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Theorem ismblfin 37167
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 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (𝐴 ∈ dom vol ↔ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )))
Distinct variable group:   𝑦,𝑏,𝐴
Proof of Theorem ismblfin
Dummy variables π‘Ž 𝑐 𝑓 𝑑 𝑒 𝑣 𝑀 π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mblfinlem4 37166 . 2 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) β†’ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < ))
2 elpwi 4613 . . . . 5 (𝑀 ∈ 𝒫 ℝ β†’ 𝑀 βŠ† ℝ)
3 elmapi 8874 . . . . . . . . . . . 12 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
4 inss1 4231 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∩ 𝐴) βŠ† 𝑀
5 ovolsscl 25435 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∩ 𝐴) βŠ† 𝑀 ∧ 𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) β†’ (vol*β€˜(𝑀 ∩ 𝐴)) ∈ ℝ)
64, 5mp3an1 1444 . . . . . . . . . . . . . . . . . . 19 ((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) β†’ (vol*β€˜(𝑀 ∩ 𝐴)) ∈ ℝ)
7 difss 4132 . . . . . . . . . . . . . . . . . . . 20 (𝑀 βˆ– 𝐴) βŠ† 𝑀
8 ovolsscl 25435 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 βˆ– 𝐴) βŠ† 𝑀 ∧ 𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) β†’ (vol*β€˜(𝑀 βˆ– 𝐴)) ∈ ℝ)
97, 8mp3an1 1444 . . . . . . . . . . . . . . . . . . 19 ((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) β†’ (vol*β€˜(𝑀 βˆ– 𝐴)) ∈ ℝ)
106, 9readdcld 11281 . . . . . . . . . . . . . . . . . 18 ((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ∈ ℝ)
1110rexrd 11302 . . . . . . . . . . . . . . . . 17 ((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ∈ ℝ*)
1211ad3antlr 729 . . . . . . . . . . . . . . . 16 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ∈ ℝ*)
13 rncoss 5979 . . . . . . . . . . . . . . . . . . 19 ran ((,) ∘ 𝑓) βŠ† ran (,)
1413unissi 4921 . . . . . . . . . . . . . . . . . 18 βˆͺ ran ((,) ∘ 𝑓) βŠ† βˆͺ ran (,)
15 unirnioo 13466 . . . . . . . . . . . . . . . . . 18 ℝ = βˆͺ ran (,)
1614, 15sseqtrri 4019 . . . . . . . . . . . . . . . . 17 βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ
17 ovolcl 25427 . . . . . . . . . . . . . . . . 17 (βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ*)
1816, 17mp1i 13 . . . . . . . . . . . . . . . 16 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ*)
19 eqid 2728 . . . . . . . . . . . . . . . . . . 19 ((abs ∘ βˆ’ ) ∘ 𝑓) = ((abs ∘ βˆ’ ) ∘ 𝑓)
20 eqid 2728 . . . . . . . . . . . . . . . . . . 19 seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) = seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))
2119, 20ovolsf 25421 . . . . . . . . . . . . . . . . . 18 (𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)):β„•βŸΆ(0[,)+∞))
22 frn 6734 . . . . . . . . . . . . . . . . . . 19 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)):β„•βŸΆ(0[,)+∞) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† (0[,)+∞))
23 icossxr 13449 . . . . . . . . . . . . . . . . . . 19 (0[,)+∞) βŠ† ℝ*
2422, 23sstrdi 3994 . . . . . . . . . . . . . . . . . 18 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)):β„•βŸΆ(0[,)+∞) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† ℝ*)
25 supxrcl 13334 . . . . . . . . . . . . . . . . . 18 (ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† ℝ* β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
2621, 24, 253syl 18 . . . . . . . . . . . . . . . . 17 (𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
2726ad2antlr 725 . . . . . . . . . . . . . . . 16 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
28 pnfge 13150 . . . . . . . . . . . . . . . . . . . . . 22 (((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ∈ ℝ* β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ +∞)
2911, 28syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ +∞)
3029ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = +∞) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ +∞)
31 simpr 483 . . . . . . . . . . . . . . . . . . . 20 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = +∞) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = +∞)
3230, 31breqtrrd 5180 . . . . . . . . . . . . . . . . . . 19 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = +∞) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
3332adantlll 716 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = +∞) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
3416, 17ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ*
35 nltpnft 13183 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ* β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = +∞ ↔ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < +∞))
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = +∞ ↔ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < +∞)
3736necon2abii 2988 . . . . . . . . . . . . . . . . . . . 20 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < +∞ ↔ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) β‰  +∞)
38 ovolge0 25430 . . . . . . . . . . . . . . . . . . . . . 22 (βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ β†’ 0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
3916, 38ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))
40 0re 11254 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
41 xrre3 13190 . . . . . . . . . . . . . . . . . . . . . 22 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ 0 ∈ ℝ) ∧ (0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < +∞)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
4234, 40, 41mpanl12 700 . . . . . . . . . . . . . . . . . . . . 21 ((0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < +∞) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
4339, 42mpan 688 . . . . . . . . . . . . . . . . . . . 20 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < +∞ β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
4437, 43sylbir 234 . . . . . . . . . . . . . . . . . . 19 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) β‰  +∞ β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
4510ad3antlr 729 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ∈ ℝ)
46 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) β†’ 𝑧 = (volβ€˜π‘Ž))
47 eleq1w 2812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = π‘Ž β†’ (𝑏 ∈ dom vol ↔ π‘Ž ∈ dom vol))
48 uniretop 24699 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ℝ = βˆͺ (topGenβ€˜ran (,))
4948cldss 22953 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ 𝑏 βŠ† ℝ)
50 dfss4 4261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 βŠ† ℝ ↔ (ℝ βˆ– (ℝ βˆ– 𝑏)) = 𝑏)
5149, 50sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (ℝ βˆ– (ℝ βˆ– 𝑏)) = 𝑏)
52 rembl 25489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ℝ ∈ dom vol
5348cldopn 22955 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (ℝ βˆ– 𝑏) ∈ (topGenβ€˜ran (,)))
54 opnmbl 25551 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((ℝ βˆ– 𝑏) ∈ (topGenβ€˜ran (,)) β†’ (ℝ βˆ– 𝑏) ∈ dom vol)
5553, 54syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (ℝ βˆ– 𝑏) ∈ dom vol)
56 difmbl 25492 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((ℝ ∈ dom vol ∧ (ℝ βˆ– 𝑏) ∈ dom vol) β†’ (ℝ βˆ– (ℝ βˆ– 𝑏)) ∈ dom vol)
5752, 55, 56sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (ℝ βˆ– (ℝ βˆ– 𝑏)) ∈ dom vol)
5851, 57eqeltrrd 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ 𝑏 ∈ dom vol)
5947, 58vtoclga 3565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ π‘Ž ∈ dom vol)
60 mblvol 25479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (π‘Ž ∈ dom vol β†’ (volβ€˜π‘Ž) = (vol*β€˜π‘Ž))
6159, 60syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (volβ€˜π‘Ž) = (vol*β€˜π‘Ž))
6246, 61sylan9eqr 2790 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))) β†’ 𝑧 = (vol*β€˜π‘Ž))
6362adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)))) β†’ 𝑧 = (vol*β€˜π‘Ž))
64 inss1 4231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑓)
