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Theorem ismblfin 33874
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 (,)))(𝑏𝐴𝑦 = (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 33873 . 2 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ))
2 elpwi 4325 . . . . 5 (𝑤 ∈ 𝒫 ℝ → 𝑤 ⊆ ℝ)
3 elmapi 8082 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4 inss1 3992 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐴) ⊆ 𝑤
5 ovolsscl 23544 . . . . . . . . . . . . . . . . . . . 20 (((𝑤𝐴) ⊆ 𝑤𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤𝐴)) ∈ ℝ)
64, 5mp3an1 1572 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤𝐴)) ∈ ℝ)
7 difss 3899 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐴) ⊆ 𝑤
8 ovolsscl 23544 . . . . . . . . . . . . . . . . . . . 20 (((𝑤𝐴) ⊆ 𝑤𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤𝐴)) ∈ ℝ)
97, 8mp3an1 1572 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → (vol*‘(𝑤𝐴)) ∈ ℝ)
106, 9readdcld 10323 . . . . . . . . . . . . . . . . . 18 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ∈ ℝ)
1110rexrd 10343 . . . . . . . . . . . . . . . . 17 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ∈ ℝ*)
1211ad3antlr 722 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ran ((,) ∘ 𝑓)) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ∈ ℝ*)
13 rncoss 5555 . . . . . . . . . . . . . . . . . . 19 ran ((,) ∘ 𝑓) ⊆ ran (,)
1413unissi 4619 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ⊆ ran (,)
15 unirnioo 12476 . . . . . . . . . . . . . . . . . 18 ℝ = ran (,)
1614, 15sseqtr4i 3798 . . . . . . . . . . . . . . . . 17 ran ((,) ∘ 𝑓) ⊆ ℝ
17 ovolcl 23536 . . . . . . . . . . . . . . . . 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 (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ran ((,) ∘ 𝑓)) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*)
19 eqid 2765 . . . . . . . . . . . . . . . . . . 19 ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
20 eqid 2765 . . . . . . . . . . . . . . . . . . 19 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
2119, 20ovolsf 23530 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
22 frn 6229 . . . . . . . . . . . . . . . . . . 19 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
23 icossxr 12460 . . . . . . . . . . . . . . . . . . 19 (0[,)+∞) ⊆ ℝ*
2422, 23syl6ss 3773 . . . . . . . . . . . . . . . . . 18 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
25 supxrcl 12347 . . . . . . . . . . . . . . . . . 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 718 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ran ((,) ∘ 𝑓)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
28 pnfge 12164 . . . . . . . . . . . . . . . . . . . . . 22 (((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ∈ ℝ* → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ +∞)
2911, 28syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ +∞)
3029ad2antrr 717 . . . . . . . . . . . . . . . . . . . 20 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) = +∞) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ +∞)
31 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) = +∞) → (vol*‘ ran ((,) ∘ 𝑓)) = +∞)
3230, 31breqtrrd 4837 . . . . . . . . . . . . . . . . . . 19 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) = +∞) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
3332adantlll 709 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) = +∞) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
3416, 17ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*
35 nltpnft 12197 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* → ((vol*‘ ran ((,) ∘ 𝑓)) = +∞ ↔ ¬ (vol*‘ ran ((,) ∘ 𝑓)) < +∞))
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ((vol*‘ ran ((,) ∘ 𝑓)) = +∞ ↔ ¬ (vol*‘ ran ((,) ∘ 𝑓)) < +∞)
3736necon2abii 2987 . . . . . . . . . . . . . . . . . . . 20 ((vol*‘ ran ((,) ∘ 𝑓)) < +∞ ↔ (vol*‘ ran ((,) ∘ 𝑓)) ≠ +∞)
38 ovolge0 23539 . . . . . . . . . . . . . . . . . . . . . 22 ( ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤ (vol*‘ ran ((,) ∘ 𝑓)))
3916, 38ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 0 ≤ (vol*‘ ran ((,) ∘ 𝑓))
40 0re 10295 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
41 xrre3 12204 . . . . . . . . . . . . . . . . . . . . . 22 ((((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ 0 ∈ ℝ) ∧ (0 ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) < +∞)) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
4234, 40, 41mpanl12 693 . . . . . . . . . . . . . . . . . . . . 21 ((0 ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) < +∞) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
4339, 42mpan 681 . . . . . . . . . . . . . . . . . . . 20 ((vol*‘ ran ((,) ∘ 𝑓)) < +∞ → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
4437, 43sylbir 226 . . . . . . . . . . . . . . . . . . 19 ((vol*‘ ran ((,) ∘ 𝑓)) ≠ +∞ → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
4510ad3antlr 722 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ∈ ℝ)
46 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) → 𝑧 = (vol‘𝑎))
47 eleq1w 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑎 → (𝑏 ∈ dom vol ↔ 𝑎 ∈ dom vol))
48 uniretop 22845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ℝ = (topGen‘ran (,))
4948cldss 21113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ)
50 dfss4 4023 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
5149, 50sylib 209 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
52 rembl 23598 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ℝ ∈ dom vol
5348cldopn 21115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran (,)))
54 opnmbl 23660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((ℝ ∖ 𝑏) ∈ (topGen‘ran (,)) → (ℝ ∖ 𝑏) ∈ dom vol)
5553, 54syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ dom vol)
