Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismblfin Structured version   Visualization version   GIF version
Theorem ismblfin 35818
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 35817 . 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 4542 . . . . 5 (𝑤 ∈ 𝒫 ℝ → 𝑤 ⊆ ℝ)
3 elmapi 8637 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4 inss1 4162 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∩ 𝐴) ⊆ 𝑤
5 ovolsscl 24650 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 ∩ 𝐴) ⊆ 𝑤 ∧ 𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ∈ ℝ)
64, 5mp3an1 1447 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ∈ ℝ)
7 difss 4066 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∖ 𝐴) ⊆ 𝑤
8 ovolsscl 24650 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 ∖ 𝐴) ⊆ 𝑤 ∧ 𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤 ∖ 𝐴)) ∈ ℝ)
97, 8mp3an1 1447 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤 ∖ 𝐴)) ∈ ℝ)
106, 9readdcld 11004 . . . . . . . . . . . . . . . . . 18 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ∈ ℝ)
1110rexrd 11025 . . . . . . . . . . . . . . . . 17 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ∈ ℝ*)
1211ad3antlr 728 . . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . . . . . . 19 ran ((,) ∘ 𝑓) ⊆ ran (,)
1413unissi 4848 . . . . . . . . . . . . . . . . . 18 ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran (,)
15 unirnioo 13181 . . . . . . . . . . . . . . . . . 18 ℝ = ∪ ran (,)
1614, 15sseqtrri 3958 . . . . . . . . . . . . . . . . 17 ∪ ran ((,) ∘ 𝑓) ⊆ ℝ
17 ovolcl 24642 . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . . 19 ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
20 eqid 2738 . . . . . . . . . . . . . . . . . . 19 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
2119, 20ovolsf 24636 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
22 frn 6607 . . . . . . . . . . . . . . . . . . 19 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
23 icossxr 13164 . . . . . . . . . . . . . . . . . . 19 (0[,)+∞) ⊆ ℝ*
2422, 23sstrdi 3933 . . . . . . . . . . . . . . . . . 18 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
25 supxrcl 13049 . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . 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 12866 . . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . 20 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) = +∞) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ +∞)
31 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) = +∞) → (vol*‘∪ ran ((,) ∘ 𝑓)) = +∞)
3230, 31breqtrrd 5102 . . . . . . . . . . . . . . . . . . 19 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) = +∞) → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
3332adantlll 715 . . . . . . . . . . . . . . . . . 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 12898 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* → ((vol*‘∪ ran ((,) ∘ 𝑓)) = +∞ ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < +∞))
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ((vol*‘∪ ran ((,) ∘ 𝑓)) = +∞ ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < +∞)
3736necon2abii 2994 . . . . . . . . . . . . . . . . . . . 20 ((vol*‘∪ ran ((,) ∘ 𝑓)) < +∞ ↔ (vol*‘∪ ran ((,) ∘ 𝑓)) ≠ +∞)
38 ovolge0 24645 . . . . . . . . . . . . . . . . . . . . . 22 (∪ ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
3916, 38ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))
40 0re 10977 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
41 xrre3 12905 . . . . . . . . . . . . . . . . . . . . . 22 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ 0 ∈ ℝ) ∧ (0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) < +∞)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
4234, 40, 41mpanl12 699 . . . . . . . . . . . . . . . . . . . . 21 ((0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) < +∞) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
4339, 42mpan 687 . . . . . . . . . . . . . . . . . . . 20 ((vol*‘∪ ran ((,) ∘ 𝑓)) < +∞ → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
4437, 43sylbir 234 . . . . . . . . . . . . . . . . . . 19 ((vol*‘∪ ran ((,) ∘ 𝑓)) ≠ +∞ → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
4510ad3antlr 728 . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) → 𝑧 = (vol‘𝑎))
47 eleq1w 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑎 → (𝑏 ∈ dom vol ↔ 𝑎 ∈ dom vol))
48 uniretop 23926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ℝ = ∪ (topGen‘ran (,))
4948cldss 22180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ)
50 dfss4 4192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
5149, 50sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
52 rembl 24704 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ℝ ∈ dom vol
5348cldopn 22182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran (,)))
54 opnmbl 24766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((ℝ ∖ 𝑏) ∈ (topGen‘ran (,)) → (ℝ ∖ 𝑏) ∈ dom vol)
5553, 54syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ dom vol)
56 difmbl 24707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((ℝ ∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
5752, 55, 56sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
5851, 57eqeltrrd 2840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol)
5947, 58vtoclga 3513 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ (Clsd‘(topGen‘ran (,))) → 𝑎 ∈ dom vol)
60 mblvol 24694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ dom vol → (vol‘𝑎) = (vol*‘𝑎))
6159, 60syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑎) = (vol*‘𝑎))
6246, 61sylan9eqr 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))) → 𝑧 = (vol*‘𝑎))
6362adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)))) → 𝑧 = (vol*‘𝑎))
64 inss1 4162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓)
65 sstr 3929 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓))
6664, 65mpan2 688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) → 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓))