65 sstr 3990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓))
6664, 65mpan2 689 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) β†’ π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓))
6766ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))) β†’ π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓))
68 ovolsscl 25435 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘Ž) ∈ ℝ)
6916, 68mp3an2 1445 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘Ž) ∈ ℝ)
7069ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (vol*β€˜π‘Ž) ∈ ℝ)
7167, 70sylan2 591 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)))) β†’ (vol*β€˜π‘Ž) ∈ ℝ)
7263, 71eqeltrd 2829 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)))) β†’ 𝑧 ∈ ℝ)
7372rexlimdvaa 3153 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) β†’ 𝑧 ∈ ℝ))
7473abssdv 4065 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} βŠ† ℝ)
75 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑦 β†’ (𝑧 = (volβ€˜π‘Ž) ↔ 𝑦 = (volβ€˜π‘Ž)))
7675anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦 β†’ ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) ↔ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (volβ€˜π‘Ž))))
7776rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑦 β†’ (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) ↔ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (volβ€˜π‘Ž))))
7877ralab 3688 . . . . . . . . . . . . . . . . . . . . . . . . 25 (βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ↔ βˆ€π‘¦(βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (volβ€˜π‘Ž)) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))))
79 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (volβ€˜π‘Ž)) β†’ 𝑦 = (volβ€˜π‘Ž))
8079, 61sylan9eqr 2790 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (volβ€˜π‘Ž))) β†’ 𝑦 = (vol*β€˜π‘Ž))
81 ovolss 25434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ) β†’ (vol*β€˜π‘Ž) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
8266, 16, 81sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) β†’ (vol*β€˜π‘Ž) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
8382ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (volβ€˜π‘Ž))) β†’ (vol*β€˜π‘Ž) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
8480, 83eqbrtrd 5174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (volβ€˜π‘Ž))) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
8584rexlimiva 3144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑦 = (volβ€˜π‘Ž)) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
8678, 85mpgbir 1793 . . . . . . . . . . . . . . . . . . . . . . . 24 βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))
87 brralrspcev 5212 . . . . . . . . . . . . . . . . . . . . . . . 24 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}𝑦 ≀ π‘₯)
8886, 87mpan2 689 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}𝑦 ≀ π‘₯)
89 retop 24698 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (topGenβ€˜ran (,)) ∈ Top
90 0cld 22962 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((topGenβ€˜ran (,)) ∈ Top β†’ βˆ… ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
9189, 90ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 βˆ… ∈ (Clsdβ€˜(topGenβ€˜ran (,)))
92 0ss 4400 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)
93 0mbl 25488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 βˆ… ∈ dom vol
94 mblvol 25479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (βˆ… ∈ dom vol β†’ (volβ€˜βˆ…) = (vol*β€˜βˆ…))
9593, 94ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (volβ€˜βˆ…) = (vol*β€˜βˆ…)
96 ovol0 25442 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (vol*β€˜βˆ…) = 0
9795, 96eqtr2i 2757 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 = (volβ€˜βˆ…)
9892, 97pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (volβ€˜βˆ…))
99 sseq1 4007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (π‘Ž = βˆ… β†’ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ↔ βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)))
100 fveq2 6902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (π‘Ž = βˆ… β†’ (volβ€˜π‘Ž) = (volβ€˜βˆ…))
101100eqeq2d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (π‘Ž = βˆ… β†’ (0 = (volβ€˜π‘Ž) ↔ 0 = (volβ€˜βˆ…)))
10299, 101anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘Ž = βˆ… β†’ ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (volβ€˜π‘Ž)) ↔ (βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (volβ€˜βˆ…))))
103102rspcev 3611 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((βˆ… ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (volβ€˜βˆ…))) β†’ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (volβ€˜π‘Ž)))
10491, 98, 103mp2an 690 . . . . . . . . . . . . . . . . . . . . . . . . . 26 βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (volβ€˜π‘Ž))
105 c0ex 11246 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ V
106 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 0 β†’ (𝑧 = (volβ€˜π‘Ž) ↔ 0 = (volβ€˜π‘Ž)))
107106anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 0 β†’ ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) ↔ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (volβ€˜π‘Ž))))
108107rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 0 β†’ (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) ↔ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (volβ€˜π‘Ž))))
109105, 108elab 3669 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ↔ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 0 = (volβ€˜π‘Ž)))
110104, 109mpbir 230 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}
111110ne0ii 4341 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} β‰  βˆ…
112 suprcl 12212 . . . . . . . . . . . . . . . . . . . . . . . 24 (({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} βŠ† ℝ ∧ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}𝑦 ≀ π‘₯) β†’ sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) ∈ ℝ)
113111, 112mp3an2 1445 . . . . . . . . . . . . . . . . . . . . . . 23 (({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} βŠ† ℝ ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}𝑦 ≀ π‘₯) β†’ sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) ∈ ℝ)
11474, 88, 113syl2anc 582 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) ∈ ℝ)
115 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) β†’ 𝑧 = (volβ€˜π‘))
116 eleq1w 2812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑐 β†’ (𝑏 ∈ dom vol ↔ 𝑐 ∈ dom vol))
117116, 58vtoclga 3565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ 𝑐 ∈ dom vol)
118 mblvol 25479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ dom vol β†’ (volβ€˜π‘) = (vol*β€˜π‘))
119117, 118syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (volβ€˜π‘) = (vol*β€˜π‘))
120115, 119sylan9eqr 2790 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))) β†’ 𝑧 = (vol*β€˜π‘))
121120adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)))) β†’ 𝑧 = (vol*β€˜π‘))
122 difss2 4134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) β†’ 𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓))
123122ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))) β†’ 𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓))
124 ovolsscl 25435 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘) ∈ ℝ)
12516, 124mp3an2 1445 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘) ∈ ℝ)
126125ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (vol*β€˜π‘) ∈ ℝ)
127123, 126sylan2 591 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)))) β†’ (vol*β€˜π‘) ∈ ℝ)
128121, 127eqeltrd 2829 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)))) β†’ 𝑧 ∈ ℝ)
129128rexlimdvaa 3153 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) β†’ 𝑧 ∈ ℝ))
130129abssdv 4065 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} βŠ† ℝ)
131 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 𝑦 β†’ (𝑧 = (volβ€˜π‘) ↔ 𝑦 = (volβ€˜π‘)))