56 difmbl 23601 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((ℝ ∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
5752, 55, 56sylancr 581 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
5851, 57eqeltrrd 2845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol)
5947, 58vtoclga 3424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ (Clsd‘(topGen‘ran (,))) → 𝑎 ∈ dom vol)
60 mblvol 23588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ dom vol → (vol‘𝑎) = (vol*‘𝑎))
6159, 60syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑎) = (vol*‘𝑎))
6246, 61sylan9eqr 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))) → 𝑧 = (vol*‘𝑎))
6362adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)))) → 𝑧 = (vol*‘𝑎))
64 inss1 3992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ran ((,) ∘ 𝑓)
65 sstr 3769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ran ((,) ∘ 𝑓)) → 𝑎 ran ((,) ∘ 𝑓))
6664, 65mpan2 682 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) → 𝑎 ran ((,) ∘ 𝑓))
6766ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))) → 𝑎 ran ((,) ∘ 𝑓))
68 ovolsscl 23544 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
6916, 68mp3an2 1573 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
7069ancoms 450 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑎 ran ((,) ∘ 𝑓)) → (vol*‘𝑎) ∈ ℝ)
7167, 70sylan2 586 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)))) → (vol*‘𝑎) ∈ ℝ)
7263, 71eqeltrd 2844 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)))) → 𝑧 ∈ ℝ)
7372rexlimdvaa 3179 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) → 𝑧 ∈ ℝ))
7473abssdv 3836 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ⊆ ℝ)
75 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑦 → (𝑧 = (vol‘𝑎) ↔ 𝑦 = (vol‘𝑎)))
7675anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦 → ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))))
7776rexbidv 3199 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3524 . . . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎)) → 𝑦 = (vol‘𝑎))
8079, 61sylan9eqr 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))) → 𝑦 = (vol*‘𝑎))
81 ovolss 23543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎 ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘𝑎) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
8266, 16, 81sylancl 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) → (vol*‘𝑎) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
8382ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))) → (vol*‘𝑎) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
8480, 83eqbrtrd 4831 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎))) → 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝑓)))
8584rexlimiva 3175 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (vol‘𝑎)) → 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝑓)))
8678, 85mpgbir 1894 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦 ≤ (vol*‘ ran ((,) ∘ 𝑓))
87 brralrspcev 4869 . . . . . . . . . . . . . . . . . . . . . . . 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))}𝑦𝑥)
8886, 87mpan2 682 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}𝑦𝑥)
89 retop 22844 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (topGen‘ran (,)) ∈ Top
90 0cld 21122 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
9189, 90ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∅ ∈ (Clsd‘(topGen‘ran (,)))
92 0ss 4134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ∅ ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴)
93 0mbl 23597 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ∅ ∈ dom vol
94 mblvol 23588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
9593, 94ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (vol‘∅) = (vol*‘∅)
96 ovol0 23551 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (vol*‘∅) = 0
9795, 96eqtr2i 2788 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 = (vol‘∅)
9892, 97pm3.2i 462 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∅ ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘∅))
99 sseq1 3786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = ∅ → (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ↔ ∅ ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴)))
100 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 = ∅ → (vol‘𝑎) = (vol‘∅))
101100eqeq2d 2775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 = ∅ → (0 = (vol‘𝑎) ↔ 0 = (vol‘∅)))
10299, 101anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = ∅ → ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎)) ↔ (∅ ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘∅))))
103102rspcev 3461 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘∅))) → ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎)))
10491, 98, 103mp2an 683 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎))
105 c0ex 10287 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ V
106 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 0 → (𝑧 = (vol‘𝑎) ↔ 0 = (vol‘𝑎)))
107106anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 0 → ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (vol‘𝑎))))
108107rexbidv 3199 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3504 . . . . . . . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}
111110ne0ii 4088 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))} ≠ ∅
112 suprcl 11237 . . . . . . . . . . . . . . . . . . . . . . . 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))}𝑦𝑥) → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) ∈ ℝ)
113111, 112mp3an2 1573 . . . . . . . . . . . . . . . . . . . . . . 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))}𝑦𝑥) → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) ∈ ℝ)