6766ad2antrl 725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))) → 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓))
68 ovolsscl 24650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
6916, 68mp3an2 1448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
7069ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘𝑎) ∈ ℝ)
7167, 70sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)))) → (vol*‘𝑎) ∈ ℝ)
7263, 71eqeltrd 2839 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)))) → 𝑧 ∈ ℝ)
7372rexlimdvaa 3214 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) → 𝑧 ∈ ℝ))
7473abssdv 4002 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ⊆ ℝ)
75 eqeq1 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑦 → (𝑧 = (vol‘𝑎) ↔ 𝑦 = (vol‘𝑎)))
7675anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦 → ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))))
7776rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3628 . . . . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎)) → 𝑦 = (vol‘𝑎))
8079, 61sylan9eqr 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))) → 𝑦 = (vol*‘𝑎))
81 ovolss 24649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘𝑎) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8266, 16, 81sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) → (vol*‘𝑎) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8382ad2antrl 725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))) → (vol*‘𝑎) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8480, 83eqbrtrd 5096 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8584rexlimiva 3210 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎)) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
8678, 85mpgbir 1802 . . . . . . . . . . . . . . . . . . . . . . . 24 ∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))
87 brralrspcev 5134 . . . . . . . . . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ 𝑥)
89 retop 23925 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (topGen‘ran (,)) ∈ Top
90 0cld 22189 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
9189, 90ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∅ ∈ (Clsd‘(topGen‘ran (,)))
92 0ss 4330 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴)
93 0mbl 24703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ∅ ∈ dom vol
94 mblvol 24694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
9593, 94ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (vol‘∅) = (vol*‘∅)
96 ovol0 24657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (vol*‘∅) = 0
9795, 96eqtr2i 2767 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 = (vol‘∅)
9892, 97pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘∅))
99 sseq1 3946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = ∅ → (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ↔ ∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴)))
100 fveq2 6774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 = ∅ → (vol‘𝑎) = (vol‘∅))
101100eqeq2d 2749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = ∅ → (0 = (vol‘𝑎) ↔ 0 = (vol‘∅)))
10299, 101anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = ∅ → ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎)) ↔ (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘∅))))
103102rspcev 3561 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘∅))) → ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎)))
10491, 98, 103mp2an 689 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎))
105 c0ex 10969 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ V
106 eqeq1 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 0 → (𝑧 = (vol‘𝑎) ↔ 0 = (vol‘𝑎)))
107106anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 0 → ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎))))
108107rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}
111110ne0ii 4271 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ≠ ∅
112 suprcl 11935 . . . . . . . . . . . . . . . . . . . . . . . 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 1448 . . . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) ∈ ℝ)
115 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) → 𝑧 = (vol‘𝑐))
116 eleq1w 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑐 → (𝑏 ∈ dom vol ↔ 𝑐 ∈ dom vol))
117116, 58vtoclga 3513 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → 𝑐 ∈ dom vol)
118 mblvol 24694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ dom vol → (vol‘𝑐) = (vol*‘𝑐))
119117, 118syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑐) = (vol*‘𝑐))
120115, 119sylan9eqr 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))) → 𝑧 = (vol*‘𝑐))
121120adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)))) → 𝑧 = (vol*‘𝑐))
122 difss2 4068 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) → 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓))
123122ad2antrl 725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))) → 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓))
124 ovolsscl 24650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
12516, 124mp3an2 1448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
126125ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘𝑐) ∈ ℝ)
127123, 126sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)))) → (vol*‘𝑐) ∈ ℝ)
128121, 127eqeltrd 2839 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)))) → 𝑧 ∈ ℝ)
129128rexlimdvaa 3214 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) → 𝑧 ∈ ℝ))
130129abssdv 4002 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ⊆ ℝ)
131 eqeq1 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑦 → (𝑧 = (vol‘𝑐) ↔ 𝑦 = (vol‘𝑐)))
132131anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦 → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))))
133132rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3628 . . . . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐)) → 𝑦 = (vol‘𝑐))
136135, 119sylan9eqr 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))) → 𝑦 = (vol*‘𝑐))