132131anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑦 β†’ ((𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) ↔ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑦 = (volβ€˜π‘))))
133132rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑦 β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑦 = (volβ€˜π‘))))
134133ralab 3688 . . . . . . . . . . . . . . . . . . . . . . . . 25 (βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ↔ βˆ€π‘¦(βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑦 = (volβ€˜π‘)) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))))
135 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑦 = (volβ€˜π‘)) β†’ 𝑦 = (volβ€˜π‘))
136135, 119sylan9eqr 2790 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑦 = (volβ€˜π‘))) β†’ 𝑦 = (vol*β€˜π‘))
137 ovolss 25434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ) β†’ (vol*β€˜π‘) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
138122, 16, 137sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) β†’ (vol*β€˜π‘) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
139138ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑦 = (volβ€˜π‘))) β†’ (vol*β€˜π‘) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
140136, 139eqbrtrd 5174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑦 = (volβ€˜π‘))) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
141140rexlimiva 3144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑦 = (volβ€˜π‘)) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
142134, 141mpgbir 1793 . . . . . . . . . . . . . . . . . . . . . . . 24 βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))
143 brralrspcev 5212 . . . . . . . . . . . . . . . . . . . . . . . 24 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 ≀ π‘₯)
144142, 143mpan2 689 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 ≀ π‘₯)
145 0ss 4400 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)
146145, 97pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 0 = (volβ€˜βˆ…))
147 sseq1 4007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = βˆ… β†’ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ↔ βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
148 fveq2 6902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = βˆ… β†’ (volβ€˜π‘) = (volβ€˜βˆ…))
149148eqeq2d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = βˆ… β†’ (0 = (volβ€˜π‘) ↔ 0 = (volβ€˜βˆ…)))
150147, 149anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = βˆ… β†’ ((𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 0 = (volβ€˜π‘)) ↔ (βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 0 = (volβ€˜βˆ…))))
151150rspcev 3611 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((βˆ… ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (βˆ… βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 0 = (volβ€˜βˆ…))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 0 = (volβ€˜π‘)))
15291, 146, 151mp2an 690 . . . . . . . . . . . . . . . . . . . . . . . . . 26 βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 0 = (volβ€˜π‘))
153 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 0 β†’ (𝑧 = (volβ€˜π‘) ↔ 0 = (volβ€˜π‘)))
154153anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = 0 β†’ ((𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) ↔ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 0 = (volβ€˜π‘))))
155154rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 0 β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 0 = (volβ€˜π‘))))
156105, 155elab 3669 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 0 = (volβ€˜π‘)))
157152, 156mpbir 230 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}
158157ne0ii 4341 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} β‰  βˆ…
159 suprcl 12212 . . . . . . . . . . . . . . . . . . . . . . . 24 (({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} βŠ† ℝ ∧ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 ≀ π‘₯) β†’ sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ) ∈ ℝ)
160158, 159mp3an2 1445 . . . . . . . . . . . . . . . . . . . . . . 23 (({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} βŠ† ℝ ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 ≀ π‘₯) β†’ sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ) ∈ ℝ)
161130, 144, 160syl2anc 582 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ) ∈ ℝ)
162114, 161readdcld 11281 . . . . . . . . . . . . . . . . . . . . 21 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) + sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < )) ∈ ℝ)
163162adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) + sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < )) ∈ ℝ)
164 simpr 483 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
1656ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(𝑀 ∩ 𝐴)) ∈ ℝ)
1669ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(𝑀 βˆ– 𝐴)) ∈ ℝ)
167 ovolsscl 25435 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
16864, 16, 167mp3an12 1447 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
169168adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)) ∈ ℝ)
170 difss 4132 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑓)
171 ovolsscl 25435 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ)
172170, 16, 171mp3an12 1447 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ)
173172adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ)
174 ssrin 4236 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ (𝑀 ∩ 𝐴) βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴))
17564, 16sstri 3991 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) βŠ† ℝ
176 ovolss 25434 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑀 ∩ 𝐴) βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) βŠ† ℝ) β†’ (vol*β€˜(𝑀 ∩ 𝐴)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)))
177174, 175, 176sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ (vol*β€˜(𝑀 ∩ 𝐴)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)))
178177ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(𝑀 ∩ 𝐴)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)))
179 ssdif 4140 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ (𝑀 βˆ– 𝐴) βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))
180170, 16sstri 3991 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† ℝ
181 ovolss 25434 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑀 βˆ– 𝐴) βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† ℝ) β†’ (vol*β€˜(𝑀 βˆ– 𝐴)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
182179, 180, 181sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ (vol*β€˜(𝑀 βˆ– 𝐴)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
183182ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(𝑀 βˆ– 𝐴)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
184165, 166, 169, 173, 178, 183le2addd 11871 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
185 dfin4 4270 . . . . . . . . . . . . . . . . . . . . . . . . 25 (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) = (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))
186185fveq2i 6905 . . . . . . . . . . . . . . . . . . . . . . . 24 (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
187186oveq1i 7436 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
188184, 187breqtrdi 5193 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
189188adantlll 716 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
190 simpll 765 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )))
191185sseq2i 4011 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ↔ π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
192191anbi1i 622 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) ↔ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ 𝑧 = (volβ€˜π‘Ž)))
193192rexbii 3091 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) ↔ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ 𝑧 = (volβ€˜π‘Ž)))