11474, 88, 113syl2anc 579 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ) ∈ ℝ)
115 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) → 𝑧 = (vol‘𝑐))
116 eleq1w 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑐 → (𝑏 ∈ dom vol ↔ 𝑐 ∈ dom vol))
117116, 58vtoclga 3424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → 𝑐 ∈ dom vol)
118 mblvol 23588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ dom vol → (vol‘𝑐) = (vol*‘𝑐))
119117, 118syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑐) = (vol*‘𝑐))
120115, 119sylan9eqr 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))) → 𝑧 = (vol*‘𝑐))
121120adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)))) → 𝑧 = (vol*‘𝑐))
122 difss2 3901 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) → 𝑐 ran ((,) ∘ 𝑓))
123122ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))) → 𝑐 ran ((,) ∘ 𝑓))
124 ovolsscl 23544 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑐 ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
12516, 124mp3an2 1573 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
126125ancoms 450 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑐 ran ((,) ∘ 𝑓)) → (vol*‘𝑐) ∈ ℝ)
127123, 126sylan2 586 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)))) → (vol*‘𝑐) ∈ ℝ)
128121, 127eqeltrd 2844 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)))) → 𝑧 ∈ ℝ)
129128rexlimdvaa 3179 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) → 𝑧 ∈ ℝ))
130129abssdv 3836 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ⊆ ℝ)
131 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑦 → (𝑧 = (vol‘𝑐) ↔ 𝑦 = (vol‘𝑐)))
132131anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦 → ((𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))))
133132rexbidv 3199 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3524 . . . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐)) → 𝑦 = (vol‘𝑐))
136135, 119sylan9eqr 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))) → 𝑦 = (vol*‘𝑐))
137 ovolss 23543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑐 ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘𝑐) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
138122, 16, 137sylancl 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) → (vol*‘𝑐) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
139138ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))) → (vol*‘𝑐) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
140136, 139eqbrtrd 4831 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐))) → 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝑓)))
141140rexlimiva 3175 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑦 = (vol‘𝑐)) → 𝑦 ≤ (vol*‘ ran ((,) ∘ 𝑓)))
142134, 141mpgbir 1894 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦 ≤ (vol*‘ ran ((,) ∘ 𝑓))
143 brralrspcev 4869 . . . . . . . . . . . . . . . . . . . . . . . 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))}𝑦𝑥)
144142, 143mpan2 682 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}𝑦𝑥)
145 0ss 4134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ∅ ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)
146145, 97pm3.2i 462 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∅ ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘∅))
147 sseq1 3786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = ∅ → (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ↔ ∅ ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))
148 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = ∅ → (vol‘𝑐) = (vol‘∅))
149148eqeq2d 2775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = ∅ → (0 = (vol‘𝑐) ↔ 0 = (vol‘∅)))
150147, 149anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = ∅ → ((𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐)) ↔ (∅ ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘∅))))
151150rspcev 3461 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘∅))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐)))
15291, 146, 151mp2an 683 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐))
153 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 0 → (𝑧 = (vol‘𝑐) ↔ 0 = (vol‘𝑐)))
154153anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 0 → ((𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 0 = (vol‘𝑐))))
155154rexbidv 3199 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3504 . . . . . . . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}
158157ne0ii 4088 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))} ≠ ∅
159 suprcl 11237 . . . . . . . . . . . . . . . . . . . . . . . 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))}𝑦𝑥) → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) ∈ ℝ)
160158, 159mp3an2 1573 . . . . . . . . . . . . . . . . . . . . . . 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))}𝑦𝑥) → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) ∈ ℝ)
161130, 144, 160syl2anc 579 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) ∈ ℝ)
162114, 161readdcld 10323 . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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 477 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
1656ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤𝐴)) ∈ ℝ)
1669ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤𝐴)) ∈ ℝ)
167 ovolsscl 23544 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((( ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
16864, 16, 167mp3an12 1575 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
169168adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
170 difss 3899 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑓)
171 ovolsscl 23544 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
172170, 16, 171mp3an12 1575 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