137 ovolss 24649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
138122, 16, 137sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
139138ad2antrl 725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
140136, 139eqbrtrd 5096 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
141140rexlimiva 3210 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐)) → 𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
142134, 141mpgbir 1802 . . . . . . . . . . . . . . . . . . . . . . . 24 ∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))
143 brralrspcev 5134 . . . . . . . . . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ 𝑥)
145 0ss 4330 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)
146145, 97pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘∅))
147 sseq1 3946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = ∅ → (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ↔ ∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
148 fveq2 6774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = ∅ → (vol‘𝑐) = (vol‘∅))
149148eqeq2d 2749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = ∅ → (0 = (vol‘𝑐) ↔ 0 = (vol‘∅)))
150147, 149anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = ∅ → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐)) ↔ (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘∅))))
151150rspcev 3561 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘∅))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐)))
15291, 146, 151mp2an 689 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐))
153 eqeq1 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 0 → (𝑧 = (vol‘𝑐) ↔ 0 = (vol‘𝑐)))
154153anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 0 → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐))))
155154rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}
158157ne0ii 4271 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ≠ ∅
159 suprcl 11935 . . . . . . . . . . . . . . . . . . . . . . . 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 1448 . . . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) ∈ ℝ)
162114, 161readdcld 11004 . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . 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 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) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
1656ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ∈ ℝ)
1669ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤 ∖ 𝐴)) ∈ ℝ)
167 ovolsscl 24650 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
16864, 16, 167mp3an12 1450 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
169168adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
170 difss 4066 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓)
171 ovolsscl 24650 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
172170, 16, 171mp3an12 1450 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
173172adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
174 ssrin 4167 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) → (𝑤 ∩ 𝐴) ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴))
17564, 16sstri 3930 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ℝ
176 ovolss 24649 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑤 ∩ 𝐴) ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)))
177174, 175, 176sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) → (vol*‘(𝑤 ∩ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)))
178177ad2antlr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ⊆ ∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤 ∩ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)))
179 ssdif 4074 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) → (𝑤 ∖ 𝐴) ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
180170, 16sstri 3930 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ
181 ovolss 24649 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑤 ∖ 𝐴) ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ) → (vol*‘(𝑤 ∖ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
182179, 180, 181sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) → (vol*‘(𝑤 ∖ 𝐴)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
183182ad2antlr 724 . . . . . . . . . . . . . . . . . . . . . . . 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 11594 . . . . . . . . . . . . . . . . . . . . . . 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 4201 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) = (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
186185fveq2i 6777 . . . . . . . . . . . . . . . . . . . . . . . 24 (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
187186oveq1i 7285 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
188184, 187breqtrdi 5115 . . . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . 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 3950 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ↔ 𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
192191anbi1i 624 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ 𝑧 = (vol‘𝑎)))
193192rexbii 3181 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . . . . . . . . . . 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 9206 . . . . . . . . . . . . . . . . . . . . . . . 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 524 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ))
197196adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ))
199198adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 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 11055 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3436 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑥 ∈ V
204 eqeq1 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = 𝑥 → (𝑧 = (vol‘𝑐) ↔ 𝑥 = (vol‘𝑐)))
205204anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = 𝑥 → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))))
206205rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