194193abbii 2798 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} = {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ 𝑧 = (volβ€˜π‘Ž))}
195194supeq1i 9478 . . . . . . . . . . . . . . . . . . . . . . . 24 sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) = sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < )
19616jctl 522 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ))
197196adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ))
198172, 180jctil 518 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† ℝ ∧ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ))
199198adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† ℝ ∧ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ))
200 ltso 11332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 < Or ℝ
201200a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ < Or ℝ)
202 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
203 vex 3477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 π‘₯ ∈ V
204 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = π‘₯ β†’ (𝑧 = (volβ€˜π‘) ↔ π‘₯ = (volβ€˜π‘)))
205204anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = π‘₯ β†’ ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘)) ↔ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ = (volβ€˜π‘))))
206205rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = π‘₯ β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ = (volβ€˜π‘))))
207203, 206elab 3669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (π‘₯ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))} ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ = (volβ€˜π‘)))
20816, 137mpan2 689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ (vol*β€˜π‘) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
209208ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ = (volβ€˜π‘))) β†’ (vol*β€˜π‘) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
21048cldss 22953 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ 𝑐 βŠ† ℝ)
211 ovolcl 25427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 βŠ† ℝ β†’ (vol*β€˜π‘) ∈ ℝ*)
212210, 211syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (vol*β€˜π‘) ∈ ℝ*)
213 xrlenlt 11317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((vol*β€˜π‘) ∈ ℝ* ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ*) β†’ ((vol*β€˜π‘) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ↔ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < (vol*β€˜π‘)))
214212, 34, 213sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ ((vol*β€˜π‘) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ↔ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < (vol*β€˜π‘)))
215214adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ = (volβ€˜π‘))) β†’ ((vol*β€˜π‘) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ↔ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < (vol*β€˜π‘)))
216 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (π‘₯ = (volβ€˜π‘) β†’ π‘₯ = (volβ€˜π‘))
217216, 119sylan9eqr 2790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ π‘₯ = (volβ€˜π‘)) β†’ π‘₯ = (vol*β€˜π‘))
218 breq2 5156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (π‘₯ = (vol*β€˜π‘) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < π‘₯ ↔ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < (vol*β€˜π‘)))
219218notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (π‘₯ = (vol*β€˜π‘) β†’ (Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < π‘₯ ↔ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < (vol*β€˜π‘)))
220217, 219syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ π‘₯ = (volβ€˜π‘)) β†’ (Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < π‘₯ ↔ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < (vol*β€˜π‘)))
221220adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ = (volβ€˜π‘))) β†’ (Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < π‘₯ ↔ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < (vol*β€˜π‘)))
222215, 221bitr4d 281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ = (volβ€˜π‘))) β†’ ((vol*β€˜π‘) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ↔ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < π‘₯))
223209, 222mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ = (volβ€˜π‘))) β†’ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < π‘₯)
224223rexlimiva 3144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ = (volβ€˜π‘)) β†’ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < π‘₯)
225207, 224sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (π‘₯ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))} β†’ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < π‘₯)
226225adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ π‘₯ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}) β†’ Β¬ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) < π‘₯)
227 retopbas 24697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ran (,) ∈ TopBases
228 bastg 22889 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (ran (,) ∈ TopBases β†’ ran (,) βŠ† (topGenβ€˜ran (,)))
229227, 228ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ran (,) βŠ† (topGenβ€˜ran (,))
23013, 229sstri 3991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ran ((,) ∘ 𝑓) βŠ† (topGenβ€˜ran (,))
231 uniopn 22819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((topGenβ€˜ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) βŠ† (topGenβ€˜ran (,))) β†’ βˆͺ ran ((,) ∘ 𝑓) ∈ (topGenβ€˜ran (,)))
23289, 230, 231mp2an 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 βˆͺ ran ((,) ∘ 𝑓) ∈ (topGenβ€˜ran (,))
233 mblfinlem2 37164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((βˆͺ ran ((,) ∘ 𝑓) ∈ (topGenβ€˜ran (,)) ∧ π‘₯ ∈ ℝ ∧ π‘₯ < (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ < (vol*β€˜π‘)))
234232, 233mp3an1 1444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((π‘₯ ∈ ℝ ∧ π‘₯ < (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ < (vol*β€˜π‘)))
235119eqcomd 2734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (vol*β€˜π‘) = (volβ€˜π‘))
236235anim1i 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ π‘₯ < (vol*β€˜π‘)) β†’ ((vol*β€˜π‘) = (volβ€˜π‘) ∧ π‘₯ < (vol*β€˜π‘)))
237236ex 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (π‘₯ < (vol*β€˜π‘) β†’ ((vol*β€˜π‘) = (volβ€˜π‘) ∧ π‘₯ < (vol*β€˜π‘))))
238237anim2d 610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ < (vol*β€˜π‘)) β†’ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜π‘) = (volβ€˜π‘) ∧ π‘₯ < (vol*β€˜π‘)))))
239 fvex 6915 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (vol*β€˜π‘) ∈ V
240 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑦 = (vol*β€˜π‘) β†’ (𝑦 = (volβ€˜π‘) ↔ (vol*β€˜π‘) = (volβ€˜π‘)))
241240anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = (vol*β€˜π‘) β†’ ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ↔ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜π‘) = (volβ€˜π‘))))
242 breq2 5156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = (vol*β€˜π‘) β†’ (π‘₯ < 𝑦 ↔ π‘₯ < (vol*β€˜π‘)))
243241, 242anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 = (vol*β€˜π‘) β†’ (((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦) ↔ ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜π‘) = (volβ€˜π‘)) ∧ π‘₯ < (vol*β€˜π‘))))
244239, 243spcev 3595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜π‘) = (volβ€˜π‘)) ∧ π‘₯ < (vol*β€˜π‘)) β†’ βˆƒπ‘¦((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦))
245244anasss 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜π‘) = (volβ€˜π‘) ∧ π‘₯ < (vol*β€˜π‘))) β†’ βˆƒπ‘¦((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦))
246238, 245syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ < (vol*β€˜π‘)) β†’ βˆƒπ‘¦((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦)))