173172adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
174 ssrin 3997 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ran ((,) ∘ 𝑓) → (𝑤𝐴) ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴))
17564, 16sstri 3770 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ℝ
176 ovolss 23543 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑤𝐴) ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ℝ) → (vol*‘(𝑤𝐴)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝐴)))
177174, 175, 176sylancl 580 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ran ((,) ∘ 𝑓) → (vol*‘(𝑤𝐴)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝐴)))
178177ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤𝐴)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝐴)))
179 ssdif 3907 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ran ((,) ∘ 𝑓) → (𝑤𝐴) ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴))
180170, 16sstri 3770 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ
181 ovolss 23543 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑤𝐴) ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ) → (vol*‘(𝑤𝐴)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)))
182179, 180, 181sylancl 580 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ran ((,) ∘ 𝑓) → (vol*‘(𝑤𝐴)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)))
183182ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(𝑤𝐴)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)))
184165, 166, 169, 173, 178, 183le2addd 10900 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝐴)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴))))
185 dfin4 4032 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ran ((,) ∘ 𝑓) ∩ 𝐴) = ( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))
186185fveq2i 6378 . . . . . . . . . . . . . . . . . . . . . . . 24 (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝐴)) = (vol*‘( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))
187186oveq1i 6852 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝐴)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴))) = ((vol*‘( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)))
188184, 187syl6breq 4850 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴))))
189188adantlll 709 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴))))
190 simpll 783 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ran ((,) ∘ 𝑓)) → ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )))
191185sseq2i 3790 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ↔ 𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))
192191anbi1i 617 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ 𝑧 = (vol‘𝑎)))
193192rexbii 3188 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2882 . . . . . . . . . . . . . . . . . . . . . . . . 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 8560 . . . . . . . . . . . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → ( ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ))
197196adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ( ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ))
198172, 180jctil 515 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ ∧ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ))
199198adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ ∧ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ))
200 ltso 10372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑥 ∈ V
204 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = 𝑥 → (𝑧 = (vol‘𝑐) ↔ 𝑥 = (vol‘𝑐)))
205204anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = 𝑥 → ((𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))))
206205rexbidv 3199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 682 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 ran ((,) ∘ 𝑓) → (vol*‘𝑐) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
209208ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → (vol*‘𝑐) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
21048cldss 21113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → 𝑐 ⊆ ℝ)
211 ovolcl 23536 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ⊆ ℝ → (vol*‘𝑐) ∈ ℝ*)
212210, 211syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (vol*‘𝑐) ∈ ℝ*)
213 xrlenlt 10357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((vol*‘𝑐) ∈ ℝ* ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*) → ((vol*‘𝑐) ≤ (vol*‘ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
214212, 34, 213sylancl 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → ((vol*‘𝑐) ≤ (vol*‘ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
215214adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑥 = (vol‘𝑐)) → 𝑥 = (vol*‘𝑐))
218 breq2 4813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑥 = (vol*‘𝑐) → ((vol*‘ ran ((,) ∘ 𝑓)) < 𝑥 ↔ (vol*‘ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
219218notbid 309 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → (¬ (vol*‘ ran ((,) ∘ 𝑓)) < 𝑥 ↔ ¬ (vol*‘ ran ((,) ∘ 𝑓)) < (vol*‘𝑐)))
222215, 221bitr4d 273 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → ((vol*‘𝑐) ≤ (vol*‘ ran ((,) ∘ 𝑓)) ↔ ¬ (vol*‘ ran ((,) ∘ 𝑓)) < 𝑥))
223209, 222mpbid 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑐 ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐))) → ¬ (vol*‘ ran ((,) ∘ 𝑓)) < 𝑥)
224223rexlimiva 3175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑥 = (vol‘𝑐)) → ¬ (vol*‘ ran ((,) ∘ 𝑓)) < 𝑥)
225207, 224sylbi 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))} → ¬ (vol*‘ ran ((,) ∘ 𝑓)) < 𝑥)
226225adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑥 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}) → ¬ (vol*‘ ran ((,) ∘ 𝑓)) < 𝑥)
227 retopbas 22843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ran (,) ∈ TopBases