209208ad2antrl 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → (vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
21048cldss 22180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → 𝑐 ⊆ ℝ)
211 ovolcl 24642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ⊆ ℝ → (vol*‘𝑐) ∈ ℝ*)
212210, 211syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (vol*‘𝑐) ∈ ℝ*)
213 xrlenlt 11040 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((vol*‘𝑐) ∈ ℝ* ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ*) → ((vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
214212, 34, 213sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → ((vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
215214adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑥 = (vol‘𝑐)) → 𝑥 = (vol*‘𝑐))
218 breq2 5078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑥 = (vol*‘𝑐) → ((vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥 ↔ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
219218notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → (¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥 ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
222215, 221bitr4d 281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → ((vol*‘𝑐) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥))
223209, 222mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥)
224223rexlimiva 3210 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐)) → ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥)
225207, 224sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))} → ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥)
226225adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑥 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}) → ¬ (vol*‘∪ ran ((,) ∘ 𝑓)) < 𝑥)
227 retopbas 23924 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ran (,) ∈ TopBases
228 bastg 22116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
229227, 228ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ran (,) ⊆ (topGen‘ran (,))
23013, 229sstri 3930 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))
231 uniopn 22046 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))) → ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)))
23289, 230, 231mp2an 689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))
233 mblfinlem2 35815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) ∧ 𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)))
234232, 233mp3an1 1447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)))
235119eqcomd 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (vol*‘𝑐) = (vol‘𝑐))
236235anim1i 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑥 < (vol*‘𝑐)) → ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐)))
237236ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (𝑥 < (vol*‘𝑐) → ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐))))
238237anim2d 612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)) → (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐)))))
239 fvex 6787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (vol*‘𝑐) ∈ V
240 eqeq1 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑦 = (vol*‘𝑐) → (𝑦 = (vol‘𝑐) ↔ (vol*‘𝑐) = (vol‘𝑐)))
241240anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = (vol*‘𝑐) → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ↔ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘𝑐) = (vol‘𝑐))))
242 breq2 5078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = (vol*‘𝑐) → (𝑥 < 𝑦 ↔ 𝑥 < (vol*‘𝑐)))
243241, 242anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 = (vol*‘𝑐) → (((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦) ↔ ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘𝑐) = (vol‘𝑐)) ∧ 𝑥 < (vol*‘𝑐))))
244239, 243spcev 3545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘𝑐) = (vol‘𝑐)) ∧ 𝑥 < (vol*‘𝑐)) → ∃𝑦((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
245244anasss 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1850 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = 𝑦 → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐))))
253252rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = 𝑦 → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐))))
254253rexab 3631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 303 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}𝑥 < 𝑦)
257256adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9216 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) = (vol*‘∪ ran ((,) ∘ 𝑓)))
259258eqcomd 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘∪ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ))
260259adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑎 → (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝑎 ⊆ ∪ ran ((,) ∘ 𝑓)))
262 fveq2 6774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑎 → (vol‘𝑐) = (vol‘𝑎))
263262eqeq2d 2749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑎 → (𝑧 = (vol‘𝑐) ↔ 𝑧 = (vol‘𝑎)))
264261, 263anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = 𝑎 → ((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))))
265264cbvrexvw 3384 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎)))
266265abbii 2808 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))} = {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))}
267266supeq1i 9206 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = 𝑧 → (𝑦 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑏)))
271270anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = 𝑧 → ((𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑏))))
272271rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑏))))
273 sseq1 3946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑐 → (𝑏 ⊆ 𝐴 ↔ 𝑐 ⊆ 𝐴))
274 fveq2 6774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑐 → (vol‘𝑏) = (vol‘𝑐))
275274eqeq2d 2749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑐 → (𝑧 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑐)))
276273, 275anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 = 𝑐 → ((𝑏 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑏)) ↔ (𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))))