247246reximia 3078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ π‘₯ < (vol*β€˜π‘)) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))βˆƒπ‘¦((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦))
248234, 247syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((π‘₯ ∈ ℝ ∧ π‘₯ < (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))βˆƒπ‘¦((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦))
249 r19.41v 3186 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦) ↔ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦))
250249exbii 1842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (βˆƒπ‘¦βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦) ↔ βˆƒπ‘¦(βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦))
251 rexcom4 3283 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))βˆƒπ‘¦((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦) ↔ βˆƒπ‘¦βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦))
252131anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = 𝑦 β†’ ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘)) ↔ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘))))
253252rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = 𝑦 β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘))))
254253rexab 3691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (βˆƒπ‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}π‘₯ < 𝑦 ↔ βˆƒπ‘¦(βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦))
255250, 251, 2543bitr4i 302 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))βˆƒπ‘¦((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (volβ€˜π‘)) ∧ π‘₯ < 𝑦) ↔ βˆƒπ‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}π‘₯ < 𝑦)
256248, 255sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((π‘₯ ∈ ℝ ∧ π‘₯ < (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}π‘₯ < 𝑦)
257256adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘₯ ∈ ℝ ∧ π‘₯ < (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))) β†’ βˆƒπ‘¦ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}π‘₯ < 𝑦)
258201, 202, 226, 257eqsupd 9488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ) = (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
259258eqcomd 2734 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ))
260259adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ))
261 sseq1 4007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = π‘Ž β†’ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ↔ π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓)))
262 fveq2 6902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = π‘Ž β†’ (volβ€˜π‘) = (volβ€˜π‘Ž))
263262eqeq2d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = π‘Ž β†’ (𝑧 = (volβ€˜π‘) ↔ 𝑧 = (volβ€˜π‘Ž)))
264261, 263anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = π‘Ž β†’ ((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘)) ↔ (π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘Ž))))
265264cbvrexvw 3233 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘)) ↔ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘Ž)))
266265abbii 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))} = {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘Ž))}
267266supeq1i 9478 . . . . . . . . . . . . . . . . . . . . . . . . . 26 sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ) = sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < )
268260, 267eqtrdi 2784 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ))
269 simpll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ))
270 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = 𝑧 β†’ (𝑦 = (volβ€˜π‘) ↔ 𝑧 = (volβ€˜π‘)))
271270anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = 𝑧 β†’ ((𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘)) ↔ (𝑏 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))))
272271rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = 𝑧 β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))))
273 sseq1 4007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑐 β†’ (𝑏 βŠ† 𝐴 ↔ 𝑐 βŠ† 𝐴))
274 fveq2 6902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑐 β†’ (volβ€˜π‘) = (volβ€˜π‘))
275274eqeq2d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = 𝑐 β†’ (𝑧 = (volβ€˜π‘) ↔ 𝑧 = (volβ€˜π‘)))
276273, 275anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 = 𝑐 β†’ ((𝑏 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘)) ↔ (𝑐 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))))
277276cbvrexvw 3233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘)))
278272, 277bitrdi 286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = 𝑧 β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))))
279278cbvabv 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 {𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))} = {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))}
280279supeq1i 9478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < ) = sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < )
281280eqeq2i 2741 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < ) ↔ (vol*β€˜π΄) = sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ))
282281biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < ) β†’ (vol*β€˜π΄) = sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ))
283282ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π΄) = sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ))
284 mblfinlem3 37165 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) ∧ (𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ) ∧ (vol*β€˜π΄) = sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† 𝐴 ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ))) β†’ sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
285197, 269, 260, 283, 284syl112anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
286 sseq1 4007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = π‘Ž β†’ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ↔ π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
287286, 263anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = π‘Ž β†’ ((𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) ↔ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))))
288287cbvrexvw 3233 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) ↔ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)))
289288abbii 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} = {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}
290289supeq1i 9478 . . . . . . . . . . . . . . . . . . . . . . . . . 26 sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < ) = sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < )
291285, 290eqtr3di 2783 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) = sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ))
292 mblfinlem3 37165 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† ℝ ∧ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) ∧ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) = sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ))) β†’ sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
293197, 199, 268, 291, 292syl112anc 1371 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
294195, 293eqtrid 2780 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
295294, 285oveq12d 7444 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) + sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < )) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
296190, 295sylan 578 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) + sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < )) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
297189, 296breqtrrd 5180 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) + sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < )))
298 ne0i 4338 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} β†’ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} β‰  βˆ…)
299110, 298mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} β‰  βˆ…)