228 bastg 21050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
229227, 228ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ran (,) ⊆ (topGen‘ran (,))
23013, 229sstri 3770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))
231 uniopn 20981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))) → ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)))
23289, 230, 231mp2an 683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))
233 mblfinlem2 33871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (( ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) ∧ 𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘ ran ((,) ∘ 𝑓))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)))
234232, 233mp3an1 1572 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘ ran ((,) ∘ 𝑓))) → ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)))
235119eqcomd 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (vol*‘𝑐) = (vol‘𝑐))
236235anim1i 608 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑐 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑥 < (vol*‘𝑐)) → ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐)))
237236ex 401 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → (𝑥 < (vol*‘𝑐) → ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐))))
238237anim2d 605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 ∈ (Clsd‘(topGen‘ran (,))) → ((𝑐 ran ((,) ∘ 𝑓) ∧ 𝑥 < (vol*‘𝑐)) → (𝑐 ran ((,) ∘ 𝑓) ∧ ((vol*‘𝑐) = (vol‘𝑐) ∧ 𝑥 < (vol*‘𝑐)))))
239 fvex 6388 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (vol*‘𝑐) ∈ V
240 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑦 = (vol*‘𝑐) → (𝑦 = (vol‘𝑐) ↔ (vol*‘𝑐) = (vol‘𝑐)))
241240anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = (vol*‘𝑐) → ((𝑐 ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ↔ (𝑐 ran ((,) ∘ 𝑓) ∧ (vol*‘𝑐) = (vol‘𝑐))))
242 breq2 4813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = (vol*‘𝑐) → (𝑥 < 𝑦𝑥 < (vol*‘𝑐)))
243241, 242anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 = (vol*‘𝑐) → (((𝑐 ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦) ↔ ((𝑐 ran ((,) ∘ 𝑓) ∧ (vol*‘𝑐) = (vol‘𝑐)) ∧ 𝑥 < (vol*‘𝑐))))
244239, 243spcev 3452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑐 ran ((,) ∘ 𝑓) ∧ (vol*‘𝑐) = (vol‘𝑐)) ∧ 𝑥 < (vol*‘𝑐)) → ∃𝑦((𝑐 ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
245244anasss 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3236 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1943 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑦𝑐 ∈ (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 3378 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))∃𝑦((𝑐 ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦) ↔ ∃𝑦𝑐 ∈ (Clsd‘(topGen‘ran (,)))((𝑐 ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐)) ∧ 𝑥 < 𝑦))
252131anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = 𝑦 → ((𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐))))
253252rexbidv 3199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = 𝑦 → (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑦 = (vol‘𝑐))))
254253rexab 3526 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ ℝ ∧ 𝑥 < (vol*‘ ran ((,) ∘ 𝑓))) → ∃𝑦 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}𝑥 < 𝑦)
257256adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8570 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) = (vol*‘ ran ((,) ∘ 𝑓)))
259258eqcomd 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ))
260259adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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 3786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑎 → (𝑐 ran ((,) ∘ 𝑓) ↔ 𝑎 ran ((,) ∘ 𝑓)))
262 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑎 → (vol‘𝑐) = (vol‘𝑎))
263262eqeq2d 2775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑎 → (𝑧 = (vol‘𝑐) ↔ 𝑧 = (vol‘𝑎)))
264261, 263anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = 𝑎 → ((𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑎 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))))
265264cbvrexv 3320 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎)))
266265abbii 2882 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑐))} = {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))}
267266supeq1i 8560 . . . . . . . . . . . . . . . . . . . . . . . . . 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, 267syl6eq 2815 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ran ((,) ∘ 𝑓) ∧ 𝑧 = (vol‘𝑎))}, ℝ, < ))
269 sseq1 3786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑎 → (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ↔ 𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))
270269, 263anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = 𝑎 → ((𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎))))
271270cbvrexv 3320 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑎)))
272271abbii 2882 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))}
273272supeq1i 8560 . . . . . . . . . . . . . . . . . . . . . . . . . 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, < )
274 simpll 783 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
275 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = 𝑧 → (𝑦 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑏)))
276275anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = 𝑧 → ((𝑏𝐴𝑦 = (vol‘𝑏)) ↔ (𝑏𝐴𝑧 = (vol‘𝑏))))
277276rexbidv 3199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑧 = (vol‘𝑏))))
278 sseq1 3786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑐 → (𝑏𝐴𝑐𝐴))
279 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑐 → (vol‘𝑏) = (vol‘𝑐))
280279eqeq2d 2775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑐 → (𝑧 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑐)))
