑢)
48473ad2antl1 1200 . 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 744 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (vol*‘𝐴) ∈ ℝ)
50 resubcl 10547 . . . . . . . . . . . . . . . . . . 19 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ 𝑢 ∈ ℝ) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
5150adantrr 688 . . . . . . . . . . . . . . . . . 18 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
52 posdif 10723 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (𝑢 < (vol*‘(𝐴𝐵)) ↔ 0 < ((vol*‘(𝐴𝐵)) − 𝑢)))
5352ancoms 455 . . . . . . . . . . . . . . . . . . . 20 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘(𝐴𝐵)) ↔ 0 < ((vol*‘(𝐴𝐵)) − 𝑢)))
5453biimpd 219 . . . . . . . . . . . . . . . . . . 19 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘(𝐴𝐵)) → 0 < ((vol*‘(𝐴𝐵)) − 𝑢)))
5554impr 442 . . . . . . . . . . . . . . . . . 18 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → 0 < ((vol*‘(𝐴𝐵)) − 𝑢))
5651, 55elrpd 12072 . . . . . . . . . . . . . . . . 17 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ+)
57 3nn 11388 . . . . . . . . . . . . . . . . . 18 3 ∈ ℕ
58 nnrp 12045 . . . . . . . . . . . . . . . . . 18 (3 ∈ ℕ → 3 ∈ ℝ+)
5957, 58ax-mp 5 . . . . . . . . . . . . . . . . 17 3 ∈ ℝ+
60 rpdivcl 12059 . . . . . . . . . . . . . . . . 17 ((((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ+ ∧ 3 ∈ ℝ+) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
6156, 59, 60sylancl 566 . . . . . . . . . . . . . . . 16 (((vol*‘(𝐴𝐵)) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
625, 61sylan 561 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
6349, 62ltsubrpd 12107 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol*‘𝐴))
6463adantr 466 . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 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 4812 . . . . . . . . . . . 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 10229 . . . . . . . . . . . . . . . . . 18 ℝ ∈ V
6867ssex 4936 . . . . . . . . . . . . . . . . 17 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
6968adantr 466 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → 𝐴 ∈ V)
70 sseq1 3775 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → (𝑣 ⊆ ℝ ↔ 𝐴 ⊆ ℝ))
71 fveq2 6332 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → (vol*‘𝑣) = (vol*‘𝐴))
7271eleq1d 2835 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
7370, 72anbi12d 608 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)))
74 sseq2 3776 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = 𝐴 → (𝑏𝑣𝑏𝐴))
7574anbi1d 607 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝐴 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝐴𝑦 = (vol‘𝑏))))
7675rexbidv 3200 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝐴 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))))
7776abbidv 2890 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} = {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))})
7877sseq1d 3781 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ⊆ ℝ))
7977neeq1d 3002 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ≠ ∅))
8077raleqdv 3293 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑧𝑥))
8180rexbidv 3200 . . . . . . . . . . . . . . . . . . 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 1547 . . . . . . . . . . . . . . . . . 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 333 . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑣𝑦 = (vol‘𝑏)) → 𝑦 = (vol‘𝑏))
8584, 36sylan9eqr 2827 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏))) → 𝑦 = (vol*‘𝑏))
8685adantl 467 . . . . . . . . . . . . . . . . . . . . 21 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏)))) → 𝑦 = (vol*‘𝑏))
87 simprl 746 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏))) → 𝑏𝑣)
88 ovolsscl 23474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏𝑣𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → (vol*‘𝑏) ∈ ℝ)
89883expb 1113 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑣 ∧ (𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ)) → (vol*‘𝑏) ∈ ℝ)
9089ancoms 455 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ 𝑏𝑣) → (vol*‘𝑏) ∈ ℝ)
9187, 90sylan2 572 . . . . . . . . . . . . . . . . . . . . 21 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏)))) → (vol*‘𝑏) ∈ ℝ)
9286, 91eqeltrd 2850 . . . . . . . . . . . . . . . . . . . 20 (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑦 = (vol‘𝑏)))) → 𝑦 ∈ ℝ)
9392rexlimdvaa 3180 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) → 𝑦 ∈ ℝ))
9493abssdv 3825 . . . . . . . . . . . . . . . . . 18 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ)
95 retop 22785 . . . . . . . . . . . . . . . . . . . . . 22 (topGen‘ran (,)) ∈ Top
96 0cld 21063 . . . . . . . . . . . . . . . . . . . . . 22 ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
9795, 96ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ (Clsd‘(topGen‘ran (,)))
98 0ss 4116 . . . . . . . . . . . . . . . . . . . . . 22 ∅ ⊆ 𝑣
99 0mbl 23527 . . . . . . . . . . . . . . . . . . . . . . . 24 ∅ ∈ dom vol
100 mblvol 23518 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
10199, 100ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (vol‘∅) = (vol*‘∅)
102 ovol0 23481 . . . . . . . . . . . . . . . . . . . . . . 23 (vol*‘∅) = 0
103101, 102eqtr2i 2794 . . . . . . . . . . . . . . . . . . . . . 22 0 = (vol‘∅)
10498, 103pm3.2i 447 . . . . . . . . . . . . . . . . . . . . 21 (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅))
105 sseq1 3775 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ∅ → (𝑏𝑣 ↔ ∅ ⊆ 𝑣))
106 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ∅ → (vol‘𝑏) = (vol‘∅))
107106eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ∅ → (0 = (vol‘𝑏) ↔ 0 = (vol‘∅)))
108105, 107anbi12d 608 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = ∅ → ((𝑏𝑣 ∧ 0 = (vol‘𝑏)) ↔ (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅))))
109108rspcev 3460 . . . . . . . . . . . . . . . . . . . . 21 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏)))
11097, 104, 109mp2an 664 . . . . . . . . . . . . . . . . . . . 20 𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏))
111 c0ex 10236 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
112 eqeq1 2775 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 0 → (𝑦 = (vol‘𝑏) ↔ 0 = (vol‘𝑏)))
113112anbi2d 606 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 0 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝑣 ∧ 0 = (vol‘𝑏))))
114113rexbidv 3200 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 0 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣 ∧ 0 = (vol‘𝑏))))
115111, 114spcev 3451 . . . . . . . . . . . . . . . . . . . 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 4101 . . . . . . . . . . . . . . . . . . . 20 ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ↔ ∃𝑦𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)))
118117biimpri 218 . . . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 = (vol‘𝑏))
121120, 36sylan9eqr 2827 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏))) → 𝑧 = (vol*‘𝑏))
122121adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑣 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏)))) → 𝑧 = (vol*‘𝑏))
123 simprl 746 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏))) → 𝑏𝑣)
124 ovolss 23473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑏𝑣𝑣 ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘𝑣))
125124ancoms 455 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 ⊆ ℝ ∧ 𝑏𝑣) → (vol*‘𝑏) ≤ (vol*‘𝑣))
126123, 125sylan2 572 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑣 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘𝑣))
127122, 126eqbrtrd 4808 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝑣𝑧 = (vol‘𝑏)))) → 𝑧 ≤ (vol*‘𝑣))
128127rexlimdvaa 3180 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ⊆ ℝ → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣)))
129128alrimiv 2007 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ⊆ ℝ → ∀𝑧(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣)))
130 eqeq1 2775 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑧 → (𝑦 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑏)))
131130anbi2d 606 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑧 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝑣𝑧 = (vol‘𝑏))))
132131rexbidv 3200 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑧 = (vol‘𝑏))))
133132ralab 3519 . . . . . . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ⊆ ℝ → ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣))
135 breq2 4790 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (vol*‘𝑣) → (𝑧𝑥𝑧 ≤ (vol*‘𝑣)))
136135ralbidv 3135 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (vol*‘𝑣) → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣)))
137136rspcev 3460 . . . . . . . . . . . . . . . . . . . 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)
138134, 137sylan2 572 . . . . . . . . . . . . . . . . . . 19 (((vol*‘𝑣) ∈ ℝ ∧ 𝑣 ⊆ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥)
139138ancoms 455 . . . . . . . . . . . . . . . . . 18 ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥)
14094, 119, 1393jca 1122 . . . . . . . . . . . . . . . . 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))
14183, 140vtoclg 3417 . . . . . . . . . . . . . . . 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)))
14269, 141mpcom 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))
143142adantr 466 . . . . . . . . . . . . . 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))
14462rpred 12075 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ)
14549, 144resubcld 10660 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
146 suprlub 11189 . . . . . . . . . . . . . 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))
147143, 145, 146syl2anc 565 . . . . . . . . . . . . 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))
148147adantr 466 . . . . . . . . . . . 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))
14966, 148mpbid 222 . . . . . . . . . . 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)
150 eqeq1 2775 . . . . . . . . . . . . . . 15 (𝑦 = 𝑣 → (𝑦 = (vol‘𝑏) ↔ 𝑣 = (vol‘𝑏)))
151150anbi2d 606 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → ((𝑏𝐴𝑦 = (vol‘𝑏)) ↔ (𝑏𝐴𝑣 = (vol‘𝑏))))
152151rexbidv 3200 . . . . . . . . . . . . 13 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏))))
153152rexab 3521 . . . . . . . . . . . 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))
154 breq2 4790 . . . . . . . . . . . . . . . . 17 (𝑣 = (vol‘𝑏) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
155154ad2antll 700 . . . . . . . . . . . . . . . 16 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑣 = (vol‘𝑏))) → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
156 sseq1 3775 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑏 → (𝑠𝐴𝑏𝐴))
157 fveq2 6332 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = 𝑏 → (vol‘𝑠) = (vol‘𝑏))
158157breq2d 4798 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑏 → (((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠) ↔ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
159156, 158anbi12d 608 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑏 → ((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ↔ (𝑏𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏))))
160159rspcev 3460 . . . . . . . . . . . . . . . . . 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)))
161160expr 444 . . . . . . . . . . . . . . . . 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))))
162161adantrr 688 . . . . . . . . . . . . . . . 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))))
163155, 162sylbid 230 . . . . . . . . . . . . . . 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))))
164163rexlimiva 3176 . . . . . . . . . . . . . 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))))
165164imp 393 . . . . . . . . . . . . 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)))
166165exlimiv 2010 . . . . . . . . . . . 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)))
167153, 166sylbi 207 . . . . . . . . . . 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)))
168149, 167syl 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)))
169168ex 397 . . . . . . . . 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))))
170169adantlr 686 . . . . . . . 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))))
171 simplrr 755 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (vol*‘𝐵) ∈ ℝ)
17262adantlr 686 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ+)
173171, 172ltsubrpd 12107 . . . . . . . . . . . . 13 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol*‘𝐵))
174173adantr 466 . . . . . . . . . . . 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))
175 simpr 471 . . . . . . . . . . . 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, < ))
176174, 175breqtrd 4812 . . . . . . . . . . 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, < ))
17767ssex 4936 . . . . . . . . . . . . . . . 16 (𝐵 ⊆ ℝ → 𝐵 ∈ V)
178177adantr 466 . . . . . . . . . . . . . . 15 ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → 𝐵 ∈ V)
179 sseq1 3775 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → (𝑣 ⊆ ℝ ↔ 𝐵 ⊆ ℝ))
180 fveq2 6332 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → (vol*‘𝑣) = (vol*‘𝐵))
181180eleq1d 2835 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐵) ∈ ℝ))
182179, 181anbi12d 608 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
183 sseq2 3776 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝐵 → (𝑏𝑣𝑏𝐵))
184183anbi1d 607 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝐵 → ((𝑏𝑣𝑦 = (vol‘𝑏)) ↔ (𝑏𝐵𝑦 = (vol‘𝑏))))
185184rexbidv 3200 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐵 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))))
186185abbidv 2890 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} = {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))})