300 ne0i 4338 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} β†’ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} β‰  βˆ…)
301157, 300mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} β‰  βˆ…)
302 eqid 2728 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)} = {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}
30374, 299, 88, 130, 301, 144, 302supadd 12220 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) + sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < )) = sup({𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}, ℝ, < ))
304 reeanv 3224 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž)) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘))) ↔ (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž)) ∧ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘))))
305 vex 3477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑒 ∈ V
306 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑒 β†’ (𝑧 = (volβ€˜π‘Ž) ↔ 𝑒 = (volβ€˜π‘Ž)))
307306anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = 𝑒 β†’ ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) ↔ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž))))
308307rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑒 β†’ (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž)) ↔ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž))))
309305, 308elab 3669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ↔ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž)))
310 vex 3477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑣 ∈ V
311 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑣 β†’ (𝑧 = (volβ€˜π‘) ↔ 𝑣 = (volβ€˜π‘)))
312311anbi2d 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = 𝑣 β†’ ((𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) ↔ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘))))
313312rexbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑣 β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘))))
314310, 313elab 3669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘)))
315309, 314anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}) ↔ (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž)) ∧ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘))))
316304, 315bitr4i 277 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž)) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘))) ↔ (𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}))
317 an4 654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ (𝑒 = (volβ€˜π‘Ž) ∧ 𝑣 = (volβ€˜π‘))) ↔ ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž)) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘))))
318 oveq12 7435 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑒 = (volβ€˜π‘Ž) ∧ 𝑣 = (volβ€˜π‘)) β†’ (𝑒 + 𝑣) = ((volβ€˜π‘Ž) + (volβ€˜π‘)))
31959adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,)))) β†’ π‘Ž ∈ dom vol)
320319ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ π‘Ž ∈ dom vol)
321117adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,)))) β†’ 𝑐 ∈ dom vol)
322321ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ 𝑐 ∈ dom vol)
323 ss2in 4239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) β†’ (π‘Ž ∩ 𝑐) βŠ† ((βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
324185ineq1i 4210 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) = ((βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))
325 incom 4203 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) = ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
326 disjdif 4475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) = βˆ…
327324, 325, 3263eqtri 2760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) = βˆ…
328323, 327sseqtrdi 4032 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) β†’ (π‘Ž ∩ 𝑐) βŠ† βˆ…)
329 ss0 4402 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘Ž ∩ 𝑐) βŠ† βˆ… β†’ (π‘Ž ∩ 𝑐) = βˆ…)
330328, 329syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) β†’ (π‘Ž ∩ 𝑐) = βˆ…)
331330adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ (π‘Ž ∩ 𝑐) = βˆ…)
33261adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,)))) β†’ (volβ€˜π‘Ž) = (vol*β€˜π‘Ž))
333332ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ (volβ€˜π‘Ž) = (vol*β€˜π‘Ž))
33466, 16jctir 519 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) β†’ (π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ))
335683expa 1115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘Ž) ∈ ℝ)
336334, 335sylan 578 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘Ž) ∈ ℝ)
337336ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴)) β†’ (vol*β€˜π‘Ž) ∈ ℝ)
338337ad2ant2r 745 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ (vol*β€˜π‘Ž) ∈ ℝ)
339333, 338eqeltrd 2829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ (volβ€˜π‘Ž) ∈ ℝ)
340119adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,)))) β†’ (volβ€˜π‘) = (vol*β€˜π‘))
341340ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ (volβ€˜π‘) = (vol*β€˜π‘))
342122, 16jctir 519 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) β†’ (𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ))
3431243expa 1115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘) ∈ ℝ)
344342, 343sylan 578 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘) ∈ ℝ)
345344ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) β†’ (vol*β€˜π‘) ∈ ℝ)
346345ad2ant2rl 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ (vol*β€˜π‘) ∈ ℝ)
347341, 346eqeltrd 2829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ (volβ€˜π‘) ∈ ℝ)
348 volun 25494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((π‘Ž ∈ dom vol ∧ 𝑐 ∈ dom vol ∧ (π‘Ž ∩ 𝑐) = βˆ…) ∧ ((volβ€˜π‘Ž) ∈ ℝ ∧ (volβ€˜π‘) ∈ ℝ)) β†’ (volβ€˜(π‘Ž βˆͺ 𝑐)) = ((volβ€˜π‘Ž) + (volβ€˜π‘)))
349320, 322, 331, 339, 347, 348syl32anc 1375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ (volβ€˜(π‘Ž βˆͺ 𝑐)) = ((volβ€˜π‘Ž) + (volβ€˜π‘)))
350 unmbl 25486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((π‘Ž ∈ dom vol ∧ 𝑐 ∈ dom vol) β†’ (π‘Ž βˆͺ 𝑐) ∈ dom vol)
35159, 117, 350syl2an 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,)))) β†’ (π‘Ž βˆͺ 𝑐) ∈ dom vol)
352 mblvol 25479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘Ž βˆͺ 𝑐) ∈ dom vol β†’ (volβ€˜(π‘Ž βˆͺ 𝑐)) = (vol*β€˜(π‘Ž βˆͺ 𝑐)))
353351, 352syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,)))) β†’ (volβ€˜(π‘Ž βˆͺ 𝑐)) = (vol*β€˜(π‘Ž βˆͺ 𝑐)))
354353ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ (volβ€˜(π‘Ž βˆͺ 𝑐)) = (vol*β€˜(π‘Ž βˆͺ 𝑐)))
355349, 354eqtr3d 2770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) β†’ ((volβ€˜π‘Ž) + (volβ€˜π‘)) = (vol*β€˜(π‘Ž βˆͺ 𝑐)))
356318, 355sylan9eqr 2790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ∧ (𝑒 = (volβ€˜π‘Ž) ∧ 𝑣 = (volβ€˜π‘))) β†’ (𝑒 + 𝑣) = (vol*β€˜(π‘Ž βˆͺ 𝑐)))
357 eqtr 2751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 = (𝑒 + 𝑣) ∧ (𝑒 + 𝑣) = (vol*β€˜(π‘Ž βˆͺ 𝑐))) β†’ 𝑦 = (vol*β€˜(π‘Ž βˆͺ 𝑐)))
358357ancoms 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑒 + 𝑣) = (vol*β€˜(π‘Ž βˆͺ 𝑐)) ∧ 𝑦 = (𝑒 + 𝑣)) β†’ 𝑦 = (vol*β€˜(π‘Ž βˆͺ 𝑐)))
359356, 358sylan 578 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ∧ (𝑒 = (volβ€˜π‘Ž) ∧ 𝑣 = (volβ€˜π‘))) ∧ 𝑦 = (𝑒 + 𝑣)) β†’ 𝑦 = (vol*β€˜(π‘Ž βˆͺ 𝑐)))
36066, 122anim12i 611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) β†’ (π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓)))
361 unss 4186 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((π‘Ž βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑐 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ↔ (π‘Ž βˆͺ 𝑐) βŠ† βˆͺ ran ((,) ∘ 𝑓))
362360, 361sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) β†’ (π‘Ž βˆͺ 𝑐) βŠ† βˆͺ ran ((,) ∘ 𝑓))
363 ovolss 25434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((π‘Ž βˆͺ 𝑐) βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ) β†’ (vol*β€˜(π‘Ž βˆͺ 𝑐)) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