281278, 280anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 = 𝑐 → ((𝑏𝐴𝑧 = (vol‘𝑏)) ↔ (𝑐𝐴𝑧 = (vol‘𝑐))))
282281cbvrexv 3320 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑧 = (vol‘𝑏)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐𝐴𝑧 = (vol‘𝑐)))
283277, 282syl6bb 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏)) ↔ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐𝐴𝑧 = (vol‘𝑐))))
284283cbvabv 2890 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} = {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐𝐴𝑧 = (vol‘𝑐))}
285284supeq1i 8560 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐𝐴𝑧 = (vol‘𝑐))}, ℝ, < )
286285eqeq2i 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ (vol*‘𝐴) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐𝐴𝑧 = (vol‘𝑐))}, ℝ, < ))
287286biimpi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) → (vol*‘𝐴) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐𝐴𝑧 = (vol‘𝑐))}, ℝ, < ))
288287ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝐴) = sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐𝐴𝑧 = (vol‘𝑐))}, ℝ, < ))
289 mblfinlem3 33872 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (,)))(𝑐𝐴𝑧 = (vol‘𝑐))}, ℝ, < ))) → sup({𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}, ℝ, < ) = (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)))
290197, 274, 260, 288, 289syl112anc 1493 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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)))
291273, 290syl5reqr 2814 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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 33872 . . . . . . . . . . . . . . . . . . . . . . . . 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 1493 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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, 293syl5eq 2811 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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, 290oveq12d 6860 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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 575 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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 4837 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (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 4085 . . . . . . . . . . . . . . . . . . . . . . . 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 4085 . . . . . . . . . . . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . . . . . . . . . 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 11245 . . . . . . . . . . . . . . . . . . . . . 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 3254 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑢 ∈ V
306 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑢 → (𝑧 = (vol‘𝑎) ↔ 𝑢 = (vol‘𝑎)))
307306anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = 𝑢 → ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎)) ↔ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑢 = (vol‘𝑎))))
308307rexbidv 3199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑣 ∈ V
311 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑣 → (𝑧 = (vol‘𝑐) ↔ 𝑣 = (vol‘𝑐)))
312311anbi2d 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = 𝑣 → ((𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐)) ↔ (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑣 = (vol‘𝑐))))
313312rexbidv 3199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 620 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 269 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑢 = (vol‘𝑎) ∧ 𝑣 = (vol‘𝑐)) → (𝑢 + 𝑣) = ((vol‘𝑎) + (vol‘𝑐)))
31959adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑎 ∈ dom vol)
320319ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → 𝑐 ∈ dom vol)
322321ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎𝑐) ⊆ (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))
324185ineq1i 3972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) = (( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴))
325 incom 3967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) = (( ran ((,) ∘ 𝑓) ∖ 𝐴) ∩ ( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))
326 disjdif 4200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (( ran ((,) ∘ 𝑓) ∖ 𝐴) ∩ ( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) = ∅
327324, 325, 3263eqtri 2791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (( ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) = ∅
328323, 327syl6sseq 3811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎𝑐) ⊆ ∅)
329 ss0 4136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎𝑐) ⊆ ∅ → (𝑎𝑐) = ∅)
330328, 329syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎𝑐) = ∅)
331330adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) → (𝑎𝑐) = ∅)
33261adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑎) = (vol*‘𝑎))
333332ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) → (𝑎 ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ))
335683expa 1147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑎 ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
336334, 335sylan 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑎) ∈ ℝ)
337336ancoms 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴)) → (vol*‘𝑎) ∈ ℝ)
338337ad2ant2r 753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘𝑐) = (vol*‘𝑐))
341340ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) → (𝑐 ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ))
3431243expa 1147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑐 ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
344342, 343sylan 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑐) ∈ ℝ)
345344ancoms 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) → (vol*‘𝑐) ∈ ℝ)