187186sseq1d 3781 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ⊆ ℝ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ⊆ ℝ))
188186neeq1d 3002 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))} ≠ ∅ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))} ≠ ∅))
189186raleqdv 3293 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝑣𝑦 = (vol‘𝑏))}𝑧𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏))}𝑧𝑥))
190189rexbidv 3200 . . . . . . . . . . . . . . . . . 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))
191187, 188, 1903anbi123d 1547 . . . . . . . . . . . . . . . . 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)))
192182, 191imbi12d 333 . . . . . . . . . . . . . . . 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))))
193192, 140vtoclg 3417 . . . . . . . . . . . . . . 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)))
194178, 193mpcom 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))
195194ad2antlr 698 . . . . . . . . . . . . 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))
196144adantlr 686 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ)
197171, 196resubcld 10660 . . . . . . . . . . . . 13 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
198 suprlub 11189 . . . . . . . . . . . . 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))
199195, 197, 198syl2anc 565 . . . . . . . . . . . 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))
200199adantr 466 . . . . . . . . . . 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))
201176, 200mpbid 222 . . . . . . . . . 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)
202150anbi2d 606 . . . . . . . . . . . . 13 (𝑦 = 𝑣 → ((𝑏𝐵𝑦 = (vol‘𝑏)) ↔ (𝑏𝐵𝑣 = (vol‘𝑏))))
203202rexbidv 3200 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐵𝑣 = (vol‘𝑏))))
204203rexab 3521 . . . . . . . . . . 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))
205 breq2 4790 . . . . . . . . . . . . . . . 16 (𝑣 = (vol‘𝑏) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
206205ad2antll 700 . . . . . . . . . . . . . . 15 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐵𝑣 = (vol‘𝑏))) → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
207 sseq1 3775 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑏 → (𝑤𝐵𝑏𝐵))
208 fveq2 6332 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑏 → (vol‘𝑤) = (vol‘𝑏))
209208breq2d 4798 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑏 → (((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤) ↔ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))
210207, 209anbi12d 608 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑏 → ((𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤)) ↔ (𝑏𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑏))))
211210rspcev 3460 . . . . . . . . . . . . . . . . 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)))
212211expr 444 . . . . . . . . . . . . . . . 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))))
213212adantrr 688 . . . . . . . . . . . . . . 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))))
214206, 213sylbid 230 . . . . . . . . . . . . . 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))))
215214rexlimiva 3176 . . . . . . . . . . . . 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))))
216215imp 393 . . . . . . . . . . . 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)))
217216exlimiv 2010 . . . . . . . . . . 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)))
218204, 217sylbi 207 . . . . . . . . . 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)))
219201, 218syl 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)))
220219ex 397 . . . . . . . 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))))
221170, 220anim12d 588 . . . . . . 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)))))
222 reeanv 3255 . . . . . . 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))))
223221, 222syl6ibr 242 . . . . . 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)))))
224 eqid 2771 . . . . . . . . . . . . . 14 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
225224ovolgelb 23468 . . . . . . . . . . . . 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))))
2262253expa 1111 . . . . . . . . . . . 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))))
22762, 226sylan2 572 . . . . . . . . . . 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))))
228227ancoms 455 . . . . . . . . . 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))))
229228an32s 623 . . . . . . . . 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))))
230 elmapi 8031 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
231 ssid 3773 . . . . . . . . . . . . . . 15 ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)
232224ovollb 23467 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
233231, 232mpan2 663 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
234233adantl 467 . . . . . . . . . . . . 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*, < ))
235 eqid 2771 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
236235, 224ovolsf 23460 . . . . . . . . . . . . . . 15 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
237 frn 6193 . . . . . . . . . . . . . . . 16 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
238 icossxr 12463 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℝ*
239237, 238syl6ss 3764 . . . . . . . . . . . . . . 15 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
240 supxrcl 12350 . . . . . . . . . . . . . . 15 (ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
241236, 239, 2403syl 18 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
242 simpr 471 . . . . . . . . . . . . . . . . 17 ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝐵) ∈ ℝ)
243 readdcl 10221 . . . . . . . . . . . . . . . . 17 (((vol*‘𝐵) ∈ ℝ ∧ (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℝ) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
244242, 144, 243syl2anr 576 . . . . . . . . . . . . . . . 16 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
245244rexrd 10291 . . . . . . . . . . . . . . 15 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ*)
246245an32s 623 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ*)
247 rncoss 5524 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ⊆ ran (,)
248247unissi 4597 . . . . . . . . . . . . . . . . 17 ran ((,) ∘ 𝑓) ⊆ ran (,)
249 unirnioo 12479 . . . . . . . . . . . . . . . . 17 ℝ = ran (,)
250248, 249sseqtr4i 3787 . . . . . . . . . . . . . . . 16 ran ((,) ∘ 𝑓) ⊆ ℝ
251 ovolcl 23466 . . . . . . . . . . . . . . . 16 ( ran ((,) ∘ 𝑓) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*)
252250, 251ax-mp 5 . . . . . . . . . . . . . . 15 (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*
253 xrletr 12194 . . . . . . . . . . . . . . 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))))