364362, 16, 363sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) β†’ (vol*β€˜(π‘Ž βˆͺ 𝑐)) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
365364ad3antlr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ∧ (𝑒 = (volβ€˜π‘Ž) ∧ 𝑣 = (volβ€˜π‘))) ∧ 𝑦 = (𝑒 + 𝑣)) β†’ (vol*β€˜(π‘Ž βˆͺ 𝑐)) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
366359, 365eqbrtrd 5174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ∧ (𝑒 = (volβ€˜π‘Ž) ∧ 𝑣 = (volβ€˜π‘))) ∧ 𝑦 = (𝑒 + 𝑣)) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
367366ex 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) ∧ (π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ∧ (𝑒 = (volβ€˜π‘Ž) ∧ 𝑣 = (volβ€˜π‘))) β†’ (𝑦 = (𝑒 + 𝑣) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))))
368367expl 456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) β†’ (((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ (𝑒 = (volβ€˜π‘Ž) ∧ 𝑣 = (volβ€˜π‘))) β†’ (𝑦 = (𝑒 + 𝑣) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))))
369317, 368biimtrrid 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (π‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑐 ∈ (Clsdβ€˜(topGenβ€˜ran (,))))) β†’ (((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž)) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘))) β†’ (𝑦 = (𝑒 + 𝑣) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))))
370369rexlimdvva 3209 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))((π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑒 = (volβ€˜π‘Ž)) ∧ (𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑣 = (volβ€˜π‘))) β†’ (𝑦 = (𝑒 + 𝑣) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))))
371316, 370biimtrrid 242 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ ((𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}) β†’ (𝑦 = (𝑒 + 𝑣) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))))
372371rexlimdvv 3208 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 = (𝑒 + 𝑣) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))))
373372alrimiv 1922 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ βˆ€π‘¦(βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 = (𝑒 + 𝑣) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))))
374 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑦 β†’ (𝑑 = (𝑒 + 𝑣) ↔ 𝑦 = (𝑒 + 𝑣)))
3753742rexbidv 3217 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑦 β†’ (βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣) ↔ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 = (𝑒 + 𝑣)))
376375ralab 3688 . . . . . . . . . . . . . . . . . . . . . . . 24 (βˆ€π‘¦ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ↔ βˆ€π‘¦(βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑦 = (𝑒 + 𝑣) β†’ 𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))))
377373, 376sylibr 233 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ βˆ€π‘¦ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
378 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))})) ∧ 𝑑 = (𝑒 + 𝑣)) β†’ 𝑑 = (𝑒 + 𝑣))
37974sselda 3982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}) β†’ 𝑒 ∈ ℝ)
380130sselda 3982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}) β†’ 𝑣 ∈ ℝ)
381 readdcl 11229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑒 + 𝑣) ∈ ℝ)
382379, 380, 381syl2an 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))})) β†’ (𝑒 + 𝑣) ∈ ℝ)
383382anandis 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))})) β†’ (𝑒 + 𝑣) ∈ ℝ)
384383adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))})) ∧ 𝑑 = (𝑒 + 𝑣)) β†’ (𝑒 + 𝑣) ∈ ℝ)
385378, 384eqeltrd 2829 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))})) ∧ 𝑑 = (𝑒 + 𝑣)) β†’ 𝑑 ∈ ℝ)
386385ex 411 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ (𝑒 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ∧ 𝑣 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))})) β†’ (𝑑 = (𝑒 + 𝑣) β†’ 𝑑 ∈ ℝ))
387386rexlimdvva 3209 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣) β†’ 𝑑 ∈ ℝ))
388387abssdv 4065 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)} βŠ† ℝ)
389 00id 11427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (0 + 0) = 0
390389eqcomi 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 = (0 + 0)
391 rspceov 7473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))} ∧ 0 ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))} ∧ 0 = (0 + 0)) β†’ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}0 = (𝑒 + 𝑣))
392110, 157, 390, 391mp3an 1457 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}0 = (𝑒 + 𝑣)
393 eqeq1 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑑 = 0 β†’ (𝑑 = (𝑒 + 𝑣) ↔ 0 = (𝑒 + 𝑣)))
3943932rexbidv 3217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑 = 0 β†’ (βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣) ↔ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}0 = (𝑒 + 𝑣)))
395105, 394spcev 3595 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}0 = (𝑒 + 𝑣) β†’ βˆƒπ‘‘βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣))
396392, 395ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 βˆƒπ‘‘βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)
397 abn0 4384 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)} β‰  βˆ… ↔ βˆƒπ‘‘βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣))
398396, 397mpbir 230 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)} β‰  βˆ…
399398a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)} β‰  βˆ…)
400 brralrspcev 5212 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ ∧ βˆ€π‘¦ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}𝑦 ≀ π‘₯)
401377, 400mpdan 685 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}𝑦 ≀ π‘₯)
402388, 399, 4013jca 1125 . . . . . . . . . . . . . . . . . . . . . . . 24 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ ({𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)} βŠ† ℝ ∧ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)} β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}𝑦 ≀ π‘₯))
403 suprleub 12218 . . . . . . . . . . . . . . . . . . . . . . . 24 ((({𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)} βŠ† ℝ ∧ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)} β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}𝑦 ≀ π‘₯) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (sup({𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}, ℝ, < ) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ↔ βˆ€π‘¦ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))))
404402, 403mpancom 686 . . . . . . . . . . . . . . . . . . . . . . 23 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (sup({𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}, ℝ, < ) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ↔ βˆ€π‘¦ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}𝑦 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))))
405377, 404mpbird 256 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ sup({𝑑 ∣ βˆƒπ‘’ ∈ {𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}βˆƒπ‘£ ∈ {𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}𝑑 = (𝑒 + 𝑣)}, ℝ, < ) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
406303, 405eqbrtrd 5174 . . . . . . . . . . . . . . . . . . . . 21 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) + sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < )) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
407406adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (sup({𝑧 ∣ βˆƒπ‘Ž ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(π‘Ž βŠ† (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) ∧ 𝑧 = (volβ€˜π‘Ž))}, ℝ, < ) + sup({𝑧 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑐 βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∧ 𝑧 = (volβ€˜π‘))}, ℝ, < )) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
40845, 163, 164, 297, 407letrd 11409 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
40944, 408sylan2 591 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) β‰  +∞) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
41033, 409pm2.61dane 3026 . . . . . . . . . . . . . . . . 17 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
411410adantlr 713 . . . . . . . . . . . . . . . 16 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
412 ssid 4004 . . . . . . . . . . . . . . . . . 18 βˆͺ ran ((,) ∘ 𝑓) βŠ† βˆͺ ran ((,) ∘ 𝑓)
41320ovollb 25428 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
414412, 413mpan2 689 . . . . . . . . . . . . . . . . 17 (𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
415414ad2antlr 725 . . . . . . . . . . . . . . . 16 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
41612, 18, 27, 411, 415xrletrd 13181 . . . . . . . . . . . . . . 15 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
417416adantr 479 . . . . . . . . . . . . . 14 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
418 simpr 483 . . . . . . . . . . . . . 14 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) β†’ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
419417, 418breqtrrd 5180 . . . . . . . . . . . . 13 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓)) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒)
420419expl 456 . . . . . . . . . . . 12 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ ((𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒))
4213, 420sylan2 591 . . . . . . . . . . 11 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) ∧ 𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)) β†’ ((𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒))
422421rexlimdva 3152 . . . . . . . . . 10 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) β†’ (βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒))
423422ralrimivw 3147 . . . . . . . . 9 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) β†’ βˆ€π‘’ ∈ ℝ* (βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒))
424 eqeq1 2732 . . . . . . . . . . . 12 (𝑣 = 𝑒 β†’ (𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ↔ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )))
425424anbi2d 628 . . . . . . . . . . 11 (𝑣 = 𝑒 β†’ ((𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) ↔ (𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))))
426425rexbidv 3176 . . . . . . . . . 10 (𝑣 = 𝑒 β†’ (βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) ↔ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))))
427426ralrab 3690 . . . . . . . . 9 (βˆ€π‘’ ∈ {𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))} ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒 ↔ βˆ€π‘’ ∈ ℝ* (βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑒 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < )) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒))
428423, 427sylibr 233 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) β†’ βˆ€π‘’ ∈ {𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))} ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒)
429 ssrab2 4077 . . . . . . . . 9 {𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))} βŠ† ℝ*
43011adantl 480 . . . . . . . . 9 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ∈ ℝ*)
431 infxrgelb 13354 . . . . . . . . 9 (({𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))} βŠ† ℝ* ∧ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ∈ ℝ*) β†’ (((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ inf({𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ) ↔ βˆ€π‘’ ∈ {𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))} ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒))
432429, 430, 431sylancr 585 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) β†’ (((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ inf({𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ) ↔ βˆ€π‘’ ∈ {𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))} ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ 𝑒))
433428, 432mpbird 256 . . . . . . 7 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ inf({𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
434 eqid 2728 . . . . . . . . 9 {𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))} = {𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}
435434ovolval 25422 . . . . . . . 8 (𝑀 βŠ† ℝ β†’ (vol*β€˜π‘€) = inf({𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
436435ad2antrl 726 . . . . . . 7 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) β†’ (vol*β€˜π‘€) = inf({𝑣 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝑀 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑣 = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
437433, 436breqtrrd 5180 . . . . . 6 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ (𝑀 βŠ† ℝ ∧ (vol*β€˜π‘€) ∈ ℝ)) β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜π‘€))
438437expr 455 . . . . 5 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ 𝑀 βŠ† ℝ) β†’ ((vol*β€˜π‘€) ∈ ℝ β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜π‘€)))
4392, 438sylan2 591 . . . 4 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) ∧ 𝑀 ∈ 𝒫 ℝ) β†’ ((vol*β€˜π‘€) ∈ ℝ β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜π‘€)))
440439ralrimiva 3143 . . 3 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) β†’ βˆ€π‘€ ∈ 𝒫 ℝ((vol*β€˜π‘€) ∈ ℝ β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜π‘€)))
441 ismbl2 25476 . . . . 5 (𝐴 ∈ dom vol ↔ (𝐴 βŠ† ℝ ∧ βˆ€π‘€ ∈ 𝒫 ℝ((vol*β€˜π‘€) ∈ ℝ β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜π‘€))))
442441baibr 535 . . . 4 (𝐴 βŠ† ℝ β†’ (βˆ€π‘€ ∈ 𝒫 ℝ((vol*β€˜π‘€) ∈ ℝ β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜π‘€)) ↔ 𝐴 ∈ dom vol))
443442ad2antrr 724 . . 3 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) β†’ (βˆ€π‘€ ∈ 𝒫 ℝ((vol*β€˜π‘€) ∈ ℝ β†’ ((vol*β€˜(𝑀 ∩ 𝐴)) + (vol*β€˜(𝑀 βˆ– 𝐴))) ≀ (vol*β€˜π‘€)) ↔ 𝐴 ∈ dom vol))
444440, 443mpbid 231 . 2 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )) β†’ 𝐴 ∈ dom vol)
4451, 444impbida 799 1 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (𝐴 ∈ dom vol ↔ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < )))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084  βˆ€wal 1531   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  {cab 2705   β‰  wne 2937  βˆ€wral 3058  βˆƒwrex 3067  {crab 3430   βˆ– cdif 3946   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4326  π’« cpw 4606  βˆͺ cuni 4912   class class class wbr 5152   Or wor 5593   Γ— cxp 5680  dom cdm 5682  ran crn 5683   ∘ ccom 5686  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426   ↑m cmap 8851  supcsup 9471  infcinf 9472  β„cr 11145  0cc0 11146  1c1 11147   + caddc 11149  +∞cpnf 11283  β„*cxr 11285   < clt 11286   ≀ cle 11287   βˆ’ cmin 11482  β„•cn 12250  (,)cioo 13364  [,)cico 13366  seqcseq 14006  abscabs 15221  topGenctg 17426  Topctop 22815  TopBasesctb 22868  Clsdccld 22940  vol*covol 25411  volcvol 25412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-disj 5118  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-oadd 8497  df-omul 8498  df-er 8731  df-map 8853  df-pm 8854  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-fi 9442  df-sup 9473  df-inf 9474  df-oi 9541  df-dju 9932  df-card 9970  df-acn 9973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-n0 12511  df-z 12597  df-uz 12861  df-q 12971  df-rp 13015  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-ioo 13368  df-ico 13370  df-icc 13371  df-fz 13525  df-fzo 13668  df-fl 13797  df-seq 14007  df-exp 14067  df-hash 14330  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472  df-rlim 15473  df-sum 15673  df-rest 17411  df-topgen 17432  df-psmet 21278  df-xmet 21279  df-met 21280  df-bl 21281  df-mopn 21282  df-top 22816  df-topon 22833  df-bases 22869  df-cld 22943  df-cmp 23311  df-conn 23336  df-ovol 25413  df-vol 25414
This theorem is referenced by: (None)
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