346345ad2ant2rl 755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 23603 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑎 ∈ dom vol ∧ 𝑐 ∈ dom vol ∧ (𝑎𝑐) = ∅) ∧ ((vol‘𝑎) ∈ ℝ ∧ (vol‘𝑐) ∈ ℝ)) → (vol‘(𝑎𝑐)) = ((vol‘𝑎) + (vol‘𝑐)))
349320, 322, 331, 339, 347, 348syl32anc 1497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol‘(𝑎𝑐)) = ((vol‘𝑎) + (vol‘𝑐)))
350 unmbl 23595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ dom vol ∧ 𝑐 ∈ dom vol) → (𝑎𝑐) ∈ dom vol)
35159, 117, 350syl2an 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (𝑎𝑐) ∈ dom vol)
352 mblvol 23588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎𝑐) ∈ dom vol → (vol‘(𝑎𝑐)) = (vol*‘(𝑎𝑐)))
353351, 352syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,)))) → (vol‘(𝑎𝑐)) = (vol*‘(𝑎𝑐)))
354353ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑎 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑐 ∈ (Clsd‘(topGen‘ran (,))))) ∧ (𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) → (vol‘(𝑎𝑐)) = (vol*‘(𝑎𝑐)))
355349, 354eqtr3d 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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*‘(𝑎𝑐)))
356318, 355sylan9eqr 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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*‘(𝑎𝑐)))
357 eqtr 2784 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 = (𝑢 + 𝑣) ∧ (𝑢 + 𝑣) = (vol*‘(𝑎𝑐))) → 𝑦 = (vol*‘(𝑎𝑐)))
358357ancoms 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑢 + 𝑣) = (vol*‘(𝑎𝑐)) ∧ 𝑦 = (𝑢 + 𝑣)) → 𝑦 = (vol*‘(𝑎𝑐)))
359356, 358sylan 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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*‘(𝑎𝑐)))
36066, 122anim12i 606 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎 ran ((,) ∘ 𝑓) ∧ 𝑐 ran ((,) ∘ 𝑓)))
361 unss 3949 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑎 ran ((,) ∘ 𝑓) ∧ 𝑐 ran ((,) ∘ 𝑓)) ↔ (𝑎𝑐) ⊆ ran ((,) ∘ 𝑓))
362360, 361sylib 209 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) → (𝑎𝑐) ⊆ ran ((,) ∘ 𝑓))
363 ovolss 23543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑎𝑐) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘(𝑎𝑐)) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
364362, 16, 363sylancl 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) → (vol*‘(𝑎𝑐)) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
365364ad3antlr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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*‘(𝑎𝑐)) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
366359, 365eqbrtrd 4831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3185 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3184 . . . . . . . . . . . . . . . . . . . . . . . . 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 2022 . . . . . . . . . . . . . . . . . . . . . . . 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 2769 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑦 → (𝑡 = (𝑢 + 𝑣) ↔ 𝑦 = (𝑢 + 𝑣)))
3753742rexbidv 3204 . . . . . . . . . . . . . . . . . . . . . . . . 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 3524 . . . . . . . . . . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑢 ∈ {𝑧 ∣ ∃𝑎 ∈ (Clsd‘(topGen‘ran (,)))(𝑎 ⊆ ( ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (vol‘𝑎))}) → 𝑢 ∈ ℝ)
380130sselda 3761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑐 ∈ (Clsd‘(topGen‘ran (,)))(𝑐 ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∧ 𝑧 = (vol‘𝑐))}) → 𝑣 ∈ ℝ)
381 readdcl 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ)
382379, 380, 381syl2an 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3185 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3836 . . . . . . . . . . . . . . . . . . . . . . . . 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 10465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (0 + 0) = 0
390389eqcomi 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 = (0 + 0)
391 rspceov 6888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1585 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑡 = 0 → (𝑡 = (𝑢 + 𝑣) ↔ 0 = (𝑢 + 𝑣)))
3943932rexbidv 3204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3452 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → ∃𝑡𝑢 ∈ {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 𝑡𝑢 ∈ {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 4119 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)} ≠ ∅ ↔ ∃𝑡𝑢 ∈ {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 222 . . . . . . . . . . . . . . . . . . . . . . . . . 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 4869 . . . . . . . . . . . . . . . . . . . . . . . . . 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)}𝑦𝑥)
401377, 400mpdan 678 . . . . . . . . . . . . . . . . . . . . . . . . 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)}𝑦𝑥)
402388, 399, 4013jca 1158 . . . . . . . . . . . . . . . . . . . . . . . 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)}𝑦𝑥))
403 suprleub 11243 . . . . . . . . . . . . . . . . . . . . . . . 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)}𝑦𝑥) ∧ (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 679 . . . . . . . . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . . . . . . . . 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 4831 . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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 10448 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
40944, 408sylan2 586 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≠ +∞) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
41033, 409pm2.61dane 3024 . . . . . . . . . . . . . . . . 17 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑤 ran ((,) ∘ 𝑓)) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
