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Theorem ismblfin 38121
Description: Measurability in terms of inner and outer measure. Proposition 7 of [Viaclovsky8] p. 3. (Contributed by Brendan Leahy, 4-Mar-2018.) (Revised by Brendan Leahy, 28-Mar-2018.)
Assertion
Ref Expression
ismblfin ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴 ∈ dom vol ↔ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )))
Distinct variable group:   𝑦,𝑏,𝐴
Proof of Theorem ismblfin
Dummy variables 𝑎 𝑐 𝑓 𝑡 𝑢 𝑣 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mblfinlem4 38120 . 2 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))
2 elpwi 4559 . . . . 5 (𝑤 ∈ 𝒫 ℝ → 𝑤 ⊆ ℝ)
3 elmapi 8824 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4 inss1 4186 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∩ 𝐴) ⊆ 𝑤
5 ovolsscl 25536 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 ∩ 𝐴) ⊆ 𝑤 ∧ 𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ∈ ℝ)
64, 5mp3an1 1468 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ∈ ℝ)
7 difss 4087 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∖ 𝐴) ⊆ 𝑤
8 ovolsscl 25536 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 ∖ 𝐴) ⊆ 𝑤 ∧ 𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤 ∖ 𝐴)) ∈ ℝ)
97, 8mp3an1 1468 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤 ∖ 𝐴)) ∈ ℝ)
106, 9readdcld 11205 . . . . . . . . . . . . . . . . . 18 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ∈ ℝ)
1110rexrd 11226 . . . . . . . . . . . . . . . . 17 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ∈ ℝ*)
1211ad3antlr 741 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ∈ ℝ*)
13 rncoss 5949 . . . . . . . . . . . . . . . . . . 19 ran ((,) ∘ 𝑓) ⊆ ran (,)
1413unissi 4871 . . . . . . . . . . . . . . . . . 18 ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran (,)
15 unirnioo 13447 . . . . . . . . . . . . . . . . . 18 ℝ = ∪ ran (,)
1614, 15sseqtrri 3983 . . . . . . . . . . . . . . . . 17 ∪ ran ((,) ∘ 𝑓) ⊆ ℝ
17 ovolcl 25528 . . . . . . . . . . . . . . . . 17 (∪ ran ((,) ∘ 𝑓) ⊆ ℝ → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ*)
1816, 17mp1i 13 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ*)
19 eqid 2761 . . . . . . . . . . . . . . . . . . 19 ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
20 eqid 2761 . . . . . . . . . . . . . . . . . . 19 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
2119, 20ovolsf 25522 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
22 frn 6694 . . . . . . . . . . . . . . . . . . 19 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
23 icossxr 13430 . . . . . . . . . . . . . . . . . . 19 (0[,)+∞) ⊆ ℝ*
2422, 23sstrdi 3946 . . . . . . . . . . . . . . . . . 18 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
25 supxrcl 13312 . . . . . . . . . . . . . . . . . 18 (ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
2621, 24, 253syl 18 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
2726ad2antlr 737 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
28 pnfge 13126 . . . . . . . . . . . . . . . . . . . . . 22 (((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ∈ ℝ* → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ +∞)
2911, 28syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ +∞)
3029ad2antrr 736 . . . . . . . . . . . . . . . . . . . 20 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) = +∞) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ +∞)
31 simpr 488 . . . . . . . . . . . . . . . . . . . 20 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) = +∞) → (vol*‘∪ ran ((,) ∘ 𝑓)) = +∞)
3230, 31breqtrrd 5125 . . . . . . . . . . . . . . . . . . 19 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) = +∞) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
3332adantlll 728 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) = +∞) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
3416, 17ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ*
35 nltpnft 13161 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* → ((vol*‘∪ ran ((,) ∘ 𝑓)) = +∞ ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < +∞))
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ((vol*‘∪ ran ((,) ∘ 𝑓)) = +∞ ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < +∞)
3736necon2abii 3006 . . . . . . . . . . . . . . . . . . . 20 ((vol*‘∪ ran ((,) ∘ 𝑓)) < +∞ ↔ (vol*‘∪ ran ((,) ∘ 𝑓)) ≠ +∞)
38 ovolge0 25531 . . . . . . . . . . . . . . . . . . . . . 22 (∪ ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
3916, 38ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))
40 0re 11177 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
41 xrre3 13168 . . . . . . . . . . . . . . . . . . . . . 22 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ 0 ∈ ℝ) ∧ (0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) < +∞)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
4234, 40, 41mpanl12 712 . . . . . . . . . . . . . . . . . . . . 21 ((0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) < +∞) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
4339, 42mpan 700 . . . . . . . . . . . . . . . . . . . 20 ((vol*‘∪ ran ((,) ∘ 𝑓)) < +∞ → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
4437, 43sylbir 237 . . . . . . . . . . . . . . . . . . 19 ((vol*‘∪ ran ((,) ∘ 𝑓)) ≠ +∞ → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
4510ad3antlr 741 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ∈ ℝ)
46 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) → 𝑧 = (vol‘𝑎))
47 eleq1w 2844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑎 → (𝑏 ∈ dom vol ↔ 𝑎 ∈ dom vol))
48 uniretop 24810 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ℝ = ∪ (topGen‘ran (,))
4948cldss 23077 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ)
50 dfss4 4219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
5149, 50sylib 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
52 rembl 25590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ℝ ∈ dom vol
5348cldopn 23079 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran (,)))
54 opnmbl 25652 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((ℝ ∖ 𝑏) ∈ (topGen‘ran (,)) → (ℝ ∖ 𝑏) ∈ dom vol)
5553, 54syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ dom vol)
56 difmbl 25593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((ℝ ∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
5752, 55, 56sylancr 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
5851, 57eqeltrrd 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol)
5947, 58vtoclga 3540 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ (Clsd‘(topGen‘ran (,))) → 𝑎 ∈ dom vol)
60 mblvol 25580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ dom vol → (vol‘𝑎) = (vol*‘𝑎))
6159, 60syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑎) = (vol*‘𝑎))
6246, 61sylan9eqr 2818 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))) → 𝑧 = (vol*‘𝑎))
6362adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)))) → 𝑧 = (vol*‘𝑎))
64 inss1 4186 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓)
65 sstr 3942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓))
6664, 65mpan2 701 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) → 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓))
6766ad2antrl 738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))) → 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓))
68 ovolsscl 25536 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
6916, 68mp3an2 1469 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
7069ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘𝑎) ∈ ℝ)
7167, 70sylan2 602 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)))) → (vol*‘𝑎) ∈ ℝ)
7263, 71eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)))) → 𝑧 ∈ ℝ)
7372rexlimdvaa 3163 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) → 𝑧 ∈ ℝ))
7473abssdv 4018 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ⊆ ℝ)
75 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑦 → (𝑧 = (vol‘𝑎) ↔ 𝑦 = (vol‘𝑎)))
7675anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦 → ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))))
7776rexbidv 3185 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑦 → (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))))
7877ralab 3654 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ∀𝑦(∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎)) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))))
79 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎)) → 𝑦 = (vol‘𝑎))
8079, 61sylan9eqr 2818 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))) → 𝑦 = (vol*‘𝑎))
81 ovolss 25535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘𝑎) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8266, 16, 81sylancl 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) → (vol*‘𝑎) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8382ad2antrl 738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))) → (vol*‘𝑎) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8480, 83eqbrtrd 5119 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8584rexlimiva 3154 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎)) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8678, 85mpgbir 1818 . . . . . . . . . . . . . . . . . . . . . . . 24 ∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))
87 brralrspcev 5157 . . . . . . . . . . . . . . . . . . . . . . . 24 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ ∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ 𝑥)
8886, 87mpan2 701 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ 𝑥)
89 retop 24809 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (topGen‘ran (,)) ∈ Top
90 0cld 23086 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
9189, 90ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∅ ∈ (Clsd‘(topGen‘ran (,)))
92 0ss 4351 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴)
93 0mbl 25589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ∅ ∈ dom vol
94 mblvol 25580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
9593, 94ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (vol‘∅) = (vol*‘∅)
96 ovol0 25543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (vol*‘∅) = 0
9795, 96eqtr2i 2785 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 = (vol‘∅)
9892, 97pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘∅))
99 sseq1 3959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = ∅ → (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ↔ ∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴)))
100 fveq2 6862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 = ∅ → (vol‘𝑎) = (vol‘∅))
101100eqeq2d 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = ∅ → (0 = (vol‘𝑎) ↔ 0 = (vol‘∅)))
10299, 101anbi12d 641 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = ∅ → ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎)) ↔ (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘∅))))
103102rspcev 3580 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘∅))) → ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎)))
10491, 98, 103mp2an 702 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎))
105 c0ex 11167 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ V
106 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 0 → (𝑧 = (vol‘𝑎) ↔ 0 = (vol‘𝑎)))
107106anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 0 → ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎))))
108107rexbidv 3185 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 0 → (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎))))
109105, 108elab 3637 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎)))
110104, 109mpbir 233 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}
111110ne0ii 4294 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ≠ ∅
112 suprcl 12146 . . . . . . . . . . . . . . . . . . . . . . . 24 (({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ⊆ ℝ ∧ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ 𝑥) → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) ∈ ℝ)
113111, 112mp3an2 1469 . . . . . . . . . . . . . . . . . . . . . . 23 (({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ 𝑥) → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) ∈ ℝ)
11474, 88, 113syl2anc 593 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) ∈ ℝ)
115 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) → 𝑧 = (vol‘𝑐))
116 eleq1w 2844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑐 → (𝑏 ∈ dom vol ↔ 𝑐 ∈ dom vol))
117116, 58vtoclga 3540 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → 𝑐 ∈ dom vol)
118 mblvol 25580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ dom vol → (vol‘𝑐) = (vol*‘𝑐))
119117, 118syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑐) = (vol*‘𝑐))
120115, 119sylan9eqr 2818 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))) → 𝑧 = (vol*‘𝑐))
121120adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)))) → 𝑧 = (vol*‘𝑐))
122 difss2 4089 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) → 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓))
123122ad2antrl 738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))) → 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓))
124 ovolsscl 25536 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
12516, 124mp3an2 1469 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
126125ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘𝑐) ∈ ℝ)
127123, 126sylan2 602 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)))) → (vol*‘𝑐) ∈ ℝ)
128121, 127eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)))) → 𝑧 ∈ ℝ)
129128rexlimdvaa 3163 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) → 𝑧 ∈ ℝ))
130129abssdv 4018 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ⊆ ℝ)
131 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑦 → (𝑧 = (vol‘𝑐) ↔ 𝑦 = (vol‘𝑐)))
132131anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦 → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))))
133132rexbidv 3185 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑦 → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))))
134133ralab 3654 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ∀𝑦(∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐)) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))))
135 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐)) → 𝑦 = (vol‘𝑐))
136135, 119sylan9eqr 2818 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))) → 𝑦 = (vol*‘𝑐))
137 ovolss 25535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
138122, 16, 137sylancl 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
139138ad2antrl 738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
140136, 139eqbrtrd 5119 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
141140rexlimiva 3154 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐)) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
142134, 141mpgbir 1818 . . . . . . . . . . . . . . . . . . . . . . . 24 ∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))
143 brralrspcev 5157 . . . . . . . . . . . . . . . . . . . . . . . 24 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ ∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ 𝑥)
144142, 143mpan2 701 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ 𝑥)
145 0ss 4351 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)
146145, 97pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘∅))
147 sseq1 3959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = ∅ → (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ↔ ∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
148 fveq2 6862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = ∅ → (vol‘𝑐) = (vol‘∅))
149148eqeq2d 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = ∅ → (0 = (vol‘𝑐) ↔ 0 = (vol‘∅)))
150147, 149anbi12d 641 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = ∅ → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐)) ↔ (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘∅))))
151150rspcev 3580 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘∅))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐)))
15291, 146, 151mp2an 702 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐))
153 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 0 → (𝑧 = (vol‘𝑐) ↔ 0 = (vol‘𝑐)))
154153anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 0 → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐))))
155154rexbidv 3185 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 0 → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐))))
156105, 155elab 3637 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐)))
157152, 156mpbir 233 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}
158157ne0ii 4294 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ≠ ∅
159 suprcl 12146 . . . . . . . . . . . . . . . . . . . . . . . 24 (({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ⊆ ℝ ∧ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ 𝑥) → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) ∈ ℝ)
160158, 159mp3an2 1469 . . . . . . . . . . . . . . . . . . . . . . 23 (({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ 𝑥) → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) ∈ ℝ)
161130, 144, 160syl2anc 593 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) ∈ ℝ)
162114, 161readdcld 11205 . . . . . . . . . . . . . . . . . . . . 21 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) + sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < )) ∈ ℝ)
163162adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) + sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < )) ∈ ℝ)
164 simpr 488 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
1656ad2antrr 736 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ∈ ℝ)
1669ad2antrr 736 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤 ∖ 𝐴)) ∈ ℝ)
167 ovolsscl 25536 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
16864, 16, 167mp3an12 1471 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
169168adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
170 difss 4087 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓)
171 ovolsscl 25536 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
172170, 16, 171mp3an12 1471 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
173172adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
174 ssrin 4191 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) → (𝑤 ∩ 𝐴) ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴))
17564, 16sstri 3943 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ℝ
176 ovolss 25535 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑤 ∩ 𝐴) ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)))
177174, 175, 176sylancl 595 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) → (vol*‘(𝑤 ∩ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)))
178177ad2antlr 737 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)))
179 ssdif 4095 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) → (𝑤 ∖ 𝐴) ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
180170, 16sstri 3943 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ
181 ovolss 25535 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑤 ∖ 𝐴) ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ) → (vol*‘(𝑤 ∖ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
182179, 180, 181sylancl 595 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) → (vol*‘(𝑤 ∖ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
183182ad2antlr 737 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤 ∖ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
184165, 166, 169, 173, 178, 183le2addd 11800 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
185 dfin4 4228 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) = (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
186185fveq2i 6865 . . . . . . . . . . . . . . . . . . . . . . . 24 (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
187186oveq1i 7401 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
188184, 187breqtrdi 5138 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
189188adantlll 728 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
190 simpll 776 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) → ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )))
191185sseq2i 3963 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ↔ 𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
192191anbi1i 633 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ 𝑧 = (vol‘𝑎)))
193192rexbii 3108 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ 𝑧 = (vol‘𝑎)))
194193abbii 2828 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} = {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ 𝑧 = (vol‘𝑎))}
195194supeq1i 9387 . . . . . . . . . . . . . . . . . . . . . . . 24 sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) = sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < )
19616jctl 531 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ))
197196adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ))
198172, 180jctil 527 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ))
199198adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ))
200 ltso 11257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 < Or ℝ
201200a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → < Or ℝ)
202 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
203 vex 3457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑥 ∈ V
204 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = 𝑥 → (𝑧 = (vol‘𝑐) ↔ 𝑥 = (vol‘𝑐)))
205204anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = 𝑥 → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))))
206205rexbidv 3185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑥 → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))))
207203, 206elab 3637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))} ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐)))
20816, 137mpan2 701 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
209208ad2antrl 738 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
21048cldss 23077 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → 𝑐 ⊆ ℝ)
211 ovolcl 25528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ⊆ ℝ → (vol*‘𝑐) ∈ ℝ*)
212210, 211syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (vol*‘𝑐) ∈ ℝ*)
213 xrlenlt 11241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((vol*‘𝑐) ∈ ℝ* ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ*) → ((vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
214212, 34, 213sylancl 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → ((vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
215214adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → ((vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
216 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑥 = (vol‘𝑐) → 𝑥 = (vol‘𝑐))
217216, 119sylan9eqr 2818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑥 = (vol‘𝑐)) → 𝑥 = (vol*‘𝑐))
218 breq2 5101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑥 = (vol*‘𝑐) → ((vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥 ↔ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
219218notbid 320 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑥 = (vol*‘𝑐) → (¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥 ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
220217, 219syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑥 = (vol‘𝑐)) → (¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥 ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
221220adantrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → (¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥 ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
222215, 221bitr4d 284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → ((vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥))
223209, 222mpbid 234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥)
224223rexlimiva 3154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐)) → ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥)
225207, 224sylbi 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))} → ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥)
226225adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑥 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}) → ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥)
227 retopbas 24808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ran (,) ∈ TopBases
228 bastg 23014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
229227, 228ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ran (,) ⊆ (topGen‘ran (,))
23013, 229sstri 3943 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))
231 uniopn 22945 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))) → ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)))
23289, 230, 231mp2an 702 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))
233 mblfinlem2 38118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) ∧ 𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)))
234232, 233mp3an1 1468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)))
235119eqcomd 2767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (vol*‘𝑐) = (vol‘𝑐))
236235anim1i 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑥 < (vol*‘𝑐)) → ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐)))
237236ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (𝑥 < (vol*‘𝑐) → ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐))))
238237anim2d 621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)) → (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐)))))
239 fvex 6875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (vol*‘𝑐) ∈ V
240 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑦 = (vol*‘𝑐) → (𝑦 = (vol‘𝑐) ↔ (vol*‘𝑐) = (vol‘𝑐)))
241240anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = (vol*‘𝑐) → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ↔ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘𝑐) = (vol‘𝑐))))
242 breq2 5101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = (vol*‘𝑐) → (𝑥 < 𝑦 ↔ 𝑥 < (vol*‘𝑐)))
243241, 242anbi12d 641 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 = (vol*‘𝑐) → (((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦) ↔ ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘𝑐) = (vol‘𝑐)) ∧ 𝑥 < (vol*‘𝑐))))
244239, 243spcev 3564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘𝑐) = (vol‘𝑐)) ∧ 𝑥 < (vol*‘𝑐)) → ∃𝑦((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
245244anasss 470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐))) → ∃𝑦((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
246238, 245syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)) → ∃𝑦((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦)))
247246reximia 3096 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))∃𝑦((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
248234, 247syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))∃𝑦((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
249 r19.41v 3191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦) ↔ (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
250249exbii 1867 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑦∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦) ↔ ∃𝑦(∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
251 rexcom4 3288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))∃𝑦((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦) ↔ ∃𝑦∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
252131anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = 𝑦 → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐))))
253252rexbidv 3185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = 𝑦 → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐))))
254253rexab 3656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}𝑥 < 𝑦 ↔ ∃𝑦(∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
255250, 251, 2543bitr4i 305 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))∃𝑦((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦) ↔ ∃𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}𝑥 < 𝑦)
256248, 255sylib 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}𝑥 < 𝑦)
257256adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘∪ ran ((,) ∘ 𝑓)))) → ∃𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}𝑥 < 𝑦)
258201, 202, 226, 257eqsupd 9397 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) = (vol*‘∪ ran ((,) ∘ 𝑓)))
259258eqcomd 2767 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘∪ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ))
260259adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘∪ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ))
261 sseq1 3959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑎 → (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓)))
262 fveq2 6862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑎 → (vol‘𝑐) = (vol‘𝑎))
263262eqeq2d 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑎 → (𝑧 = (vol‘𝑐) ↔ 𝑧 = (vol‘𝑎)))
264261, 263anbi12d 641 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = 𝑎 → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))))
265264cbvrexvw 3240 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎)))
266265abbii 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))} = {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))}
267266supeq1i 9387 . . . . . . . . . . . . . . . . . . . . . . . . . 26 sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) = sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < )
268260, 267eqtrdi 2812 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘∪ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ))
269 simpll 776 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
270 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = 𝑧 → (𝑦 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑏)))
271270anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = 𝑧 → ((𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑏))))
272271rexbidv 3185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑏))))
273 sseq1 3959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑐 → (𝑏 ⊆ 𝐴 ↔ 𝑐 ⊆ 𝐴))
274 fveq2 6862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑐 → (vol‘𝑏) = (vol‘𝑐))
275274eqeq2d 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑐 → (𝑧 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑐)))
276273, 275anbi12d 641 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 = 𝑐 → ((𝑏 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑏)) ↔ (𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))))
277276cbvrexvw 3240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑏)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐)))
278272, 277bitrdi 289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))))
279278cbvabv 2831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} = {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))}
280279supeq1i 9387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))}, ℝ, < )
281280eqeq2i 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ (vol*‘𝐴) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ))
282281biimpi 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) → (vol*‘𝐴) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ))
283282ad2antlr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝐴) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ))
284 mblfinlem3 38119 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) ∧ (vol*‘𝐴) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ))) → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
285197, 269, 260, 283, 284syl112anc 1392 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
286 sseq1 3959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑎 → (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ↔ 𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
287286, 263anbi12d 641 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = 𝑎 → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎))))
288287cbvrexvw 3240 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎)))
289288abbii 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} = {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎))}
290289supeq1i 9387 . . . . . . . . . . . . . . . . . . . . . . . . . 26 sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) = sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < )
291285, 290eqtr3di 2811 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ))
292 mblfinlem3 38119 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ))) → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
293197, 199, 268, 291, 292syl112anc 1392 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
294195, 293eqtrid 2808 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
295294, 285oveq12d 7409 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) + sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < )) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
296190, 295sylan 589 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) + sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < )) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
297189, 296breqtrrd 5125 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) + sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < )))
298 ne0i 4291 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} → {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ≠ ∅)
299110, 298mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ≠ ∅)
300 ne0i 4291 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} → {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ≠ ∅)
301157, 300mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ≠ ∅)
302 eqid 2761 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)} = {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}
30374, 299, 88, 130, 301, 144, 302supadd 12154 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) + sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < )) = sup({𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}, ℝ, < ))
304 reeanv 3233 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎)) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))) ↔ (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎)) ∧ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))))
305 vex 3457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑢 ∈ V
306 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑢 → (𝑧 = (vol‘𝑎) ↔ 𝑢 = (vol‘𝑎)))
307306anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = 𝑢 → ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎))))
308307rexbidv 3185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑢 → (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎))))
309305, 308elab 3637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎)))
310 vex 3457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑣 ∈ V
311 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑣 → (𝑧 = (vol‘𝑐) ↔ 𝑣 = (vol‘𝑐)))
312311anbi2d 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = 𝑣 → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))))
313312rexbidv 3185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑣 → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))))
314310, 313elab 3637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐)))
315309, 314anbi12i 637 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}) ↔ (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎)) ∧ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))))
316304, 315bitr4i 280 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎)) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))) ↔ (𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}))
317 an4 666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ (𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐))) ↔ ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎)) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))))
318 oveq12 7400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐)) → (𝑢 + 𝑣) = ((vol‘𝑎) + (vol‘𝑐)))
31959adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑎 ∈ dom vol)
320319ad2antlr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → 𝑎 ∈ dom vol)
321117adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑐 ∈ dom vol)
322321ad2antlr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → 𝑐 ∈ dom vol)
323 ss2in 4194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ∩ 𝑐) ⊆ ((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
324185ineq1i 4166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = ((∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
325 incom 4159 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
326 disjdif 4423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) = ∅
327324, 325, 3263eqtri 2788 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = ∅
328323, 327sseqtrdi 3974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ∩ 𝑐) ⊆ ∅)
329 ss0 4353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∩ 𝑐) ⊆ ∅ → (𝑎 ∩ 𝑐) = ∅)
330328, 329syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ∩ 𝑐) = ∅)
331330adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (𝑎 ∩ 𝑐) = ∅)
33261adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑎) = (vol*‘𝑎))
333332ad2antlr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol‘𝑎) = (vol*‘𝑎))
33466, 16jctir 528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) → (𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ))
335683expa 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
336334, 335sylan 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
337336ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) → (vol*‘𝑎) ∈ ℝ)
338337ad2ant2r 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol*‘𝑎) ∈ ℝ)
339333, 338eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol‘𝑎) ∈ ℝ)
340119adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑐) = (vol*‘𝑐))
341340ad2antlr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol‘𝑐) = (vol*‘𝑐))
342122, 16jctir 528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) → (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ))
3431243expa 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
344342, 343sylan 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
345344ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (vol*‘𝑐) ∈ ℝ)
346345ad2ant2rl 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol*‘𝑐) ∈ ℝ)
347341, 346eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol‘𝑐) ∈ ℝ)
348 volun 25595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑎 ∈ dom vol ∧ 𝑐 ∈ dom vol ∧ (𝑎 ∩ 𝑐) = ∅) ∧ ((vol‘𝑎) ∈ ℝ ∧ (vol‘𝑐) ∈ ℝ)) → (vol‘(𝑎 ∪ 𝑐)) = ((vol‘𝑎) + (vol‘𝑐)))
349320, 322, 331, 339, 347, 348syl32anc 1396 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol‘(𝑎 ∪ 𝑐)) = ((vol‘𝑎) + (vol‘𝑐)))
350 unmbl 25587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ dom vol ∧ 𝑐 ∈ dom vol) → (𝑎 ∪ 𝑐) ∈ dom vol)
35159, 117, 350syl2an 605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (𝑎 ∪ 𝑐) ∈ dom vol)
352 mblvol 25580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∪ 𝑐) ∈ dom vol → (vol‘(𝑎 ∪ 𝑐)) = (vol*‘(𝑎 ∪ 𝑐)))
353351, 352syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘(𝑎 ∪ 𝑐)) = (vol*‘(𝑎 ∪ 𝑐)))
354353ad2antlr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol‘(𝑎 ∪ 𝑐)) = (vol*‘(𝑎 ∪ 𝑐)))
355349, 354eqtr3d 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → ((vol‘𝑎) + (vol‘𝑐)) = (vol*‘(𝑎 ∪ 𝑐)))
356318, 355sylan9eqr 2818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ∧ (𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐))) → (𝑢 + 𝑣) = (vol*‘(𝑎 ∪ 𝑐)))
357 eqtr 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 = (𝑢 + 𝑣) ∧ (𝑢 + 𝑣) = (vol*‘(𝑎 ∪ 𝑐))) → 𝑦 = (vol*‘(𝑎 ∪ 𝑐)))
358357ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑢 + 𝑣) = (vol*‘(𝑎 ∪ 𝑐)) ∧ 𝑦 = (𝑢 + 𝑣)) → 𝑦 = (vol*‘(𝑎 ∪ 𝑐)))
359356, 358sylan 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ∧ (𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐))) ∧ 𝑦 = (𝑢 + 𝑣)) → 𝑦 = (vol*‘(𝑎 ∪ 𝑐)))
36066, 122anim12i 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓)))
361 unss 4140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓)) ↔ (𝑎 ∪ 𝑐) ⊆ ∪ ran ((,) ∘ 𝑓))
362360, 361sylib 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ∪ 𝑐) ⊆ ∪ ran ((,) ∘ 𝑓))
363 ovolss 25535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑎 ∪ 𝑐) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘(𝑎 ∪ 𝑐)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
364362, 16, 363sylancl 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (vol*‘(𝑎 ∪ 𝑐)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
365364ad3antlr 741 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ∧ (𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐))) ∧ 𝑦 = (𝑢 + 𝑣)) → (vol*‘(𝑎 ∪ 𝑐)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
366359, 365eqbrtrd 5119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ∧ (𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐))) ∧ 𝑦 = (𝑢 + 𝑣)) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
367366ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ∧ (𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐))) → (𝑦 = (𝑢 + 𝑣) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))))
368367expl 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) → (((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ (𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐))) → (𝑦 = (𝑢 + 𝑣) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))))
369317, 368biimtrrid 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) → (((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎)) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))) → (𝑦 = (𝑢 + 𝑣) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))))
370369rexlimdvva 3218 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎)) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))) → (𝑦 = (𝑢 + 𝑣) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))))
371316, 370biimtrrid 245 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ((𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}) → (𝑦 = (𝑢 + 𝑣) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))))
372371rexlimdvv 3217 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 = (𝑢 + 𝑣) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))))
373372alrimiv 1946 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ∀𝑦(∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 = (𝑢 + 𝑣) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))))
374 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑦 → (𝑡 = (𝑢 + 𝑣) ↔ 𝑦 = (𝑢 + 𝑣)))
3753742rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑦 → (∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣) ↔ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 = (𝑢 + 𝑣)))
376375ralab 3654 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑦 ∈ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ∀𝑦(∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 = (𝑢 + 𝑣) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))))
377373, 376sylibr 236 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ∀𝑦 ∈ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
378 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))})) ∧ 𝑡 = (𝑢 + 𝑣)) → 𝑡 = (𝑢 + 𝑣))
37974sselda 3934 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}) → 𝑢 ∈ ℝ)
380130sselda 3934 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}) → 𝑣 ∈ ℝ)
381 readdcl 11150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ)
382379, 380, 381syl2an 605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))})) → (𝑢 + 𝑣) ∈ ℝ)
383382anandis 688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))})) → (𝑢 + 𝑣) ∈ ℝ)
384383adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))})) ∧ 𝑡 = (𝑢 + 𝑣)) → (𝑢 + 𝑣) ∈ ℝ)
385378, 384eqeltrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))})) ∧ 𝑡 = (𝑢 + 𝑣)) → 𝑡 ∈ ℝ)
386385ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))})) → (𝑡 = (𝑢 + 𝑣) → 𝑡 ∈ ℝ))
387386rexlimdvva 3218 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣) → 𝑡 ∈ ℝ))
388387abssdv 4018 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)} ⊆ ℝ)
389 00id 11352 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (0 + 0) = 0
390389eqcomi 2770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 = (0 + 0)
391 rspceov 7440 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ∧ 0 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ∧ 0 = (0 + 0)) → ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}0 = (𝑢 + 𝑣))
392110, 157, 390, 391mp3an 1481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}0 = (𝑢 + 𝑣)
393 eqeq1 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑡 = 0 → (𝑡 = (𝑢 + 𝑣) ↔ 0 = (𝑢 + 𝑣)))
3943932rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑡 = 0 → (∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣) ↔ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}0 = (𝑢 + 𝑣)))
395105, 394spcev 3564 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}0 = (𝑢 + 𝑣) → ∃𝑡∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣))
396392, 395ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∃𝑡∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)
397 abn0 4335 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)} ≠ ∅ ↔ ∃𝑡∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣))
398396, 397mpbir 233 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)} ≠ ∅
399398a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)} ≠ ∅)
400 brralrspcev 5157 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ ∀𝑦 ∈ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}𝑦 ≤ 𝑥)
401377, 400mpdan 697 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}𝑦 ≤ 𝑥)
402388, 399, 4013jca 1140 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ({𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)} ⊆ ℝ ∧ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}𝑦 ≤ 𝑥))
403 suprleub 12152 . . . . . . . . . . . . . . . . . . . . . . . 24 ((({𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)} ⊆ ℝ ∧ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}𝑦 ≤ 𝑥) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (sup({𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}, ℝ, < ) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ∀𝑦 ∈ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))))
404402, 403mpancom 698 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (sup({𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}, ℝ, < ) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ∀𝑦 ∈ {𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))))
405377, 404mpbird 259 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑡 ∣ ∃𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}∃𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑡 = (𝑢 + 𝑣)}, ℝ, < ) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
406303, 405eqbrtrd 5119 . . . . . . . . . . . . . . . . . . . . 21 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) + sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < )) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
407406adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) + sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < )) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
40845, 163, 164, 297, 407letrd 11334 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
40944, 408sylan2 602 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≠ +∞) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
41033, 409pm2.61dane 3043 . . . . . . . . . . . . . . . . 17 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
411410adantlr 725 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
412 ssid 3956 . . . . . . . . . . . . . . . . . 18 ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ((,) ∘ 𝑓)
41320ovollb 25529 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
414412, 413mpan2 701 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
415414ad2antlr 737 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
41612, 18, 27, 411, 415xrletrd 13158 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
417416adantr 484 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
418 simpr 488 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
419417, 418breqtrrd 5125 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢)
420419expl 461 . . . . . . . . . . . 12 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢))
4213, 420sylan2 602 . . . . . . . . . . 11 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → ((𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢))
422421rexlimdva 3162 . . . . . . . . . 10 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢))
423422ralrimivw 3157 . . . . . . . . 9 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → ∀𝑢 ∈ ℝ* (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢))
424 eqeq1 2765 . . . . . . . . . . . 12 (𝑣 = 𝑢 → (𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))
425424anbi2d 639 . . . . . . . . . . 11 (𝑣 = 𝑢 → ((𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))))
426425rexbidv 3185 . . . . . . . . . 10 (𝑣 = 𝑢 → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))))
427426ralrab 3655 . . . . . . . . 9 (∀𝑢 ∈ {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢 ↔ ∀𝑢 ∈ ℝ* (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢))
428423, 427sylibr 236 . . . . . . . 8 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → ∀𝑢 ∈ {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢)
429 ssrab2 4031 . . . . . . . . 9 {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ ℝ*
43011adantl 485 . . . . . . . . 9 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ∈ ℝ*)
431 infxrgelb 13333 . . . . . . . . 9 (({𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ ℝ* ∧ ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ∈ ℝ*) → (((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ inf({𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ) ↔ ∀𝑢 ∈ {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢))
432429, 430, 431sylancr 596 . . . . . . . 8 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → (((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ inf({𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ) ↔ ∀𝑢 ∈ {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ 𝑢))
433428, 432mpbird 259 . . . . . . 7 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ inf({𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
434 eqid 2761 . . . . . . . . 9 {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} = {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
435434ovolval 25523 . . . . . . . 8 (𝑤 ⊆ ℝ → (vol*‘𝑤) = inf({𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
436435ad2antrl 738 . . . . . . 7 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → (vol*‘𝑤) = inf({𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
437433, 436breqtrrd 5125 . . . . . 6 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘𝑤))
438437expr 460 . . . . 5 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ 𝑤 ⊆ ℝ) → ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘𝑤)))
4392, 438sylan2 602 . . . 4 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ 𝑤 ∈ 𝒫 ℝ) → ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘𝑤)))
440439ralrimiva 3153 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → ∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘𝑤)))
441 ismbl2 25577 . . . . 5 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘𝑤))))
442441baibr 544 . . . 4 (𝐴 ⊆ ℝ → (∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘𝑤)) ↔ 𝐴 ∈ dom vol))
443442ad2antrr 736 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → (∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘𝑤)) ↔ 𝐴 ∈ dom vol))
444440, 443mpbid 234 . 2 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → 𝐴 ∈ dom vol)
4451, 444impbida 810 1 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴 ∈ dom vol ↔ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  {crab 3413   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4552  ∪ cuni 4862   class class class wbr 5097   Or wor 5550   × cxp 5641  dom cdm 5643  ran crn 5644   ∘ ccom 5647  ⟶wf 6512  ‘cfv 6516  (class class class)co 7391   ↑m cmap 8802  supcsup 9380  infcinf 9381  ℝcr 11066  0cc0 11067  1c1 11068   + caddc 11070  +∞cpnf 11207  ℝ*cxr 11209   < clt 11210   ≤ cle 11211   − cmin 11408  ℕcn 12204  (,)cioo 13343  [,)cico 13345  seqcseq 14008  abscabs 15252  topGenctg 17457  Topctop 22941  TopBasesctb 22993  Clsdccld 23064  vol*covol 25512  volcvol 25513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-inf2 9590  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144  ax-pre-sup 11145
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-disj 5065  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-of 7655  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-oadd 8435  df-omul 8436  df-er 8672  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fi 9351  df-sup 9382  df-inf 9383  df-oi 9452  df-dju 9853  df-card 9891  df-acn 9894  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-div 11839  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-n0 12476  df-z 12563  df-uz 12834  df-q 12944  df-rp 12988  df-xneg 13108  df-xadd 13109  df-xmul 13110  df-ioo 13347  df-ico 13349  df-icc 13350  df-fz 13507  df-fzo 13654  df-fl 13796  df-seq 14009  df-exp 14069  df-hash 14338  df-cj 15117  df-re 15118  df-im 15119  df-sqrt 15253  df-abs 15254  df-clim 15506  df-rlim 15507  df-sum 15705  df-rest 17442  df-topgen 17463  df-psmet 21404  df-xmet 21405  df-met 21406  df-bl 21407  df-mopn 21408  df-top 22942  df-topon 22959  df-bases 22994  df-cld 23067  df-cmp 23435  df-conn 23460  df-ovol 25514  df-vol 25515
This theorem is referenced by: (None)
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