254252, 253mp3an1 1559 . . . . . . . . . . . . . 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))))
255241, 246, 254syl2anr 576 . . . . . . . . . . . . 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))))
256234, 255mpand 667 . . . . . . . . . . . 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))))
257230, 256sylan2 572 . . . . . . . . . . 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))))
258257anim2d 591 . . . . . . . . . 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)))))
259258reximdva 3165 . . . . . . . . 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)))))
260229, 259mpd 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))))
261 rexex 3150 . . . . . . . 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))))
262260, 261syl 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))))
26316cldss 21054 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ⊆ ℝ)
264 indif2 4019 . . . . . . . . . . . . . . . . . 18 (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) = ((𝑠 ∩ ℝ) ∖ ran ((,) ∘ 𝑓))
265 df-ss 3737 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ⊆ ℝ ↔ (𝑠 ∩ ℝ) = 𝑠)
266265biimpi 206 . . . . . . . . . . . . . . . . . . 19 (𝑠 ⊆ ℝ → (𝑠 ∩ ℝ) = 𝑠)
267266difeq1d 3878 . . . . . . . . . . . . . . . . . 18 (𝑠 ⊆ ℝ → ((𝑠 ∩ ℝ) ∖ ran ((,) ∘ 𝑓)) = (𝑠 ran ((,) ∘ 𝑓)))
268264, 267syl5eq 2817 . . . . . . . . . . . . . . . . 17 (𝑠 ⊆ ℝ → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) = (𝑠 ran ((,) ∘ 𝑓)))
269263, 268syl 17 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) = (𝑠 ran ((,) ∘ 𝑓)))
270 retopbas 22784 . . . . . . . . . . . . . . . . . . . . 21 ran (,) ∈ TopBases
271 bastg 20991 . . . . . . . . . . . . . . . . . . . . 21 (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
272270, 271ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ran (,) ⊆ (topGen‘ran (,))
273247, 272sstri 3761 . . . . . . . . . . . . . . . . . . 19 ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))
274 uniopn 20922 . . . . . . . . . . . . . . . . . . 19 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))) → ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)))
27595, 273, 274mp2an 664 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))
27616opncld 21058 . . . . . . . . . . . . . . . . . 18 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))) → (ℝ ∖ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
27795, 275, 276mp2an 664 . . . . . . . . . . . . . . . . 17 (ℝ ∖ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,)))
278 incld 21068 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (ℝ ∖ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) ∈ (Clsd‘(topGen‘ran (,))))
279277, 278mpan2 663 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑓))) ∈ (Clsd‘(topGen‘ran (,))))
280269, 279eqeltrrd 2851 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
281280adantr 466 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))))
282281ad2antlr 698 . . . . . . . . . . . . 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 (,))))
283 simprll 756 . . . . . . . . . . . . . 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)
284 simplll 750 . . . . . . . . . . . . . 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))
285283, 284ssdif2d 3900 . . . . . . . . . . . . 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))
286 fveq2 6332 . . . . . . . . . . . . . . . . 17 ((𝑠 ran ((,) ∘ 𝑓)) = 𝑏 → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))
287286eqcoms 2779 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))
288287biantrud 515 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → (𝑏 ⊆ (𝐴𝐵) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))))
289 sseq1 3775 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → (𝑏 ⊆ (𝐴𝐵) ↔ (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)))
290288, 289bitr3d 270 . . . . . . . . . . . . . 14 (𝑏 = (𝑠 ran ((,) ∘ 𝑓)) → ((𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)) ↔ (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)))
291290rspcev 3460 . . . . . . . . . . . . 13 (((𝑠 ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)))
292282, 285, 291syl2anc 565 . . . . . . . . . . . 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)))
293292adantlll 689 . . . . . . . . . . 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)))
294 difss 3888 . . . . . . . . . . . . . . . 16 ((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) ⊆ (𝐴𝐵)
295294, 3sstri 3761 . . . . . . . . . . . . . . 15 ((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴
296 ovolsscl 23474 . . . . . . . . . . . . . . 15 ((((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
297295, 296mp3an1 1559 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
298297ad5antr 708 . . . . . . . . . . . . 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)
2995ad5antr 708 . . . . . . . . . . . . 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)
300 simpl 468 . . . . . . . . . . . . . 14 ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵))) → 𝑢 ∈ ℝ)
301300ad4antlr 706 . . . . . . . . . . . . 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)
302 difdif2 4033 . . . . . . . . . . . . . . 15 ((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))) = (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))
303302fveq2i 6335 . . . . . . . . . . . . . 14 (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓)))) = (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))))
304 difss 3888 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝐵)
305304, 3sstri 3761 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∖ 𝑠) ⊆ 𝐴
306 inss1 3981 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵)
307306, 3sstri 3761 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ 𝐴
308305, 307unssi 3939 . . . . . . . . . . . . . . . . 17 (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ 𝐴
309 ovolsscl 23474 . . . . . . . . . . . . . . . . 17 (((((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ∈ ℝ)
310308, 309mp3an1 1559 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)))) ∈ ℝ)
311310ad5antr 708 . . . . . . . . . . . . . . 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)
312 difss 3888 . . . . . . . . . . . . . . . . . 18 (𝐴𝑠) ⊆ 𝐴
313 ovolsscl 23474 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑠) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
314312, 313mp3an1 1559 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
315314ad5antr 708 . . . . . . . . . . . . . . . 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)
316171, 196readdcld 10271 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) ∈ ℝ)
317316, 252jctil 503 . . . . . . . . . . . . . . . . . . 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))
318 simpr 471 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3))) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))
319 ovolge0 23469 . . . . . . . . . . . . . . . . . . . . 21 ( ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤ (vol*‘ ran ((,) ∘ 𝑓)))
320250, 319ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 0 ≤ (vol*‘ ran ((,) ∘ 𝑓))
321318, 320jctil 503 . . . . . . . . . . . . . . . . . . 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))))
322 xrrege0 12210 . . . . . . . . . . . . . . . . . . 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)
323317, 321, 322syl2an 575 . . . . . . . . . . . . . . . . . 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)
324 difss 3888 . . . . . . . . . . . . . . . . . . 19 ( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ran ((,) ∘ 𝑓)
325 ovolsscl 23474 . . . . . . . . . . . . . . . . . . 19 ((( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)
326324, 250, 325mp3an12 1562 . . . . . . . . . . . . . . . . . 18 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)
327323, 326syl 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)
328327ad2antrr 697 . . . . . . . . . . . . . . . 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)
329315, 328readdcld 10271 . . . . . . . . . . . . . . 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)
3305, 50sylan 561 . . . . . . . . . . . . . . . . . 18 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑢 ∈ ℝ) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
331330adantrr 688 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
332331adantlr 686 . . . . . . . . . . . . . . . 16 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℝ)
333332ad3antrrr 701 . . . . . . . . . . . . . . 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)
334 ssdifss 3892 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ⊆ ℝ → (𝐴𝑠) ⊆ ℝ)
335324, 250sstri 3761 . . . . . . . . . . . . . . . . . . . . 21 ( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ
336 unss 3938 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑠) ⊆ ℝ ∧ ( ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ) ↔ ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ)
337334, 335, 336sylanblc 569 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ⊆ ℝ → ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ)
338 ovolcl 23466 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ*)
339337, 338syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐴 ⊆ ℝ → (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ*)
340339ad4antr 704 . . . . . . . . . . . . . . . . . 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*)
341314ad3antrrr 701 . . . . . . . . . . . . . . . . . . 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)
342341, 327readdcld 10271 . . . . . . . . . . . . . . . . . 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)
343 ovolge0 23469 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) ⊆ ℝ → 0 ≤ (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))))
344337, 343syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤))))
345344ad4antr 704 . . . . . . . . . . . . . . . . . 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))))
346334adantr 466 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴𝑠) ⊆ ℝ)
347346, 314jca 495 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((𝐴𝑠) ⊆ ℝ ∧ (vol*‘(𝐴𝑠)) ∈ ℝ))
348347ad3antrrr 701 . . . . . . . . . . . . . . . . . . 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))
349327, 335jctil 503 . . . . . . . . . . . . . . . . . . 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))
350 ovolun 23487 . . . . . . . . . . . . . . . . . . 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))))
351348, 349, 350syl2anc 565 . . . . . . . . . . . . . . . . . 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))))
352 xrrege0 12210 . . . . . . . . . . . . . . . . . 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)
353340, 342, 345, 351, 352syl22anc 1477 . . . . . . . . . . . . . . . . 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)
354353ad2antrr 697 . . . . . . . . . . . . . . . 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)
355 ssdif 3896 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝑠))
3563, 355ax-mp 5 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝑠)
357 incom 3956 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) = ( ran ((,) ∘ 𝑓) ∩ (𝐴𝐵))
358 indif2 4019 . . . . . . . . . . . . . . . . . . . 20 ( ran ((,) ∘ 𝑓) ∩ (𝐴𝐵)) = (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∖ 𝐵)
359357, 358eqtri 2793 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) = (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∖ 𝐵)
360 inss1 3981 . . . . . . . . . . . . . . . . . . . . 21 ( ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ran ((,) ∘ 𝑓)
361360a1i 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))
362 simprrl 758 . . . . . . . . . . . . . . . . . . . 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)
363361, 362ssdif2d 3900 . . . . . . . . . . . . . . . . . . 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))
364359, 363syl5eqss 3798 . . . . . . . . . . . . . . . . . 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))
365 unss12 3936 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝐵) ∖ 𝑠) ⊆ (𝐴𝑠) ∧ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓)) ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝑤)) → (((𝐴𝐵) ∖ 𝑠) ∪ ((𝐴𝐵) ∩ ran ((,) ∘ 𝑓))) ⊆ ((𝐴𝑠) ∪ ( ran ((,) ∘ 𝑓) ∖ 𝑤)))
366356, 364, 365sylancr 567 . . . . . . . . . . . . . . . . 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)))
367337ad6antr 712 . . . . . . . . . . . . . . . . 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)
368 ovolss 23473 . . . . . . . . . . . . . . . . 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))))
369366, 367, 368syl2anc 565 . . . . . . . . . . . . . . . 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))))
370334ad6antr 712 . . . . . . . . . . . . . . . . 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)
371328, 335jctil 503 . . . . . . . . . . . . . . . . 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))
372370, 315, 371, 350syl21anc 1475 . . . . . . . . . . . . . . . 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))))
373311, 354, 329, 369, 372letrd 10396 . . . . . . . . . . . . . . 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))))
374196ad3antrrr 701 . . . . . . . . . . . . . . . . 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)
375196, 196readdcld 10271 . . . . . . . . . . . . . . . . . 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)
376375ad3antrrr 701 . . . . . . . . . . . . . . . . 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)
377 eleq1w 2833 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = 𝑠 → (𝑏 ∈ dom vol ↔ 𝑠 ∈ dom vol))
378377, 34vtoclga 3423 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ∈ dom vol)
379 mblvol 23518 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ dom vol → (vol‘𝑠) = (vol*‘𝑠))
380378, 379syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑠) = (vol*‘𝑠))
381380adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑠) = (vol*‘𝑠))
382 sseqin2 3968 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠𝐴 ↔ (𝐴𝑠) = 𝑠)
383382biimpi 206 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑠𝐴 → (𝐴𝑠) = 𝑠)
384383eqcomd 2777 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠𝐴𝑠 = (𝐴𝑠))
385384fveq2d 6336 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠𝐴 → (vol*‘𝑠) = (vol*‘(𝐴𝑠)))
386385ad2antrr 697 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → (vol*‘𝑠) = (vol*‘(𝐴𝑠)))
387381, 386sylan9eq 2825 . . . . . . . . . . . . . . . . . . . . 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)))
388387oveq2d 6809 . . . . . . . . . . . . . . . . . . . 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))))
389388adantll 685 . . . . . . . . . . . . . . . . . . 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))))
390378adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑠 ∈ dom vol)
391 simplll 750 . . . . . . . . . . . . . . . . . . . . . 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))
392 mblsplit 23520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝐴) = ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))))
393392eqcomd 2777 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))) = (vol*‘𝐴))
3943933expb 1113 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ dom vol ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴𝑠))) = (vol*‘𝐴))
395390, 391, 394syl2anr 576 . . . . . . . . . . . . . . . . . . . . 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))
396395adantr 466 . . . . . . . . . . . . . . . . . . . 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))
397 simp-6r 770 . . . . . . . . . . . . . . . . . . . . . 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)
398397recnd 10270 . . . . . . . . . . . . . . . . . . . . 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)
399 inss1 3981 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑠) ⊆ 𝐴
400 ovolsscl 23474 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴𝑠) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
401399, 400mp3an1 1559 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
402401recnd 10270 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℂ)
403402ad5antr 708 . . . . . . . . . . . . . . . . . . . . 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)
404314recnd 10270 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℂ)
405404ad5antr 708 . . . . . . . . . . . . . . . . . . . . 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)
406398, 403, 405subaddd 10612 . . . . . . . . . . . . . . . . . . . 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)))
407396, 406mpbird 247 . . . . . . . . . . . . . . . . . . 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)))
408389, 407eqtrd 2805 . . . . . . . . . . . . . . . . . 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)))
409381ad2antlr 698 . . . . . . . . . . . . . . . . . . . 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))
410 simpll 742 . . . . . . . . . . . . . . . . . . . . 21 (((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → 𝑠𝐴)
411 simp-4l 760 . . . . . . . . . . . . . . . . . . . . 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))
412 ovolsscl 23474 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝑠) ∈ ℝ)
4134123expb 1113 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠𝐴 ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → (vol*‘𝑠) ∈ ℝ)
414410, 411, 413syl2anr 576 . . . . . . . . . . . . . . . . . . . 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)
415409, 414eqeltrd 2850 . . . . . . . . . . . . . . . . . . 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)
416 simprlr 757 . . . . . . . . . . . . . . . . . . 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))
417397, 374, 415, 416ltsub23d 10834 . . . . . . . . . . . . . . . . . 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))
418408, 417eqbrtrrd 4810 . . . . . . . . . . . . . . . . 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))
419323recnd 10270 . . . . . . . . . . . . . . . . . . . . 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)
420419ad2antrr 697 . . . . . . . . . . . . . . . . . . . 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)
421242ad5antlr 710 . . . . . . . . . . . . . . . . . . . . 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)
422421recnd 10270 . . . . . . . . . . . . . . . . . . . 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)
423 eleq1w 2833 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 = 𝑤 → (𝑏 ∈ dom vol ↔ 𝑤 ∈ dom vol))
424423, 34vtoclga 3423 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ∈ (Clsd‘(topGen‘ran (,))) → 𝑤 ∈ dom vol)
425 mblvol 23518 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ∈ dom vol → (vol‘𝑤) = (vol*‘𝑤))
426424, 425syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑤) = (vol*‘𝑤))
427426adantl 467 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑤) = (vol*‘𝑤))
428427ad2antlr 698 . . . . . . . . . . . . . . . . . . . . . 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))
429 simprl 746 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑠𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → 𝑤𝐵)
430 simp-4r 762 . . . . . . . . . . . . . . . . . . . . . . 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))
431 ovolsscl 23474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤𝐵𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘𝑤) ∈ ℝ)
4324313expb 1113 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑤𝐵 ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝑤) ∈ ℝ)
433429, 430, 432syl2anr 576 . . . . . . . . . . . . . . . . . . . . . 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)
434428, 433eqeltrd 2850 . . . . . . . . . . . . . . . . . . . . 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)
435434recnd 10270 . . . . . . . . . . . . . . . . . . . 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)
436420, 422, 435npncand 10618 . . . . . . . . . . . . . . . . . . 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)))
437 simplrl 754 . . . . . . . . . . . . . . . . . . . . . . . 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))
438429, 437sylan9ssr 3766 . . . . . . . . . . . . . . . . . . . . . . 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))
439 sseqin2 3968 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ran ((,) ∘ 𝑓) ↔ ( ran ((,) ∘ 𝑓) ∩ 𝑤) = 𝑤)
440438, 439sylib 208 . . . . . . . . . . . . . . . . . . . . . 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)
441440fveq2d 6336 . . . . . . . . . . . . . . . . . . . . 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))
442428, 441eqtr4d 2808 . . . . . . . . . . . . . . . . . . . 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)))
443442oveq2d 6809 . . . . . . . . . . . . . . . . . . 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))))
444424adantl 467 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑤 ∈ dom vol)
445323, 250jctil 503 . . . . . . . . . . . . . . . . . . . . . 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))
446 mblsplit 23520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘ ran ((,) ∘ 𝑓)) = ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))))
447446eqcomd 2777 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑤 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘ ran ((,) ∘ 𝑓)))
4484473expb 1113 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑤 ∈ dom vol ∧ ( ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)) → ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘ ran ((,) ∘ 𝑓)))
449444, 445, 448syl2anr 576 . . . . . . . . . . . . . . . . . . . . 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)))
450449adantr 466 . . . . . . . . . . . . . . . . . . . 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)))
451 inss1 3981 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ran ((,) ∘ 𝑓)
452 ovolsscl 23474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ)
453451, 250, 452mp3an12 1562 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ)
454323, 453syl 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)
455454recnd 10270 . . . . . . . . . . . . . . . . . . . . . 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)
456327recnd 10270 . . . . . . . . . . . . . . . . . . . . . 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)
457419, 455, 456subaddd 10612 . . . . . . . . . . . . . . . . . . . . 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))))
458457ad2antrr 697 . . . . . . . . . . . . . . . . . . . 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))))
459450, 458mpbird 247 . . . . . . . . . . . . . . . . . . 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)))
460436, 443, 4593eqtrd 2809 . . . . . . . . . . . . . . . . . 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)))
461242ad3antlr 702 . . . . . . . . . . . . . . . . . . . . 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)
462323, 461resubcld 10660 . . . . . . . . . . . . . . . . . . . 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)
463462ad2antrr 697 . . . . . . . . . . . . . . . . . . 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)
464421, 434resubcld 10660 . . . . . . . . . . . . . . . . . . 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)
465 simprr 748 . . . . . . . . . . . . . . . . . . . . 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)))
466196adantr 466 . . . . . . . . . . . . . . . . . . . . . 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)
467323, 461, 466lesubadd2d 10828 . . . . . . . . . . . . . . . . . . . . 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))))
468465, 467mpbird 247 . . . . . . . . . . . . . . . . . . . 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))
469468ad2antrr 697 . . . . . . . . . . . . . . . . . . 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))
470 simprrr 759 . . . . . . . . . . . . . . . . . . . 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))
471421, 374, 434, 470ltsub23d 10834 . . . . . . . . . . . . . . . . . . 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))
472463, 464, 374, 374, 469, 471leltaddd 10851 . . . . . . . . . . . . . . . . . 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)))
473460, 472eqbrtrrd 4810 . . . . . . . . . . . . . . . . 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)))
474315, 328, 374, 376, 418, 473lt2addd 10852 . . . . . . . . . . . . . . . 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))))
475 df-3 11282 . . . . . . . . . . . . . . . . . . . . . 22 3 = (2 + 1)
476 2cn 11293 . . . . . . . . . . . . . . . . . . . . . . 23 2 ∈ ℂ
477 ax-1cn 10196 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℂ
478476, 477addcomi 10429 . . . . . . . . . . . . . . . . . . . . . 22 (2 + 1) = (1 + 2)
479475, 478eqtri 2793 . . . . . . . . . . . . . . . . . . . . 21 3 = (1 + 2)
480479oveq1i 6803 . . . . . . . . . . . . . . . . . . . 20 (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((1 + 2) · (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
48162rpcnd 12077 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℂ)
482 adddir 10233 . . . . . . . . . . . . . . . . . . . . . . 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))))
483477, 476, 482mp3an12 1562 . . . . . . . . . . . . . . . . . . . . . 22 ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) ∈ ℂ → ((1 + 2) · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((1 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) + (2 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3))))
484481, 483syl 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))))
485481mulid2d 10260 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (1 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = (((vol*‘(𝐴𝐵)) − 𝑢) / 3))
4864812timesd 11477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (2 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((((vol*‘(𝐴𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴𝐵)) − 𝑢) / 3)))
487485, 486oveq12d 6811 . . . . . . . . . . . . . . . . . . . . 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))))
488484, 487eqtrd 2805 . . . . . . . . . . . . . . . . . . . 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))))
489480, 488syl5eq 2817 . . . . . . . . . . . . . . . . . . 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))))
490331recnd 10270 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → ((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℂ)
491 3cn 11297 . . . . . . . . . . . . . . . . . . . . 21 3 ∈ ℂ
492 3ne0 11317 . . . . . . . . . . . . . . . . . . . . 21 3 ≠ 0
493 divcan2 10895 . . . . . . . . . . . . . . . . . . . . 21 ((((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴𝐵)) − 𝑢))
494491, 492, 493mp3an23 1564 . . . . . . . . . . . . . . . . . . . 20 (((vol*‘(𝐴𝐵)) − 𝑢) ∈ ℂ → (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴𝐵)) − 𝑢))
495490, 494syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴𝐵)))) → (3 · (((vol*‘(𝐴𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴𝐵)) − 𝑢))
496489, 495eqtr3d 2807 . . . . . . . . . . . . . . . . . 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))
497496adantlr 686 . . . . . . . . . . . . . . . . 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))
498497ad3antrrr 701 . . . . . . . . . . . . . . . 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))
499474, 498breqtrd 4812 . . . . . . . . . . . . . . 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))
500311, 329, 333, 373, 499lelttrd 10397 . . . . . . . . . . . . . 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))
501303, 500syl5eqbr 4821 . . . . . . . . . . . . 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))
502298, 299, 301, 501ltsub13d 10835 . . . . . . . . . . . 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))))))
503285adantlll 689 . . . . . . . . . . . . . . 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))
504 sseqin2 3968 . . . . . . . . . . . . . . 15 ((𝑠 ran ((,) ∘ 𝑓)) ⊆ (𝐴𝐵) ↔ ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) = (𝑠 ran ((,) ∘ 𝑓)))
505503, 504sylib 208 . . . . . . . . . . . . . 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)))
506505fveq2d 6336 . . . . . . . . . . . . 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))))
507 opnmbl 23590 . . . . . . . . . . . . . . . . . . 19 ( ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) → ran ((,) ∘ 𝑓) ∈ dom vol)
508275, 507ax-mp 5 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ∈ dom vol
509 difmbl 23531 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ∈ dom vol) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
510378, 508, 509sylancl 566 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
511510adantr 466 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol)
512511ad2antlr 698 . . . . . . . . . . . . . . 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)
51313adantr 466 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴𝐵) ⊆ ℝ)
514513, 5jca 495 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ))
515514ad5antr 708 . . . . . . . . . . . . . . 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))
516 mblsplit 23520 . . . . . . . . . . . . . . . . 17 (((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol ∧ (𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))))
5175163expb 1113 . . . . . . . . . . . . . . . 16 (((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol ∧ ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ)) → (vol*‘(𝐴𝐵)) = ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))))
518517eqcomd 2777 . . . . . . . . . . . . . . 15 (((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol ∧ ((𝐴𝐵) ⊆ ℝ ∧ (vol*‘(𝐴𝐵)) ∈ ℝ)) → ((vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴𝐵) ∖ (𝑠 ran ((,) ∘ 𝑓))))) = (vol*‘(𝐴𝐵)))
519512, 515, 518syl2anc 565 . . . . . . . . . . . . . 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)))
520299recnd 10270 . . . . . . . . . . . . . . 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)
521298recnd 10270 . . . . . . . . . . . . . . 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)
522 inss1 3981 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) ⊆ (𝐴𝐵)
523522, 3sstri 3761 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴
524 ovolsscl 23474 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
525523, 524mp3an1 1559 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝐵) ∩ (𝑠 ran ((,) ∘ 𝑓)))) ∈ ℝ)
526525ad5antr 708 . . . . . . . . . . . . . . . 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)
527526recnd 10270 . . . . . . . . . . . . . . 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)
528520, 521, 527subadd2d 10613 . . . . . . . . . . . . . 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))))
529519, 528mpbird 247 . . . . . . . . . . . . 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)))))
530 mblvol 23518 . . . . . . . . . . . . . . . . 17 ((𝑠 ran ((,) ∘ 𝑓)) ∈ dom vol → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
531509, 530syl 17 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ∈ dom vol) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
532378, 508, 531sylancl 566 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
533532adantr 466 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ran ((,) ∘ 𝑓))))
534533ad2antlr 698 . . . . . . . . . . . . 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))))
535506, 529, 5343eqtr4rd 2816 . . . . . . . . . . . 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))))))
536502, 535breqtrrd 4814 . . . . . . . . . . 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))))
537 fvex 6342 . . . . . . . . . . . 12 (vol‘(𝑠 ran ((,) ∘ 𝑓))) ∈ V
538 eqeq1 2775 . . . . . . . . . . . . . . 15 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → (𝑣 = (vol‘𝑏) ↔ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏)))
539538anbi2d 606 . . . . . . . . . . . . . 14 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → ((𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ (vol‘(𝑠 ran ((,) ∘ 𝑓))) = (vol‘𝑏))))
540539rexbidv 3200 . . . . . . . . . . . . 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))))
541 breq2 4790 . . . . . . . . . . . . 13 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑓))) → (𝑢 < 𝑣𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑓)))))
542540, 541anbi12d 608 . . . . . . . . . . . 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))))))
543537, 542spcev 3451 . . . . . . . . . . 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))
544293, 536, 543syl2anc 565 . . . . . . . . . 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))
545150anbi2d 606 . . . . . . . . . . . 12 (𝑦 = 𝑣 → ((𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏))))
546545rexbidv 3200 . . . . . . . . . . 11 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴𝐵) ∧ 𝑣 = (vol‘𝑏))))
547546rexab 3521 . . . . . . . . . 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))
548544, 547sylibr 224 . . . . . . . . 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)
549548ex 397 . . . . . . . 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))
550549rexlimdvva 3186 . . . . . . 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))
551262, 550exlimddv 2015 . . . . . 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))
552223, 551syld 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))
553552exp31 406 . . . 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))))
554553com34 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))))
5555543imp1 1440 . 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)
5562, 6, 48, 555eqsupd 8519 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 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063   cuni 4574   class class class wbr 4786   Or wor 5169   × cxp 5247  dom cdm 5249  ran crn 5250  ccom 5253  wf 6027  cfv 6031  (class class class)co 6793  𝑚 cmap 8009  supcsup 8502  cc 10136  cr 10137  0cc0 10138  1c1 10139   + caddc 10141   · cmul 10143  +∞cpnf 10273  *cxr 10275   < clt 10276  cle 10277  cmin 10468   / cdiv 10886  cn 11222  2c2 11272  3c3 11273  +crp 12035  (,)cioo 12380  [,)cico 12382  seqcseq 13008  abscabs 14182  topGenctg 16306  Topctop 20918  TopBasesctb 20970  Clsdccld 21041  vol*covol 23450  volcvol 23451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-disj 4755  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-acn 8968  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-rlim 14428  df-sum 14625  df-rest 16291  df-topgen 16312  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-top 20919  df-topon 20936  df-bases 20971  df-cld 21044  df-cmp 21411  df-ovol 23452  df-vol 23453
This theorem is referenced by:  ismblfin  33783
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