277276cbvrexvw 3384 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑏)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐)))
278272, 277bitrdi 287 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))))
279278cbvabv 2811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} = {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ 𝐴 ∧ 𝑧 = (vol‘𝑐))}
280279supeq1i 9206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 35816 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑎 → (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ↔ 𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
287286, 263anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = 𝑎 → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎))))
288287cbvrexvw 3384 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎)))
289288abbii 2808 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9206 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . . . . . . . . . . . . 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 35816 . . . . . . . . . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . . . . . . . . . . . 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 7293 . . . . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . . . . 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 5102 . . . . . . . . . . . . . . . . . . . 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 4268 . . . . . . . . . . . . . . . . . . . . . . . 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 4268 . . . . . . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . . . . . . 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 11943 . . . . . . . . . . . . . . . . . . . . . 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 3294 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3436 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑢 ∈ V
306 eqeq1 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑢 → (𝑧 = (vol‘𝑎) ↔ 𝑢 = (vol‘𝑎)))
307306anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = 𝑢 → ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎))))
308307rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3436 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑣 ∈ V
311 eqeq1 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑣 → (𝑧 = (vol‘𝑐) ↔ 𝑣 = (vol‘𝑐)))
312311anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = 𝑣 → ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))))
313312rexbidv 3226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 627 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 277 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐)) → (𝑢 + 𝑣) = ((vol‘𝑎) + (vol‘𝑐)))
31959adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑎 ∈ dom vol)
320319ad2antlr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑐 ∈ dom vol)
322321ad2antlr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ∩ 𝑐) ⊆ ((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
324185ineq1i 4142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = ((∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
325 incom 4135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
326 disjdif 4405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) = ∅
327324, 325, 3263eqtri 2770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = ∅
328323, 327sseqtrdi 3971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ∩ 𝑐) ⊆ ∅)
329 ss0 4332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∩ 𝑐) ⊆ ∅ → (𝑎 ∩ 𝑐) = ∅)
330328, 329syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ∩ 𝑐) = ∅)
331330adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) → (𝑎 ∩ 𝑐) = ∅)
33261adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑎) = (vol*‘𝑎))
333332ad2antlr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 521 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) → (𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ))
335683expa 1117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
336334, 335sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
337336ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴)) → (vol*‘𝑎) ∈ ℝ)
338337ad2ant2r 744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑐) = (vol*‘𝑐))
341340ad2antlr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 521 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) → (𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ))
3431243expa 1117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑐 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
344342, 343sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
345344ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (vol*‘𝑐) ∈ ℝ)
346345ad2ant2rl 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1377 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24701 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ dom vol ∧ 𝑐 ∈ dom vol) → (𝑎 ∪ 𝑐) ∈ dom vol)
35159, 117, 350syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (𝑎 ∪ 𝑐) ∈ dom vol)
352 mblvol 24694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 = (𝑢 + 𝑣) ∧ (𝑢 + 𝑣) = (vol*‘(𝑎 ∪ 𝑐))) → 𝑦 = (vol*‘(𝑎 ∪ 𝑐)))
358357ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑢 + 𝑣) = (vol*‘(𝑎 ∪ 𝑐)) ∧ 𝑦 = (𝑢 + 𝑣)) → 𝑦 = (vol*‘(𝑎 ∪ 𝑐)))
359356, 358sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓)))
361 unss 4118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑎 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑐 ⊆ ∪ ran ((,) ∘ 𝑓)) ↔ (𝑎 ∪ 𝑐) ⊆ ∪ ran ((,) ∘ 𝑓))
362360, 361sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ∪ 𝑐) ⊆ ∪ ran ((,) ∘ 𝑓))
363 ovolss 24649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑎 ∪ 𝑐) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘(𝑎 ∪ 𝑐)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
364362, 16, 363sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) → (vol*‘(𝑎 ∪ 𝑐)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