411410adantlr 706 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ran ((,) ∘ 𝑓)) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
412 ssid 3783 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)
41320ovollb 23537 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
414412, 413mpan2 682 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
415414ad2antlr 718 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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 12195 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ran ((,) ∘ 𝑓)) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
417416adantr 472 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
418 simpr 477 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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 4837 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑤 ran ((,) ∘ 𝑓)) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢)
420419expl 449 . . . . . . . . . . . 12 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝑤 ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢))
4213, 420sylan2 586 . . . . . . . . . . 11 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)) → ((𝑤 ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢))
422421rexlimdva 3178 . . . . . . . . . 10 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑤 ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢))
423422ralrimivw 3114 . . . . . . . . 9 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢))
424 eqeq1 2769 . . . . . . . . . . . 12 (𝑣 = 𝑢 → (𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))
425424anbi2d 622 . . . . . . . . . . 11 (𝑣 = 𝑢 → ((𝑤 ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ (𝑤 ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))))
426425rexbidv 3199 . . . . . . . . . 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 3525 . . . . . . . . 9 (∀𝑢 ∈ {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑤 ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢 ↔ ∀𝑢 ∈ ℝ* (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑤 ran ((,) ∘ 𝑓) ∧ 𝑢 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢))
428423, 427sylibr 225 . . . . . . . 8 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢)
429 ssrab2 3847 . . . . . . . . 9 {𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑤 ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ ℝ*
43011adantl 473 . . . . . . . . 9 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ∈ ℝ*)
431 infxrgelb 12367 . . . . . . . . 9 (({𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑤 ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ ℝ* ∧ ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ∈ ℝ*) → (((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 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*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢))
432429, 430, 431sylancr 581 . . . . . . . 8 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → (((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 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*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ 𝑢))
433428, 432mpbird 248 . . . . . . 7 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ inf({𝑣 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑤 ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
434 eqid 2765 . . . . . . . . 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 23531 . . . . . . . 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 719 . . . . . . 7 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (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 4837 . . . . . 6 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ (𝑤 ⊆ ℝ ∧ (vol*‘𝑤) ∈ ℝ)) → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘𝑤))
438437expr 448 . . . . 5 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ 𝑤 ⊆ ℝ) → ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘𝑤)))
4392, 438sylan2 586 . . . 4 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) ∧ 𝑤 ∈ 𝒫 ℝ) → ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘𝑤)))
440439ralrimiva 3113 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) → ∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘𝑤)))
441 ismbl2 23585 . . . . 5 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘𝑤))))
442441baibr 532 . . . 4 (𝐴 ⊆ ℝ → (∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘𝑤)) ↔ 𝐴 ∈ dom vol))
443442ad2antrr 717 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) → (∀𝑤 ∈ 𝒫 ℝ((vol*‘𝑤) ∈ ℝ → ((vol*‘(𝑤𝐴)) + (vol*‘(𝑤𝐴))) ≤ (vol*‘𝑤)) ↔ 𝐴 ∈ dom vol))
444440, 443mpbid 223 . 2 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )) → 𝐴 ∈ dom vol)
4451, 444impbida 835 1 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴 ∈ dom vol ↔ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315   cuni 4594   class class class wbr 4809   Or wor 5197   × cxp 5275  dom cdm 5277  ran crn 5278  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  supcsup 8553  infcinf 8554  cr 10188  0cc0 10189  1c1 10190   + caddc 10192  +∞cpnf 10325  *cxr 10327   < clt 10328  cle 10329  cmin 10520  cn 11274  (,)cioo 12377  [,)cico 12379  seqcseq 13008  abscabs 14259  topGenctg 16364  Topctop 20977  TopBasesctb 21029  Clsdccld 21100  vol*covol 23520  volcvol 23521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-disj 4778  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-omul 7769  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-acn 9019  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-ioo 12381  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-rlim 14505  df-sum 14702  df-rest 16349  df-topgen 16370  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-cmp 21470  df-conn 21495  df-ovol 23522  df-vol 23523
This theorem is referenced by: (None)
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