365364ad3antlr 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5096 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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, 368syl5bir 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3223 . . . . . . . . . . . . . . . . . . . . . . . . . . 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, 370syl5bir 242 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3222 . . . . . . . . . . . . . . . . . . . . . . . . 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 1930 . . . . . . . . . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑦 → (𝑡 = (𝑢 + 𝑣) ↔ 𝑦 = (𝑢 + 𝑣)))
3753742rexbidv 3229 . . . . . . . . . . . . . . . . . . . . . . . . 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 3628 . . . . . . . . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3921 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}) → 𝑢 ∈ ℝ)
380130sselda 3921 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}) → 𝑣 ∈ ℝ)
381 readdcl 10954 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ)
382379, 380, 381syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3223 . . . . . . . . . . . . . . . . . . . . . . . . . 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 4002 . . . . . . . . . . . . . . . . . . . . . . . . 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 11150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (0 + 0) = 0
390389eqcomi 2747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 = (0 + 0)
391 rspceov 7322 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1460 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑡 = 0 → (𝑡 = (𝑢 + 𝑣) ↔ 0 = (𝑢 + 𝑣)))
3943932rexbidv 3229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3545 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4314 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5134 . . . . . . . . . . . . . . . . . . . . . . . . . 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 684 . . . . . . . . . . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . . . . . . . . . . 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 11941 . . . . . . . . . . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . . . . . . . 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 5096 . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . 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 11132 . . . . . . . . . . . . . . . . . . 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 593 . . . . . . . . . . . . . . . . . 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 3032 . . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . . 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 3943 . . . . . . . . . . . . . . . . . 18 ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ((,) ∘ 𝑓)
41320ovollb 24643 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
414412, 413mpan2 688 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
415414ad2antlr 724 . . . . . . . . . . . . . . . 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 12896 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 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 5102 . . . . . . . . . . . . 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 458 . . . . . . . . . . . 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 593 . . . . . . . . . . 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 3213 . . . . . . . . . 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 3104 . . . . . . . . 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 2742 . . . . . . . . . . . 12 (𝑣 = 𝑢 → (𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))
425424anbi2d 629 . . . . . . . . . . 11 (𝑣 = 𝑢 → ((𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))))
426425rexbidv 3226 . . . . . . . . . 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 3630 . . . . . . . . 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 233 . . . . . . . 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 4013 . . . . . . . . 9 {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ ℝ*
43011adantl 482 . . . . . . . . 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 13069 . . . . . . . . 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 587 . . . . . . . 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 256 . . . . . . 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 2738 . . . . . . . . 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 24637 . . . . . . . 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 725 . . . . . . 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 5102 . . . . . 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 457 . . . . 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 593 . . . 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 3103 . . 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 24691 . . . . 5 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘𝑤))))
442441baibr 537 . . . 4 (𝐴 ⊆ ℝ → (∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤 ∩ 𝐴)) + (vol*‘(𝑤 ∖ 𝐴))) ≤ (vol*‘𝑤)) ↔ 𝐴 ∈ dom vol))
443442ad2antrr 723 . . 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 231 . 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 798 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 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839   class class class wbr 5074   Or wor 5502   × cxp 5587  dom cdm 5589  ran crn 5590   ∘ ccom 5593  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑m cmap 8615  supcsup 9199  infcinf 9200  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874  +∞cpnf 11006  ℝ*cxr 11008   < clt 11009   ≤ cle 11010   − cmin 11205  ℕcn 11973  (,)cioo 13079  [,)cico 13081  seqcseq 13721  abscabs 14945  topGenctg 17148  Topctop 22042  TopBasesctb 22095  Clsdccld 22167  vol*covol 24626  volcvol 24627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-omul 8302  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-acn 9700  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170  df-cmp 22538  df-conn 22563  df-ovol 24628  df-vol 24629
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator