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Theorem mblfinlem2 36515
Description: Lemma for ismblfin 36518, effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different definition of outer measure. Note that unlike the main theorem, this holds for sets of infinite measure. (Contributed by Brendan Leahy, 21-Feb-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
Assertion
Ref Expression
mblfinlem2 ((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
Distinct variable groups:   𝐴,𝑠   𝑀,𝑠

Proof of Theorem mblfinlem2
Dummy variables π‘Ž 𝑏 𝑐 𝑓 π‘š 𝑛 𝑝 𝑑 𝑒 𝑣 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 retop 24270 . . . 4 (topGenβ€˜ran (,)) ∈ Top
2 0cld 22534 . . . 4 ((topGenβ€˜ran (,)) ∈ Top β†’ βˆ… ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
31, 2ax-mp 5 . . 3 βˆ… ∈ (Clsdβ€˜(topGenβ€˜ran (,)))
4 simpl3 1194 . . . . 5 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝐴 = βˆ…) β†’ 𝑀 < (vol*β€˜π΄))
5 fveq2 6889 . . . . . 6 (𝐴 = βˆ… β†’ (vol*β€˜π΄) = (vol*β€˜βˆ…))
65adantl 483 . . . . 5 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝐴 = βˆ…) β†’ (vol*β€˜π΄) = (vol*β€˜βˆ…))
74, 6breqtrd 5174 . . . 4 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝐴 = βˆ…) β†’ 𝑀 < (vol*β€˜βˆ…))
8 0ss 4396 . . . 4 βˆ… βŠ† 𝐴
97, 8jctil 521 . . 3 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝐴 = βˆ…) β†’ (βˆ… βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜βˆ…)))
10 sseq1 4007 . . . . 5 (𝑠 = βˆ… β†’ (𝑠 βŠ† 𝐴 ↔ βˆ… βŠ† 𝐴))
11 fveq2 6889 . . . . . 6 (𝑠 = βˆ… β†’ (vol*β€˜π‘ ) = (vol*β€˜βˆ…))
1211breq2d 5160 . . . . 5 (𝑠 = βˆ… β†’ (𝑀 < (vol*β€˜π‘ ) ↔ 𝑀 < (vol*β€˜βˆ…)))
1310, 12anbi12d 632 . . . 4 (𝑠 = βˆ… β†’ ((𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )) ↔ (βˆ… βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜βˆ…))))
1413rspcev 3613 . . 3 ((βˆ… ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (βˆ… βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜βˆ…))) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
153, 9, 14sylancr 588 . 2 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝐴 = βˆ…) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
16 mblfinlem1 36514 . . . 4 ((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝐴 β‰  βˆ…) β†’ βˆƒπ‘“ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
17163ad2antl1 1186 . . 3 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝐴 β‰  βˆ…) β†’ βˆƒπ‘“ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
18 simpl3 1194 . . . . . . . . 9 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ 𝑀 < (vol*β€˜π΄))
19 f1ofo 6838 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ 𝑓:ℕ–ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
20 rnco2 6250 . . . . . . . . . . . . . . . . 17 ran ([,] ∘ 𝑓) = ([,] β€œ ran 𝑓)
21 forn 6806 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ–ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ran 𝑓 = {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
2221imaeq2d 6058 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ([,] β€œ ran 𝑓) = ([,] β€œ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}))
2320, 22eqtrid 2785 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ran ([,] ∘ 𝑓) = ([,] β€œ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}))
2423unieqd 4922 . . . . . . . . . . . . . . 15 (𝑓:ℕ–ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆͺ ran ([,] ∘ 𝑓) = βˆͺ ([,] β€œ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}))
2519, 24syl 17 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆͺ ran ([,] ∘ 𝑓) = βˆͺ ([,] β€œ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}))
2625adantl 483 . . . . . . . . . . . . 13 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆͺ ran ([,] ∘ 𝑓) = βˆͺ ([,] β€œ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}))
27 oveq1 7413 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑒 β†’ (π‘₯ / (2↑𝑦)) = (𝑒 / (2↑𝑦)))
28 oveq1 7413 . . . . . . . . . . . . . . . . . 18 (π‘₯ = 𝑒 β†’ (π‘₯ + 1) = (𝑒 + 1))
2928oveq1d 7421 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑒 β†’ ((π‘₯ + 1) / (2↑𝑦)) = ((𝑒 + 1) / (2↑𝑦)))
3027, 29opeq12d 4881 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑒 β†’ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩ = ⟨(𝑒 / (2↑𝑦)), ((𝑒 + 1) / (2↑𝑦))⟩)
31 oveq2 7414 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑣 β†’ (2↑𝑦) = (2↑𝑣))
3231oveq2d 7422 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑣 β†’ (𝑒 / (2↑𝑦)) = (𝑒 / (2↑𝑣)))
3331oveq2d 7422 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑣 β†’ ((𝑒 + 1) / (2↑𝑦)) = ((𝑒 + 1) / (2↑𝑣)))
3432, 33opeq12d 4881 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑣 β†’ ⟨(𝑒 / (2↑𝑦)), ((𝑒 + 1) / (2↑𝑦))⟩ = ⟨(𝑒 / (2↑𝑣)), ((𝑒 + 1) / (2↑𝑣))⟩)
3530, 34cbvmpov 7501 . . . . . . . . . . . . . . 15 (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) = (𝑒 ∈ β„€, 𝑣 ∈ β„•0 ↦ ⟨(𝑒 / (2↑𝑣)), ((𝑒 + 1) / (2↑𝑣))⟩)
36 fveq2 6889 . . . . . . . . . . . . . . . . . . 19 (π‘Ž = 𝑧 β†’ ([,]β€˜π‘Ž) = ([,]β€˜π‘§))
3736sseq1d 4013 . . . . . . . . . . . . . . . . . 18 (π‘Ž = 𝑧 β†’ (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) ↔ ([,]β€˜π‘§) βŠ† ([,]β€˜π‘)))
38 eqeq1 2737 . . . . . . . . . . . . . . . . . 18 (π‘Ž = 𝑧 β†’ (π‘Ž = 𝑐 ↔ 𝑧 = 𝑐))
3937, 38imbi12d 345 . . . . . . . . . . . . . . . . 17 (π‘Ž = 𝑧 β†’ ((([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐) ↔ (([,]β€˜π‘§) βŠ† ([,]β€˜π‘) β†’ 𝑧 = 𝑐)))
4039ralbidv 3178 . . . . . . . . . . . . . . . 16 (π‘Ž = 𝑧 β†’ (βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐) ↔ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘§) βŠ† ([,]β€˜π‘) β†’ 𝑧 = 𝑐)))
4140cbvrabv 3443 . . . . . . . . . . . . . . 15 {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} = {𝑧 ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘§) βŠ† ([,]β€˜π‘) β†’ 𝑧 = 𝑐)}
42 ssrab2 4077 . . . . . . . . . . . . . . . 16 {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} βŠ† ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩)
4342a1i 11 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) β†’ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} βŠ† ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩))
4435, 41, 43dyadmbllem 25108 . . . . . . . . . . . . . 14 ((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) β†’ βˆͺ ([,] β€œ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴}) = βˆͺ ([,] β€œ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}))
4544adantr 482 . . . . . . . . . . . . 13 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆͺ ([,] β€œ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴}) = βˆͺ ([,] β€œ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}))
4626, 45eqtr4d 2776 . . . . . . . . . . . 12 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆͺ ran ([,] ∘ 𝑓) = βˆͺ ([,] β€œ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴}))
47 opnmbllem0 36513 . . . . . . . . . . . . . 14 (𝐴 ∈ (topGenβ€˜ran (,)) β†’ βˆͺ ([,] β€œ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴}) = 𝐴)
48473ad2ant1 1134 . . . . . . . . . . . . 13 ((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) β†’ βˆͺ ([,] β€œ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴}) = 𝐴)
4948adantr 482 . . . . . . . . . . . 12 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆͺ ([,] β€œ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴}) = 𝐴)
5046, 49eqtrd 2773 . . . . . . . . . . 11 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆͺ ran ([,] ∘ 𝑓) = 𝐴)
5150fveq2d 6893 . . . . . . . . . 10 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ (vol*β€˜βˆͺ ran ([,] ∘ 𝑓)) = (vol*β€˜π΄))
52 f1of 6831 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ 𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
53 ssrab2 4077 . . . . . . . . . . . . . 14 {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} βŠ† {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴}
5435dyadf 25100 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩):(β„€ Γ— β„•0)⟢( ≀ ∩ (ℝ Γ— ℝ))
55 frn 6722 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩):(β„€ Γ— β„•0)⟢( ≀ ∩ (ℝ Γ— ℝ)) β†’ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) βŠ† ( ≀ ∩ (ℝ Γ— ℝ)))
5654, 55ax-mp 5 . . . . . . . . . . . . . . 15 ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) βŠ† ( ≀ ∩ (ℝ Γ— ℝ))
5742, 56sstri 3991 . . . . . . . . . . . . . 14 {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} βŠ† ( ≀ ∩ (ℝ Γ— ℝ))
5853, 57sstri 3991 . . . . . . . . . . . . 13 {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} βŠ† ( ≀ ∩ (ℝ Γ— ℝ))
59 fss 6732 . . . . . . . . . . . . 13 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} βŠ† ( ≀ ∩ (ℝ Γ— ℝ))) β†’ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
6052, 58, 59sylancl 587 . . . . . . . . . . . 12 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
6153, 42sstri 3991 . . . . . . . . . . . . . . . . . . . 20 {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} βŠ† ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩)
62 ffvelcdm 7081 . . . . . . . . . . . . . . . . . . . 20 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ (π‘“β€˜π‘š) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
6361, 62sselid 3980 . . . . . . . . . . . . . . . . . . 19 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ (π‘“β€˜π‘š) ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩))
6463adantrr 716 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ (π‘“β€˜π‘š) ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩))
65 ffvelcdm 7081 . . . . . . . . . . . . . . . . . . . 20 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ (π‘“β€˜π‘§) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
6661, 65sselid 3980 . . . . . . . . . . . . . . . . . . 19 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ (π‘“β€˜π‘§) ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩))
6766adantrl 715 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ (π‘“β€˜π‘§) ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩))
6835dyaddisj 25105 . . . . . . . . . . . . . . . . . 18 (((π‘“β€˜π‘š) ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∧ (π‘“β€˜π‘§) ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩)) β†’ (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š)) ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
6964, 67, 68syl2anc 585 . . . . . . . . . . . . . . . . 17 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š)) ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
7052, 69sylan 581 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š)) ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
71 df-3or 1089 . . . . . . . . . . . . . . . 16 ((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š)) ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) ↔ ((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š))) ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
7270, 71sylib 217 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ ((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š))) ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
73 elrabi 3677 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘“β€˜π‘§) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ (π‘“β€˜π‘§) ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴})
74 fveq2 6889 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘Ž = (π‘“β€˜π‘š) β†’ ([,]β€˜π‘Ž) = ([,]β€˜(π‘“β€˜π‘š)))
7574sseq1d 4013 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘Ž = (π‘“β€˜π‘š) β†’ (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) ↔ ([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜π‘)))
76 eqeq1 2737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘Ž = (π‘“β€˜π‘š) β†’ (π‘Ž = 𝑐 ↔ (π‘“β€˜π‘š) = 𝑐))
7775, 76imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘Ž = (π‘“β€˜π‘š) β†’ ((([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐) ↔ (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘š) = 𝑐)))
7877ralbidv 3178 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘Ž = (π‘“β€˜π‘š) β†’ (βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐) ↔ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘š) = 𝑐)))
7978elrab 3683 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘“β€˜π‘š) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ↔ ((π‘“β€˜π‘š) ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∧ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘š) = 𝑐)))
8079simprbi 498 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘“β€˜π‘š) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘š) = 𝑐))
81 fveq2 6889 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (π‘“β€˜π‘§) β†’ ([,]β€˜π‘) = ([,]β€˜(π‘“β€˜π‘§)))
8281sseq2d 4014 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (π‘“β€˜π‘§) β†’ (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜π‘) ↔ ([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§))))
83 eqeq2 2745 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (π‘“β€˜π‘§) β†’ ((π‘“β€˜π‘š) = 𝑐 ↔ (π‘“β€˜π‘š) = (π‘“β€˜π‘§)))
8482, 83imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = (π‘“β€˜π‘§) β†’ ((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘š) = 𝑐) ↔ (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) β†’ (π‘“β€˜π‘š) = (π‘“β€˜π‘§))))
8584rspcva 3611 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘“β€˜π‘§) ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∧ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘š) = 𝑐)) β†’ (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) β†’ (π‘“β€˜π‘š) = (π‘“β€˜π‘§)))
8673, 80, 85syl2anr 598 . . . . . . . . . . . . . . . . . . . . 21 (((π‘“β€˜π‘š) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘“β€˜π‘§) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ (([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) β†’ (π‘“β€˜π‘š) = (π‘“β€˜π‘§)))
87 elrabi 3677 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘“β€˜π‘š) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ (π‘“β€˜π‘š) ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴})
88 fveq2 6889 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘Ž = (π‘“β€˜π‘§) β†’ ([,]β€˜π‘Ž) = ([,]β€˜(π‘“β€˜π‘§)))
8988sseq1d 4013 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘Ž = (π‘“β€˜π‘§) β†’ (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) ↔ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜π‘)))
90 eqeq1 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘Ž = (π‘“β€˜π‘§) β†’ (π‘Ž = 𝑐 ↔ (π‘“β€˜π‘§) = 𝑐))
9189, 90imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘Ž = (π‘“β€˜π‘§) β†’ ((([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐) ↔ (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘§) = 𝑐)))
9291ralbidv 3178 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘Ž = (π‘“β€˜π‘§) β†’ (βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐) ↔ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘§) = 𝑐)))
9392elrab 3683 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘“β€˜π‘§) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ↔ ((π‘“β€˜π‘§) ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∧ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘§) = 𝑐)))
9493simprbi 498 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘“β€˜π‘§) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘§) = 𝑐))
95 fveq2 6889 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = (π‘“β€˜π‘š) β†’ ([,]β€˜π‘) = ([,]β€˜(π‘“β€˜π‘š)))
9695sseq2d 4014 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (π‘“β€˜π‘š) β†’ (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜π‘) ↔ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š))))
97 eqeq2 2745 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (π‘“β€˜π‘š) β†’ ((π‘“β€˜π‘§) = 𝑐 ↔ (π‘“β€˜π‘§) = (π‘“β€˜π‘š)))
9896, 97imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (π‘“β€˜π‘š) β†’ ((([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘§) = 𝑐) ↔ (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š)) β†’ (π‘“β€˜π‘§) = (π‘“β€˜π‘š))))
9998rspcva 3611 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘“β€˜π‘š) ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∧ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜π‘) β†’ (π‘“β€˜π‘§) = 𝑐)) β†’ (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š)) β†’ (π‘“β€˜π‘§) = (π‘“β€˜π‘š)))
10087, 94, 99syl2an 597 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘“β€˜π‘š) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘“β€˜π‘§) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š)) β†’ (π‘“β€˜π‘§) = (π‘“β€˜π‘š)))
101 eqcom 2740 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘“β€˜π‘§) = (π‘“β€˜π‘š) ↔ (π‘“β€˜π‘š) = (π‘“β€˜π‘§))
102100, 101syl6ib 251 . . . . . . . . . . . . . . . . . . . . 21 (((π‘“β€˜π‘š) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘“β€˜π‘§) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ (([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š)) β†’ (π‘“β€˜π‘š) = (π‘“β€˜π‘§)))
10386, 102jaod 858 . . . . . . . . . . . . . . . . . . . 20 (((π‘“β€˜π‘š) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘“β€˜π‘§) ∈ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ ((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š))) β†’ (π‘“β€˜π‘š) = (π‘“β€˜π‘§)))
10462, 65, 103syl2an 597 . . . . . . . . . . . . . . . . . . 19 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) ∧ (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•)) β†’ ((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š))) β†’ (π‘“β€˜π‘š) = (π‘“β€˜π‘§)))
105104anandis 677 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ ((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š))) β†’ (π‘“β€˜π‘š) = (π‘“β€˜π‘§)))
10652, 105sylan 581 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ ((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š))) β†’ (π‘“β€˜π‘š) = (π‘“β€˜π‘§)))
107 f1of1 6830 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ 𝑓:ℕ–1-1β†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
108 f1veqaeq 7253 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ–1-1β†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ ((π‘“β€˜π‘š) = (π‘“β€˜π‘§) β†’ π‘š = 𝑧))
109107, 108sylan 581 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ ((π‘“β€˜π‘š) = (π‘“β€˜π‘§) β†’ π‘š = 𝑧))
110106, 109syld 47 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ ((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š))) β†’ π‘š = 𝑧))
111110orim1d 965 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ (((([,]β€˜(π‘“β€˜π‘š)) βŠ† ([,]β€˜(π‘“β€˜π‘§)) ∨ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ([,]β€˜(π‘“β€˜π‘š))) ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) β†’ (π‘š = 𝑧 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…)))
11272, 111mpd 15 . . . . . . . . . . . . . 14 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ (π‘š = 𝑧 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
113112ralrimivva 3201 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆ€π‘š ∈ β„• βˆ€π‘§ ∈ β„• (π‘š = 𝑧 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
114 eqeq1 2737 . . . . . . . . . . . . . . . . 17 (π‘š = 𝑧 β†’ (π‘š = 𝑝 ↔ 𝑧 = 𝑝))
115 2fveq3 6894 . . . . . . . . . . . . . . . . . . 19 (π‘š = 𝑧 β†’ ((,)β€˜(π‘“β€˜π‘š)) = ((,)β€˜(π‘“β€˜π‘§)))
116115ineq1d 4211 . . . . . . . . . . . . . . . . . 18 (π‘š = 𝑧 β†’ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))) = (((,)β€˜(π‘“β€˜π‘§)) ∩ ((,)β€˜(π‘“β€˜π‘))))
117116eqeq1d 2735 . . . . . . . . . . . . . . . . 17 (π‘š = 𝑧 β†’ ((((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ… ↔ (((,)β€˜(π‘“β€˜π‘§)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…))
118114, 117orbi12d 918 . . . . . . . . . . . . . . . 16 (π‘š = 𝑧 β†’ ((π‘š = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…) ↔ (𝑧 = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘§)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…)))
119118ralbidv 3178 . . . . . . . . . . . . . . 15 (π‘š = 𝑧 β†’ (βˆ€π‘ ∈ β„• (π‘š = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…) ↔ βˆ€π‘ ∈ β„• (𝑧 = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘§)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…)))
120119cbvralvw 3235 . . . . . . . . . . . . . 14 (βˆ€π‘š ∈ β„• βˆ€π‘ ∈ β„• (π‘š = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…) ↔ βˆ€π‘§ ∈ β„• βˆ€π‘ ∈ β„• (𝑧 = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘§)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…))
121 eqeq2 2745 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑝 β†’ (π‘š = 𝑧 ↔ π‘š = 𝑝))
122 2fveq3 6894 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑝 β†’ ((,)β€˜(π‘“β€˜π‘§)) = ((,)β€˜(π‘“β€˜π‘)))
123122ineq2d 4212 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑝 β†’ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))))
124123eqeq1d 2735 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑝 β†’ ((((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ… ↔ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…))
125121, 124orbi12d 918 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑝 β†’ ((π‘š = 𝑧 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) ↔ (π‘š = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…)))
126125cbvralvw 3235 . . . . . . . . . . . . . . 15 (βˆ€π‘§ ∈ β„• (π‘š = 𝑧 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) ↔ βˆ€π‘ ∈ β„• (π‘š = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…))
127126ralbii 3094 . . . . . . . . . . . . . 14 (βˆ€π‘š ∈ β„• βˆ€π‘§ ∈ β„• (π‘š = 𝑧 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) ↔ βˆ€π‘š ∈ β„• βˆ€π‘ ∈ β„• (π‘š = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…))
128122disjor 5128 . . . . . . . . . . . . . 14 (Disj 𝑧 ∈ β„• ((,)β€˜(π‘“β€˜π‘§)) ↔ βˆ€π‘§ ∈ β„• βˆ€π‘ ∈ β„• (𝑧 = 𝑝 ∨ (((,)β€˜(π‘“β€˜π‘§)) ∩ ((,)β€˜(π‘“β€˜π‘))) = βˆ…))
129120, 127, 1283bitr4ri 304 . . . . . . . . . . . . 13 (Disj 𝑧 ∈ β„• ((,)β€˜(π‘“β€˜π‘§)) ↔ βˆ€π‘š ∈ β„• βˆ€π‘§ ∈ β„• (π‘š = 𝑧 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
130113, 129sylibr 233 . . . . . . . . . . . 12 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ Disj 𝑧 ∈ β„• ((,)β€˜(π‘“β€˜π‘§)))
131 eqid 2733 . . . . . . . . . . . 12 seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) = seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))
13260, 130, 131uniiccvol 25089 . . . . . . . . . . 11 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ (vol*β€˜βˆͺ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
133132adantl 483 . . . . . . . . . 10 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ (vol*β€˜βˆͺ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
13451, 133eqtr3d 2775 . . . . . . . . 9 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ (vol*β€˜π΄) = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
13518, 134breqtrd 5174 . . . . . . . 8 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ 𝑀 < sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
136 absf 15281 . . . . . . . . . . . 12 abs:β„‚βŸΆβ„
137 subf 11459 . . . . . . . . . . . 12 βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚
138 fco 6739 . . . . . . . . . . . 12 ((abs:β„‚βŸΆβ„ ∧ βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚) β†’ (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„)
139136, 137, 138mp2an 691 . . . . . . . . . . 11 (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„
140 zre 12559 . . . . . . . . . . . . . . . . . . 19 (π‘₯ ∈ β„€ β†’ π‘₯ ∈ ℝ)
141 2re 12283 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
142 reexpcl 14041 . . . . . . . . . . . . . . . . . . . . 21 ((2 ∈ ℝ ∧ 𝑦 ∈ β„•0) β†’ (2↑𝑦) ∈ ℝ)
143141, 142mpan 689 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ β„•0 β†’ (2↑𝑦) ∈ ℝ)
144 2cn 12284 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ β„‚
145 2ne0 12313 . . . . . . . . . . . . . . . . . . . . 21 2 β‰  0
146 nn0z 12580 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ β„•0 β†’ 𝑦 ∈ β„€)
147 expne0i 14057 . . . . . . . . . . . . . . . . . . . . 21 ((2 ∈ β„‚ ∧ 2 β‰  0 ∧ 𝑦 ∈ β„€) β†’ (2↑𝑦) β‰  0)
148144, 145, 146, 147mp3an12i 1466 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ β„•0 β†’ (2↑𝑦) β‰  0)
149143, 148jca 513 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ β„•0 β†’ ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) β‰  0))
150 redivcl 11930 . . . . . . . . . . . . . . . . . . . . 21 ((π‘₯ ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) β‰  0) β†’ (π‘₯ / (2↑𝑦)) ∈ ℝ)
151 peano2re 11384 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯ ∈ ℝ β†’ (π‘₯ + 1) ∈ ℝ)
152 redivcl 11930 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ + 1) ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) β‰  0) β†’ ((π‘₯ + 1) / (2↑𝑦)) ∈ ℝ)
153151, 152syl3an1 1164 . . . . . . . . . . . . . . . . . . . . 21 ((π‘₯ ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) β‰  0) β†’ ((π‘₯ + 1) / (2↑𝑦)) ∈ ℝ)
154150, 153opelxpd 5714 . . . . . . . . . . . . . . . . . . . 20 ((π‘₯ ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) β‰  0) β†’ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩ ∈ (ℝ Γ— ℝ))
1551543expb 1121 . . . . . . . . . . . . . . . . . . 19 ((π‘₯ ∈ ℝ ∧ ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) β‰  0)) β†’ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩ ∈ (ℝ Γ— ℝ))
156140, 149, 155syl2an 597 . . . . . . . . . . . . . . . . . 18 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•0) β†’ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩ ∈ (ℝ Γ— ℝ))
157156rgen2 3198 . . . . . . . . . . . . . . . . 17 βˆ€π‘₯ ∈ β„€ βˆ€π‘¦ ∈ β„•0 ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩ ∈ (ℝ Γ— ℝ)
158 eqid 2733 . . . . . . . . . . . . . . . . . 18 (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) = (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩)
159158fmpo 8051 . . . . . . . . . . . . . . . . 17 (βˆ€π‘₯ ∈ β„€ βˆ€π‘¦ ∈ β„•0 ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩ ∈ (ℝ Γ— ℝ) ↔ (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩):(β„€ Γ— β„•0)⟢(ℝ Γ— ℝ))
160157, 159mpbi 229 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩):(β„€ Γ— β„•0)⟢(ℝ Γ— ℝ)
161 frn 6722 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩):(β„€ Γ— β„•0)⟢(ℝ Γ— ℝ) β†’ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) βŠ† (ℝ Γ— ℝ))
162160, 161ax-mp 5 . . . . . . . . . . . . . . 15 ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) βŠ† (ℝ Γ— ℝ)
16342, 162sstri 3991 . . . . . . . . . . . . . 14 {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} βŠ† (ℝ Γ— ℝ)
16453, 163sstri 3991 . . . . . . . . . . . . 13 {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} βŠ† (ℝ Γ— ℝ)
165 ax-resscn 11164 . . . . . . . . . . . . . 14 ℝ βŠ† β„‚
166 xpss12 5691 . . . . . . . . . . . . . 14 ((ℝ βŠ† β„‚ ∧ ℝ βŠ† β„‚) β†’ (ℝ Γ— ℝ) βŠ† (β„‚ Γ— β„‚))
167165, 165, 166mp2an 691 . . . . . . . . . . . . 13 (ℝ Γ— ℝ) βŠ† (β„‚ Γ— β„‚)
168164, 167sstri 3991 . . . . . . . . . . . 12 {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} βŠ† (β„‚ Γ— β„‚)
169 fss 6732 . . . . . . . . . . . 12 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} βŠ† (β„‚ Γ— β„‚)) β†’ 𝑓:β„•βŸΆ(β„‚ Γ— β„‚))
170168, 169mpan2 690 . . . . . . . . . . 11 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ 𝑓:β„•βŸΆ(β„‚ Γ— β„‚))
171 fco 6739 . . . . . . . . . . 11 (((abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„ ∧ 𝑓:β„•βŸΆ(β„‚ Γ— β„‚)) β†’ ((abs ∘ βˆ’ ) ∘ 𝑓):β„•βŸΆβ„)
172139, 170, 171sylancr 588 . . . . . . . . . 10 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ((abs ∘ βˆ’ ) ∘ 𝑓):β„•βŸΆβ„)
173 nnuz 12862 . . . . . . . . . . 11 β„• = (β„€β‰₯β€˜1)
174 1z 12589 . . . . . . . . . . . 12 1 ∈ β„€
175174a1i 11 . . . . . . . . . . 11 (((abs ∘ βˆ’ ) ∘ 𝑓):β„•βŸΆβ„ β†’ 1 ∈ β„€)
176 ffvelcdm 7081 . . . . . . . . . . 11 ((((abs ∘ βˆ’ ) ∘ 𝑓):β„•βŸΆβ„ ∧ 𝑛 ∈ β„•) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘›) ∈ ℝ)
177173, 175, 176serfre 13994 . . . . . . . . . 10 (((abs ∘ βˆ’ ) ∘ 𝑓):β„•βŸΆβ„ β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)):β„•βŸΆβ„)
178 frn 6722 . . . . . . . . . . 11 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)):β„•βŸΆβ„ β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† ℝ)
179 ressxr 11255 . . . . . . . . . . 11 ℝ βŠ† ℝ*
180178, 179sstrdi 3994 . . . . . . . . . 10 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)):β„•βŸΆβ„ β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† ℝ*)
18152, 172, 177, 1804syl 19 . . . . . . . . 9 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† ℝ*)
182 rexr 11257 . . . . . . . . . 10 (𝑀 ∈ ℝ β†’ 𝑀 ∈ ℝ*)
1831823ad2ant2 1135 . . . . . . . . 9 ((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) β†’ 𝑀 ∈ ℝ*)
184 supxrlub 13301 . . . . . . . . 9 ((ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† ℝ* ∧ 𝑀 ∈ ℝ*) β†’ (𝑀 < sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ↔ βˆƒπ‘§ ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))𝑀 < 𝑧))
185181, 183, 184syl2anr 598 . . . . . . . 8 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ (𝑀 < sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ↔ βˆƒπ‘§ ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))𝑀 < 𝑧))
186135, 185mpbid 231 . . . . . . 7 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆƒπ‘§ ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))𝑀 < 𝑧)
187 seqfn 13975 . . . . . . . . . 10 (1 ∈ β„€ β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) Fn (β„€β‰₯β€˜1))
188174, 187ax-mp 5 . . . . . . . . 9 seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) Fn (β„€β‰₯β€˜1)
189173fneq2i 6645 . . . . . . . . 9 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) Fn β„• ↔ seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) Fn (β„€β‰₯β€˜1))
190188, 189mpbir 230 . . . . . . . 8 seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) Fn β„•
191 breq2 5152 . . . . . . . . 9 (𝑧 = (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) β†’ (𝑀 < 𝑧 ↔ 𝑀 < (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›)))
192191rexrn 7086 . . . . . . . 8 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) Fn β„• β†’ (βˆƒπ‘§ ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))𝑀 < 𝑧 ↔ βˆƒπ‘› ∈ β„• 𝑀 < (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›)))
193190, 192ax-mp 5 . . . . . . 7 (βˆƒπ‘§ ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))𝑀 < 𝑧 ↔ βˆƒπ‘› ∈ β„• 𝑀 < (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
194186, 193sylib 217 . . . . . 6 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆƒπ‘› ∈ β„• 𝑀 < (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
19560ffvelcdmda 7084 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ (π‘“β€˜π‘§) ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
196 0le0 12310 . . . . . . . . . . . . . . . . . 18 0 ≀ 0
197 df-br 5149 . . . . . . . . . . . . . . . . . 18 (0 ≀ 0 ↔ ⟨0, 0⟩ ∈ ≀ )
198196, 197mpbi 229 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ ≀
199 0re 11213 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
200 opelxpi 5713 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ 0 ∈ ℝ) β†’ ⟨0, 0⟩ ∈ (ℝ Γ— ℝ))
201199, 199, 200mp2an 691 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ (ℝ Γ— ℝ)
202 elin 3964 . . . . . . . . . . . . . . . . 17 (⟨0, 0⟩ ∈ ( ≀ ∩ (ℝ Γ— ℝ)) ↔ (⟨0, 0⟩ ∈ ≀ ∧ ⟨0, 0⟩ ∈ (ℝ Γ— ℝ)))
203198, 201, 202mpbir2an 710 . . . . . . . . . . . . . . . 16 ⟨0, 0⟩ ∈ ( ≀ ∩ (ℝ Γ— ℝ))
204 ifcl 4573 . . . . . . . . . . . . . . . 16 (((π‘“β€˜π‘§) ∈ ( ≀ ∩ (ℝ Γ— ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≀ ∩ (ℝ Γ— ℝ))) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
205195, 203, 204sylancl 587 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
206205fmpttd 7112 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)):β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
207 df-ov 7409 . . . . . . . . . . . . . . . . . . . . . 22 (0(,)0) = ((,)β€˜βŸ¨0, 0⟩)
208 iooid 13349 . . . . . . . . . . . . . . . . . . . . . 22 (0(,)0) = βˆ…
209207, 208eqtr3i 2763 . . . . . . . . . . . . . . . . . . . . 21 ((,)β€˜βŸ¨0, 0⟩) = βˆ…
210209ineq1i 4208 . . . . . . . . . . . . . . . . . . . 20 (((,)β€˜βŸ¨0, 0⟩) ∩ ((,)β€˜(π‘“β€˜π‘§))) = (βˆ… ∩ ((,)β€˜(π‘“β€˜π‘§)))
211 0in 4393 . . . . . . . . . . . . . . . . . . . 20 (βˆ… ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…
212210, 211eqtri 2761 . . . . . . . . . . . . . . . . . . 19 (((,)β€˜βŸ¨0, 0⟩) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…
213212olci 865 . . . . . . . . . . . . . . . . . 18 (π‘š = 𝑧 ∨ (((,)β€˜βŸ¨0, 0⟩) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…)
214 ineq1 4205 . . . . . . . . . . . . . . . . . . . . 21 (((,)β€˜(π‘“β€˜π‘š)) = if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) β†’ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))))
215214eqeq1d 2735 . . . . . . . . . . . . . . . . . . . 20 (((,)β€˜(π‘“β€˜π‘š)) = if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) β†’ ((((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ… ↔ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
216215orbi2d 915 . . . . . . . . . . . . . . . . . . 19 (((,)β€˜(π‘“β€˜π‘š)) = if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) β†’ ((π‘š = 𝑧 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) ↔ (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…)))
217 ineq1 4205 . . . . . . . . . . . . . . . . . . . . 21 (((,)β€˜βŸ¨0, 0⟩) = if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) β†’ (((,)β€˜βŸ¨0, 0⟩) ∩ ((,)β€˜(π‘“β€˜π‘§))) = (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))))
218217eqeq1d 2735 . . . . . . . . . . . . . . . . . . . 20 (((,)β€˜βŸ¨0, 0⟩) = if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) β†’ ((((,)β€˜βŸ¨0, 0⟩) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ… ↔ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
219218orbi2d 915 . . . . . . . . . . . . . . . . . . 19 (((,)β€˜βŸ¨0, 0⟩) = if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) β†’ ((π‘š = 𝑧 ∨ (((,)β€˜βŸ¨0, 0⟩) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) ↔ (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…)))
220216, 219ifboth 4567 . . . . . . . . . . . . . . . . . 18 (((π‘š = 𝑧 ∨ (((,)β€˜(π‘“β€˜π‘š)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) ∧ (π‘š = 𝑧 ∨ (((,)β€˜βŸ¨0, 0⟩) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…)) β†’ (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
221112, 213, 220sylancl 587 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…))
222209ineq2i 4209 . . . . . . . . . . . . . . . . . . 19 (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜βŸ¨0, 0⟩)) = (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ βˆ…)
223 in0 4391 . . . . . . . . . . . . . . . . . . 19 (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ βˆ…) = βˆ…
224222, 223eqtri 2761 . . . . . . . . . . . . . . . . . 18 (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜βŸ¨0, 0⟩)) = βˆ…
225224olci 865 . . . . . . . . . . . . . . . . 17 (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜βŸ¨0, 0⟩)) = βˆ…)
226 ineq2 4206 . . . . . . . . . . . . . . . . . . . 20 (((,)β€˜(π‘“β€˜π‘§)) = if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩)) β†’ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))))
227226eqeq1d 2735 . . . . . . . . . . . . . . . . . . 19 (((,)β€˜(π‘“β€˜π‘§)) = if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩)) β†’ ((if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ… ↔ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))) = βˆ…))
228227orbi2d 915 . . . . . . . . . . . . . . . . . 18 (((,)β€˜(π‘“β€˜π‘§)) = if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩)) β†’ ((π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) ↔ (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))) = βˆ…)))
229 ineq2 4206 . . . . . . . . . . . . . . . . . . . 20 (((,)β€˜βŸ¨0, 0⟩) = if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩)) β†’ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜βŸ¨0, 0⟩)) = (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))))
230229eqeq1d 2735 . . . . . . . . . . . . . . . . . . 19 (((,)β€˜βŸ¨0, 0⟩) = if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩)) β†’ ((if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜βŸ¨0, 0⟩)) = βˆ… ↔ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))) = βˆ…))
231230orbi2d 915 . . . . . . . . . . . . . . . . . 18 (((,)β€˜βŸ¨0, 0⟩) = if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩)) β†’ ((π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜βŸ¨0, 0⟩)) = βˆ…) ↔ (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))) = βˆ…)))
232228, 231ifboth 4567 . . . . . . . . . . . . . . . . 17 (((π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜(π‘“β€˜π‘§))) = βˆ…) ∧ (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ ((,)β€˜βŸ¨0, 0⟩)) = βˆ…)) β†’ (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))) = βˆ…))
233221, 225, 232sylancl 587 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (π‘š ∈ β„• ∧ 𝑧 ∈ β„•)) β†’ (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))) = βˆ…))
234233ralrimivva 3201 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆ€π‘š ∈ β„• βˆ€π‘§ ∈ β„• (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))) = βˆ…))
235 disjeq2 5117 . . . . . . . . . . . . . . . . 17 (βˆ€π‘š ∈ β„• ((,)β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)) = if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) β†’ (Disj π‘š ∈ β„• ((,)β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)) ↔ Disj π‘š ∈ β„• if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩))))
236 eleq1w 2817 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = π‘š β†’ (𝑧 ∈ (1...𝑛) ↔ π‘š ∈ (1...𝑛)))
237 fveq2 6889 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = π‘š β†’ (π‘“β€˜π‘§) = (π‘“β€˜π‘š))
238236, 237ifbieq1d 4552 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = π‘š β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) = if(π‘š ∈ (1...𝑛), (π‘“β€˜π‘š), ⟨0, 0⟩))
239 eqid 2733 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) = (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))
240 fvex 6902 . . . . . . . . . . . . . . . . . . . . 21 (π‘“β€˜π‘š) ∈ V
241 opex 5464 . . . . . . . . . . . . . . . . . . . . 21 ⟨0, 0⟩ ∈ V
242240, 241ifex 4578 . . . . . . . . . . . . . . . . . . . 20 if(π‘š ∈ (1...𝑛), (π‘“β€˜π‘š), ⟨0, 0⟩) ∈ V
243238, 239, 242fvmpt 6996 . . . . . . . . . . . . . . . . . . 19 (π‘š ∈ β„• β†’ ((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š) = if(π‘š ∈ (1...𝑛), (π‘“β€˜π‘š), ⟨0, 0⟩))
244243fveq2d 6893 . . . . . . . . . . . . . . . . . 18 (π‘š ∈ β„• β†’ ((,)β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)) = ((,)β€˜if(π‘š ∈ (1...𝑛), (π‘“β€˜π‘š), ⟨0, 0⟩)))
245 fvif 6905 . . . . . . . . . . . . . . . . . 18 ((,)β€˜if(π‘š ∈ (1...𝑛), (π‘“β€˜π‘š), ⟨0, 0⟩)) = if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩))
246244, 245eqtrdi 2789 . . . . . . . . . . . . . . . . 17 (π‘š ∈ β„• β†’ ((,)β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)) = if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)))
247235, 246mprg 3068 . . . . . . . . . . . . . . . 16 (Disj π‘š ∈ β„• ((,)β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)) ↔ Disj π‘š ∈ β„• if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)))
248 eleq1w 2817 . . . . . . . . . . . . . . . . . 18 (π‘š = 𝑧 β†’ (π‘š ∈ (1...𝑛) ↔ 𝑧 ∈ (1...𝑛)))
249248, 115ifbieq1d 4552 . . . . . . . . . . . . . . . . 17 (π‘š = 𝑧 β†’ if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩)))
250249disjor 5128 . . . . . . . . . . . . . . . 16 (Disj π‘š ∈ β„• if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ↔ βˆ€π‘š ∈ β„• βˆ€π‘§ ∈ β„• (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))) = βˆ…))
251247, 250bitri 275 . . . . . . . . . . . . . . 15 (Disj π‘š ∈ β„• ((,)β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)) ↔ βˆ€π‘š ∈ β„• βˆ€π‘§ ∈ β„• (π‘š = 𝑧 ∨ (if(π‘š ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘š)), ((,)β€˜βŸ¨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)β€˜(π‘“β€˜π‘§)), ((,)β€˜βŸ¨0, 0⟩))) = βˆ…))
252234, 251sylibr 233 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ Disj π‘š ∈ β„• ((,)β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)))
253 eqid 2733 . . . . . . . . . . . . . 14 seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))) = seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))
254206, 252, 253uniiccvol 25089 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ (vol*β€˜βˆͺ ran ([,] ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))) = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))), ℝ*, < ))
255254adantr 482 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜βˆͺ ran ([,] ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))) = sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))), ℝ*, < ))
256 rexpssxrxp 11256 . . . . . . . . . . . . . . . . . . . . 21 (ℝ Γ— ℝ) βŠ† (ℝ* Γ— ℝ*)
257164, 256sstri 3991 . . . . . . . . . . . . . . . . . . . 20 {π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} βŠ† (ℝ* Γ— ℝ*)
258257, 65sselid 3980 . . . . . . . . . . . . . . . . . . 19 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ (π‘“β€˜π‘§) ∈ (ℝ* Γ— ℝ*))
259 0xr 11258 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℝ*
260 opelxpi 5713 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ ⟨0, 0⟩ ∈ (ℝ* Γ— ℝ*))
261259, 259, 260mp2an 691 . . . . . . . . . . . . . . . . . . 19 ⟨0, 0⟩ ∈ (ℝ* Γ— ℝ*)
262 ifcl 4573 . . . . . . . . . . . . . . . . . . 19 (((π‘“β€˜π‘§) ∈ (ℝ* Γ— ℝ*) ∧ ⟨0, 0⟩ ∈ (ℝ* Γ— ℝ*)) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) ∈ (ℝ* Γ— ℝ*))
263258, 261, 262sylancl 587 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) ∈ (ℝ* Γ— ℝ*))
264 eqidd 2734 . . . . . . . . . . . . . . . . . 18 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) = (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))
265 iccf 13422 . . . . . . . . . . . . . . . . . . . 20 [,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
266265a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ [,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*)
267266feqmptd 6958 . . . . . . . . . . . . . . . . . 18 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ [,] = (π‘š ∈ (ℝ* Γ— ℝ*) ↦ ([,]β€˜π‘š)))
268 fveq2 6889 . . . . . . . . . . . . . . . . . 18 (π‘š = if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) β†’ ([,]β€˜π‘š) = ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))
269263, 264, 267, 268fmptco 7124 . . . . . . . . . . . . . . . . 17 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ([,] ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))) = (𝑧 ∈ β„• ↦ ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))
27052, 269syl 17 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ([,] ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))) = (𝑧 ∈ β„• ↦ ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))
271270rneqd 5936 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ran ([,] ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))) = ran (𝑧 ∈ β„• ↦ ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))
272271unieqd 4922 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆͺ ran ([,] ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))) = βˆͺ ran (𝑧 ∈ β„• ↦ ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))
273 peano2nn 12221 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ β„• β†’ (𝑛 + 1) ∈ β„•)
274273, 173eleqtrdi 2844 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ β„• β†’ (𝑛 + 1) ∈ (β„€β‰₯β€˜1))
275 fzouzsplit 13664 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 + 1) ∈ (β„€β‰₯β€˜1) β†’ (β„€β‰₯β€˜1) = ((1..^(𝑛 + 1)) βˆͺ (β„€β‰₯β€˜(𝑛 + 1))))
276274, 275syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ β„• β†’ (β„€β‰₯β€˜1) = ((1..^(𝑛 + 1)) βˆͺ (β„€β‰₯β€˜(𝑛 + 1))))
277173, 276eqtrid 2785 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ β„• β†’ β„• = ((1..^(𝑛 + 1)) βˆͺ (β„€β‰₯β€˜(𝑛 + 1))))
278 nnz 12576 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ β„• β†’ 𝑛 ∈ β„€)
279 fzval3 13698 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ β„€ β†’ (1...𝑛) = (1..^(𝑛 + 1)))
280278, 279syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ β„• β†’ (1...𝑛) = (1..^(𝑛 + 1)))
281280uneq1d 4162 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ β„• β†’ ((1...𝑛) βˆͺ (β„€β‰₯β€˜(𝑛 + 1))) = ((1..^(𝑛 + 1)) βˆͺ (β„€β‰₯β€˜(𝑛 + 1))))
282277, 281eqtr4d 2776 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ β„• β†’ β„• = ((1...𝑛) βˆͺ (β„€β‰₯β€˜(𝑛 + 1))))
283 fvif 6905 . . . . . . . . . . . . . . . . . 18 ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩))
284283a1i 11 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ β„• β†’ ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)))
285282, 284iuneq12d 5025 . . . . . . . . . . . . . . . 16 (𝑛 ∈ β„• β†’ βˆͺ 𝑧 ∈ β„• ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) = βˆͺ 𝑧 ∈ ((1...𝑛) βˆͺ (β„€β‰₯β€˜(𝑛 + 1)))if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)))
286 fvex 6902 . . . . . . . . . . . . . . . . 17 ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) ∈ V
287286dfiun3 5964 . . . . . . . . . . . . . . . 16 βˆͺ 𝑧 ∈ β„• ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) = βˆͺ ran (𝑧 ∈ β„• ↦ ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))
288 iunxun 5097 . . . . . . . . . . . . . . . 16 βˆͺ 𝑧 ∈ ((1...𝑛) βˆͺ (β„€β‰₯β€˜(𝑛 + 1)))if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)) = (βˆͺ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)))
289285, 287, 2883eqtr3g 2796 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ βˆͺ ran (𝑧 ∈ β„• ↦ ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))) = (βˆͺ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩))))
290 iftrue 4534 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (1...𝑛) β†’ if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)) = ([,]β€˜(π‘“β€˜π‘§)))
291290iuneq2i 5018 . . . . . . . . . . . . . . . . 17 βˆͺ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)) = βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))
292291a1i 11 . . . . . . . . . . . . . . . 16 (𝑛 ∈ β„• β†’ βˆͺ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)) = βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)))
293 uznfz 13581 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1)) β†’ Β¬ 𝑧 ∈ (1...((𝑛 + 1) βˆ’ 1)))
294293adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ β„• ∧ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))) β†’ Β¬ 𝑧 ∈ (1...((𝑛 + 1) βˆ’ 1)))
295 nncn 12217 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ β„• β†’ 𝑛 ∈ β„‚)
296 ax-1cn 11165 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ β„‚
297 pncan 11463 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝑛 + 1) βˆ’ 1) = 𝑛)
298295, 296, 297sylancl 587 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ β„• β†’ ((𝑛 + 1) βˆ’ 1) = 𝑛)
299298oveq2d 7422 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ β„• β†’ (1...((𝑛 + 1) βˆ’ 1)) = (1...𝑛))
300299eleq2d 2820 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ β„• β†’ (𝑧 ∈ (1...((𝑛 + 1) βˆ’ 1)) ↔ 𝑧 ∈ (1...𝑛)))
301300notbid 318 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ β„• β†’ (Β¬ 𝑧 ∈ (1...((𝑛 + 1) βˆ’ 1)) ↔ Β¬ 𝑧 ∈ (1...𝑛)))
302301adantr 482 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ β„• ∧ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))) β†’ (Β¬ 𝑧 ∈ (1...((𝑛 + 1) βˆ’ 1)) ↔ Β¬ 𝑧 ∈ (1...𝑛)))
303294, 302mpbid 231 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ β„• ∧ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))) β†’ Β¬ 𝑧 ∈ (1...𝑛))
304303iffalsed 4539 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ β„• ∧ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))) β†’ if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)) = ([,]β€˜βŸ¨0, 0⟩))
305304iuneq2dv 5021 . . . . . . . . . . . . . . . 16 (𝑛 ∈ β„• β†’ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)) = βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩))
306292, 305uneq12d 4164 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ (βˆͺ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]β€˜(π‘“β€˜π‘§)), ([,]β€˜βŸ¨0, 0⟩))) = (βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩)))
307289, 306eqtrd 2773 . . . . . . . . . . . . . 14 (𝑛 ∈ β„• β†’ βˆͺ ran (𝑧 ∈ β„• ↦ ([,]β€˜if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))) = (βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩)))
308272, 307sylan9eq 2793 . . . . . . . . . . . . 13 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ βˆͺ ran ([,] ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))) = (βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩)))
309308fveq2d 6893 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜βˆͺ ran ([,] ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))) = (vol*β€˜(βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩))))
310 xrltso 13117 . . . . . . . . . . . . . . 15 < Or ℝ*
311310a1i 11 . . . . . . . . . . . . . 14 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ < Or ℝ*)
312 elnnuz 12863 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ β„• ↔ 𝑛 ∈ (β„€β‰₯β€˜1))
313312biimpi 215 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ β„• β†’ 𝑛 ∈ (β„€β‰₯β€˜1))
314313adantl 483 . . . . . . . . . . . . . . . 16 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ (β„€β‰₯β€˜1))
315 elfznn 13527 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (1...𝑛) β†’ 𝑒 ∈ β„•)
316172ffvelcdmda 7084 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑒 ∈ β„•) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘’) ∈ ℝ)
317315, 316sylan2 594 . . . . . . . . . . . . . . . . 17 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑒 ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘’) ∈ ℝ)
318317adantlr 714 . . . . . . . . . . . . . . . 16 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑒 ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘’) ∈ ℝ)
319 readdcl 11190 . . . . . . . . . . . . . . . . 17 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑒 + 𝑣) ∈ ℝ)
320319adantl 483 . . . . . . . . . . . . . . . 16 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ (𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ)) β†’ (𝑒 + 𝑣) ∈ ℝ)
321314, 318, 320seqcl 13985 . . . . . . . . . . . . . . 15 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ∈ ℝ)
322321rexrd 11261 . . . . . . . . . . . . . 14 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ∈ ℝ*)
323 eqidd 2734 . . . . . . . . . . . . . . . . . . . . 21 (π‘š ∈ (1...𝑛) β†’ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) = (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))
324 iftrue 4534 . . . . . . . . . . . . . . . . . . . . . 22 (π‘š ∈ (1...𝑛) β†’ if(π‘š ∈ (1...𝑛), (π‘“β€˜π‘š), ⟨0, 0⟩) = (π‘“β€˜π‘š))
325238, 324sylan9eqr 2795 . . . . . . . . . . . . . . . . . . . . 21 ((π‘š ∈ (1...𝑛) ∧ 𝑧 = π‘š) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) = (π‘“β€˜π‘š))
326 elfznn 13527 . . . . . . . . . . . . . . . . . . . . 21 (π‘š ∈ (1...𝑛) β†’ π‘š ∈ β„•)
327240a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (π‘š ∈ (1...𝑛) β†’ (π‘“β€˜π‘š) ∈ V)
328323, 325, 326, 327fvmptd 7003 . . . . . . . . . . . . . . . . . . . 20 (π‘š ∈ (1...𝑛) β†’ ((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š) = (π‘“β€˜π‘š))
329328adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ (1...𝑛)) β†’ ((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š) = (π‘“β€˜π‘š))
330329fveq2d 6893 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ (1...𝑛)) β†’ ((abs ∘ βˆ’ )β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)) = ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘š)))
331 fvex 6902 . . . . . . . . . . . . . . . . . . . . . 22 (π‘“β€˜π‘§) ∈ V
332331, 241ifex 4578 . . . . . . . . . . . . . . . . . . . . 21 if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) ∈ V
333332, 239fnmpti 6691 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) Fn β„•
334 fvco2 6986 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) Fn β„• ∧ π‘š ∈ β„•) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = ((abs ∘ βˆ’ )β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)))
335333, 326, 334sylancr 588 . . . . . . . . . . . . . . . . . . 19 (π‘š ∈ (1...𝑛) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = ((abs ∘ βˆ’ )β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)))
336335adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = ((abs ∘ βˆ’ )β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)))
337 ffn 6715 . . . . . . . . . . . . . . . . . . 19 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ 𝑓 Fn β„•)
338 fvco2 6986 . . . . . . . . . . . . . . . . . . 19 ((𝑓 Fn β„• ∧ π‘š ∈ β„•) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) = ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘š)))
339337, 326, 338syl2an 597 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) = ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘š)))
340330, 336, 3393eqtr4d 2783 . . . . . . . . . . . . . . . . 17 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š))
341340adantlr 714 . . . . . . . . . . . . . . . 16 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ π‘š ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š))
342314, 341seqfveq 13989 . . . . . . . . . . . . . . 15 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘›) = (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
343174a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ 1 ∈ β„€)
344168, 65sselid 3980 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ (π‘“β€˜π‘§) ∈ (β„‚ Γ— β„‚))
345 0cn 11203 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ β„‚
346 opelxpi 5713 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ β„‚ ∧ 0 ∈ β„‚) β†’ ⟨0, 0⟩ ∈ (β„‚ Γ— β„‚))
347345, 345, 346mp2an 691 . . . . . . . . . . . . . . . . . . . . . 22 ⟨0, 0⟩ ∈ (β„‚ Γ— β„‚)
348 ifcl 4573 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘“β€˜π‘§) ∈ (β„‚ Γ— β„‚) ∧ ⟨0, 0⟩ ∈ (β„‚ Γ— β„‚)) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) ∈ (β„‚ Γ— β„‚))
349344, 347, 348sylancl 587 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) ∈ (β„‚ Γ— β„‚))
350349fmpttd 7112 . . . . . . . . . . . . . . . . . . . 20 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)):β„•βŸΆ(β„‚ Γ— β„‚))
351 fco 6739 . . . . . . . . . . . . . . . . . . . 20 (((abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„ ∧ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)):β„•βŸΆ(β„‚ Γ— β„‚)) β†’ ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))):β„•βŸΆβ„)
352139, 350, 351sylancr 588 . . . . . . . . . . . . . . . . . . 19 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))):β„•βŸΆβ„)
353352ffvelcdmda 7084 . . . . . . . . . . . . . . . . . 18 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) ∈ ℝ)
354173, 343, 353serfre 13994 . . . . . . . . . . . . . . . . 17 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))):β„•βŸΆβ„)
355354ffnd 6716 . . . . . . . . . . . . . . . 16 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))) Fn β„•)
356 fnfvelrn 7080 . . . . . . . . . . . . . . . 16 ((seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))) Fn β„• ∧ 𝑛 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘›) ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))))
357355, 356sylan 581 . . . . . . . . . . . . . . 15 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘›) ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))))
358342, 357eqeltrrd 2835 . . . . . . . . . . . . . 14 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))))
359354frnd 6723 . . . . . . . . . . . . . . . . 17 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))) βŠ† ℝ)
360359adantr 482 . . . . . . . . . . . . . . . 16 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))) βŠ† ℝ)
361360sselda 3982 . . . . . . . . . . . . . . 15 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ π‘š ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))) β†’ π‘š ∈ ℝ)
362321adantr 482 . . . . . . . . . . . . . . 15 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ π‘š ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ∈ ℝ)
363 readdcl 11190 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘š ∈ ℝ ∧ 𝑒 ∈ ℝ) β†’ (π‘š + 𝑒) ∈ ℝ)
364363adantl 483 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ (π‘š ∈ ℝ ∧ 𝑒 ∈ ℝ)) β†’ (π‘š + 𝑒) ∈ ℝ)
365 recn 11197 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘š ∈ ℝ β†’ π‘š ∈ β„‚)
366 recn 11197 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑒 ∈ ℝ β†’ 𝑒 ∈ β„‚)
367 recn 11197 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 ∈ ℝ β†’ 𝑣 ∈ β„‚)
368 addass 11194 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘š ∈ β„‚ ∧ 𝑒 ∈ β„‚ ∧ 𝑣 ∈ β„‚) β†’ ((π‘š + 𝑒) + 𝑣) = (π‘š + (𝑒 + 𝑣)))
369365, 366, 367, 368syl3an 1161 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘š ∈ ℝ ∧ 𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ ((π‘š + 𝑒) + 𝑣) = (π‘š + (𝑒 + 𝑣)))
370369adantl 483 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ (π‘š ∈ ℝ ∧ 𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ)) β†’ ((π‘š + 𝑒) + 𝑣) = (π‘š + (𝑒 + 𝑣)))
371 nnltp1le 12615 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) β†’ (𝑛 < 𝑑 ↔ (𝑛 + 1) ≀ 𝑑))
372371biimpa 478 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (𝑛 + 1) ≀ 𝑑)
373273nnzd 12582 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ β„• β†’ (𝑛 + 1) ∈ β„€)
374 nnz 12576 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 ∈ β„• β†’ 𝑑 ∈ β„€)
375 eluz 12833 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑛 + 1) ∈ β„€ ∧ 𝑑 ∈ β„€) β†’ (𝑑 ∈ (β„€β‰₯β€˜(𝑛 + 1)) ↔ (𝑛 + 1) ≀ 𝑑))
376373, 374, 375syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) β†’ (𝑑 ∈ (β„€β‰₯β€˜(𝑛 + 1)) ↔ (𝑛 + 1) ≀ 𝑑))
377376adantr 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (𝑑 ∈ (β„€β‰₯β€˜(𝑛 + 1)) ↔ (𝑛 + 1) ≀ 𝑑))
378372, 377mpbird 257 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ 𝑑 ∈ (β„€β‰₯β€˜(𝑛 + 1)))
379378adantlll 717 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ 𝑑 ∈ (β„€β‰₯β€˜(𝑛 + 1)))
380313ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ 𝑛 ∈ (β„€β‰₯β€˜1))
381 simplll 774 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ 𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
382 elfznn 13527 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘š ∈ (1...𝑑) β†’ π‘š ∈ β„•)
383381, 382, 353syl2an 597 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ (1...𝑑)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) ∈ ℝ)
384364, 370, 379, 380, 383seqsplit 13998 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) = ((seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘›) + (seq(𝑛 + 1)( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘)))
385342ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘›) = (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
386 elfzelz 13498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (π‘š ∈ ((𝑛 + 1)...𝑑) β†’ π‘š ∈ β„€)
387386adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ π‘š ∈ β„€)
388 0red 11214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ 0 ∈ ℝ)
389273nnred 12224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑛 ∈ β„• β†’ (𝑛 + 1) ∈ ℝ)
390389ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (𝑛 + 1) ∈ ℝ)
391386zred 12663 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (π‘š ∈ ((𝑛 + 1)...𝑑) β†’ π‘š ∈ ℝ)
392391adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ π‘š ∈ ℝ)
393273nngt0d 12258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑛 ∈ β„• β†’ 0 < (𝑛 + 1))
394393ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ 0 < (𝑛 + 1))
395 elfzle1 13501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (π‘š ∈ ((𝑛 + 1)...𝑑) β†’ (𝑛 + 1) ≀ π‘š)
396395adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (𝑛 + 1) ≀ π‘š)
397388, 390, 392, 394, 396ltletrd 11371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ 0 < π‘š)
398 elnnz 12565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (π‘š ∈ β„• ↔ (π‘š ∈ β„€ ∧ 0 < π‘š))
399387, 397, 398sylanbrc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ π‘š ∈ β„•)
400333, 399, 334sylancr 588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = ((abs ∘ βˆ’ )β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)))
401 eqidd 2734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) = (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))
402 nnre 12216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑛 ∈ β„• β†’ 𝑛 ∈ ℝ)
403402adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ 𝑛 ∈ ℝ)
404389adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (𝑛 + 1) ∈ ℝ)
405391adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ π‘š ∈ ℝ)
406402ltp1d 12141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑛 ∈ β„• β†’ 𝑛 < (𝑛 + 1))
407406adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ 𝑛 < (𝑛 + 1))
408395adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (𝑛 + 1) ≀ π‘š)
409403, 404, 405, 407, 408ltletrd 11371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ 𝑛 < π‘š)
410409adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) ∧ 𝑧 = π‘š) β†’ 𝑛 < π‘š)
411403, 405ltnled 11358 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (𝑛 < π‘š ↔ Β¬ π‘š ≀ 𝑛))
412 breq1 5151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (π‘š = 𝑧 β†’ (π‘š ≀ 𝑛 ↔ 𝑧 ≀ 𝑛))
413412equcoms 2024 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑧 = π‘š β†’ (π‘š ≀ 𝑛 ↔ 𝑧 ≀ 𝑛))
414413notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 = π‘š β†’ (Β¬ π‘š ≀ 𝑛 ↔ Β¬ 𝑧 ≀ 𝑛))
415411, 414sylan9bb 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) ∧ 𝑧 = π‘š) β†’ (𝑛 < π‘š ↔ Β¬ 𝑧 ≀ 𝑛))
416410, 415mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) ∧ 𝑧 = π‘š) β†’ Β¬ 𝑧 ≀ 𝑛)
417 elfzle2 13502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 ∈ (1...𝑛) β†’ 𝑧 ≀ 𝑛)
418416, 417nsyl 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) ∧ 𝑧 = π‘š) β†’ Β¬ 𝑧 ∈ (1...𝑛))
419418iffalsed 4539 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) ∧ 𝑧 = π‘š) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) = ⟨0, 0⟩)
420386adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ π‘š ∈ β„€)
421 0red 11214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ 0 ∈ ℝ)
422393adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ 0 < (𝑛 + 1))
423421, 404, 405, 422, 408ltletrd 11371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ 0 < π‘š)
424420, 423, 398sylanbrc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ π‘š ∈ β„•)
425241a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ ⟨0, 0⟩ ∈ V)
426401, 419, 424, 425fvmptd 7003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∈ β„• ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ ((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š) = ⟨0, 0⟩)
427426ad4ant14 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ ((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š) = ⟨0, 0⟩)
428427fveq2d 6893 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ ((abs ∘ βˆ’ )β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)) = ((abs ∘ βˆ’ )β€˜βŸ¨0, 0⟩))
429400, 428eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = ((abs ∘ βˆ’ )β€˜βŸ¨0, 0⟩))
430 fvco3 6988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (( βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚ ∧ ⟨0, 0⟩ ∈ (β„‚ Γ— β„‚)) β†’ ((abs ∘ βˆ’ )β€˜βŸ¨0, 0⟩) = (absβ€˜( βˆ’ β€˜βŸ¨0, 0⟩)))
431137, 347, 430mp2an 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((abs ∘ βˆ’ )β€˜βŸ¨0, 0⟩) = (absβ€˜( βˆ’ β€˜βŸ¨0, 0⟩))
432 df-ov 7409 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 βˆ’ 0) = ( βˆ’ β€˜βŸ¨0, 0⟩)
433 0m0e0 12329 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 βˆ’ 0) = 0
434432, 433eqtr3i 2763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ( βˆ’ β€˜βŸ¨0, 0⟩) = 0
435434fveq2i 6892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (absβ€˜( βˆ’ β€˜βŸ¨0, 0⟩)) = (absβ€˜0)
436 abs0 15229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (absβ€˜0) = 0
437435, 436eqtri 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (absβ€˜( βˆ’ β€˜βŸ¨0, 0⟩)) = 0
438431, 437eqtri 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((abs ∘ βˆ’ )β€˜βŸ¨0, 0⟩) = 0
439429, 438eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = 0)
440 elfzuz 13494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (π‘š ∈ ((𝑛 + 1)...𝑑) β†’ π‘š ∈ (β„€β‰₯β€˜(𝑛 + 1)))
441 c0ex 11205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 ∈ V
442441fvconst2 7202 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (π‘š ∈ (β„€β‰₯β€˜(𝑛 + 1)) β†’ (((β„€β‰₯β€˜(𝑛 + 1)) Γ— {0})β€˜π‘š) = 0)
443440, 442syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘š ∈ ((𝑛 + 1)...𝑑) β†’ (((β„€β‰₯β€˜(𝑛 + 1)) Γ— {0})β€˜π‘š) = 0)
444443adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (((β„€β‰₯β€˜(𝑛 + 1)) Γ— {0})β€˜π‘š) = 0)
445439, 444eqtr4d 2776 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ ((𝑛 + 1)...𝑑)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = (((β„€β‰₯β€˜(𝑛 + 1)) Γ— {0})β€˜π‘š))
446378, 445seqfveq 13989 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq(𝑛 + 1)( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) = (seq(𝑛 + 1)( + , ((β„€β‰₯β€˜(𝑛 + 1)) Γ— {0}))β€˜π‘‘))
447 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (β„€β‰₯β€˜(𝑛 + 1)) = (β„€β‰₯β€˜(𝑛 + 1))
448447ser0 14017 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 ∈ (β„€β‰₯β€˜(𝑛 + 1)) β†’ (seq(𝑛 + 1)( + , ((β„€β‰₯β€˜(𝑛 + 1)) Γ— {0}))β€˜π‘‘) = 0)
449378, 448syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq(𝑛 + 1)( + , ((β„€β‰₯β€˜(𝑛 + 1)) Γ— {0}))β€˜π‘‘) = 0)
450446, 449eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq(𝑛 + 1)( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) = 0)
451450adantlll 717 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq(𝑛 + 1)( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) = 0)
452385, 451oveq12d 7424 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ ((seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘›) + (seq(𝑛 + 1)( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘)) = ((seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) + 0))
453172ffvelcdmda 7084 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) ∈ ℝ)
454326, 453sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) ∈ ℝ)
455454adantlr 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ π‘š ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) ∈ ℝ)
456 readdcl 11190 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘š ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (π‘š + 𝑣) ∈ ℝ)
457456adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ (π‘š ∈ ℝ ∧ 𝑣 ∈ ℝ)) β†’ (π‘š + 𝑣) ∈ ℝ)
458314, 455, 457seqcl 13985 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ∈ ℝ)
459458ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ∈ ℝ)
460459recnd 11239 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ∈ β„‚)
461460addridd 11411 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ ((seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) + 0) = (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
462452, 461eqtrd 2773 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ ((seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘›) + (seq(𝑛 + 1)( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘)) = (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
463384, 462eqtrd 2773 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) = (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
464453ad5ant15 758 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ β„•) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) ∈ ℝ)
465326, 464sylan2 594 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) ∧ π‘š ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) ∈ ℝ)
466380, 465, 364seqcl 13985 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ∈ ℝ)
467466leidd 11777 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
468463, 467eqbrtrd 5170 . . . . . . . . . . . . . . . . . . 19 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑛 < 𝑑) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
469 elnnuz 12863 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 ∈ β„• ↔ 𝑑 ∈ (β„€β‰₯β€˜1))
470469biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ β„• β†’ 𝑑 ∈ (β„€β‰₯β€˜1))
471470ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) β†’ 𝑑 ∈ (β„€β‰₯β€˜1))
472 eqidd 2734 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)) = (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))
473 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) ∧ 𝑧 = π‘š) β†’ 𝑧 = π‘š)
474 elfzle1 13501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (π‘š ∈ (1...𝑑) β†’ 1 ≀ π‘š)
475474adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ 1 ≀ π‘š)
476382nnred 12224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (π‘š ∈ (1...𝑑) β†’ π‘š ∈ ℝ)
477476adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ π‘š ∈ ℝ)
478 nnre 12216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑑 ∈ β„• β†’ 𝑑 ∈ ℝ)
479478ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ 𝑑 ∈ ℝ)
480402ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ 𝑛 ∈ ℝ)
481 elfzle2 13502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (π‘š ∈ (1...𝑑) β†’ π‘š ≀ 𝑑)
482481adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ π‘š ≀ 𝑑)
483 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ 𝑑 ≀ 𝑛)
484477, 479, 480, 482, 483letrd 11368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ π‘š ≀ 𝑛)
485 elfzelz 13498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (π‘š ∈ (1...𝑑) β†’ π‘š ∈ β„€)
486278ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) β†’ 𝑛 ∈ β„€)
487 elfz 13487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((π‘š ∈ β„€ ∧ 1 ∈ β„€ ∧ 𝑛 ∈ β„€) β†’ (π‘š ∈ (1...𝑛) ↔ (1 ≀ π‘š ∧ π‘š ≀ 𝑛)))
488174, 487mp3an2 1450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((π‘š ∈ β„€ ∧ 𝑛 ∈ β„€) β†’ (π‘š ∈ (1...𝑛) ↔ (1 ≀ π‘š ∧ π‘š ≀ 𝑛)))
489485, 486, 488syl2anr 598 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ (π‘š ∈ (1...𝑛) ↔ (1 ≀ π‘š ∧ π‘š ≀ 𝑛)))
490475, 484, 489mpbir2and 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ π‘š ∈ (1...𝑛))
491490ad5ant2345 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ π‘š ∈ (1...𝑛))
492491adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) ∧ 𝑧 = π‘š) β†’ π‘š ∈ (1...𝑛))
493473, 492eqeltrd 2834 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) ∧ 𝑧 = π‘š) β†’ 𝑧 ∈ (1...𝑛))
494 iftrue 4534 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ (1...𝑛) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) = (π‘“β€˜π‘§))
495493, 494syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) ∧ 𝑧 = π‘š) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) = (π‘“β€˜π‘§))
496237adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) ∧ 𝑧 = π‘š) β†’ (π‘“β€˜π‘§) = (π‘“β€˜π‘š))
497495, 496eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) ∧ 𝑧 = π‘š) β†’ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩) = (π‘“β€˜π‘š))
498382adantl 483 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ π‘š ∈ β„•)
499240a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ (π‘“β€˜π‘š) ∈ V)
500472, 497, 498, 499fvmptd 7003 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ ((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š) = (π‘“β€˜π‘š))
501500fveq2d 6893 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ ((abs ∘ βˆ’ )β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)) = ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘š)))
502333, 382, 334sylancr 588 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘š ∈ (1...𝑑) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = ((abs ∘ βˆ’ )β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)))
503502adantl 483 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = ((abs ∘ βˆ’ )β€˜((𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))β€˜π‘š)))
504 simplll 774 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) β†’ 𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)})
505 fvco3 6988 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) = ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘š)))
506504, 382, 505syl2an 597 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) = ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘š)))
507501, 503, 5063eqtr4d 2783 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑑)) β†’ (((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))β€˜π‘š) = (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š))
508471, 507seqfveq 13989 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) = (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘‘))
509 eluz 12833 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 ∈ β„€ ∧ 𝑛 ∈ β„€) β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘‘) ↔ 𝑑 ≀ 𝑛))
510374, 278, 509syl2anr 598 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘‘) ↔ 𝑑 ≀ 𝑛))
511510biimpar 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ β„• ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘‘))
512511adantlll 717 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘‘))
513504, 326, 453syl2an 597 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ (1...𝑛)) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) ∈ ℝ)
514 elfzelz 13498 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘š ∈ ((𝑑 + 1)...𝑛) β†’ π‘š ∈ β„€)
515514adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 ∈ β„• ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ π‘š ∈ β„€)
516 0red 11214 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 ∈ β„• ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ 0 ∈ ℝ)
517 peano2nn 12221 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑑 ∈ β„• β†’ (𝑑 + 1) ∈ β„•)
518517nnred 12224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑 ∈ β„• β†’ (𝑑 + 1) ∈ ℝ)
519518adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 ∈ β„• ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ (𝑑 + 1) ∈ ℝ)
520514zred 12663 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘š ∈ ((𝑑 + 1)...𝑛) β†’ π‘š ∈ ℝ)
521520adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 ∈ β„• ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ π‘š ∈ ℝ)
522517nngt0d 12258 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑 ∈ β„• β†’ 0 < (𝑑 + 1))
523522adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 ∈ β„• ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ 0 < (𝑑 + 1))
524 elfzle1 13501 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘š ∈ ((𝑑 + 1)...𝑛) β†’ (𝑑 + 1) ≀ π‘š)
525524adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 ∈ β„• ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ (𝑑 + 1) ≀ π‘š)
526516, 519, 521, 523, 525ltletrd 11371 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 ∈ β„• ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ 0 < π‘š)
527515, 526, 398sylanbrc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 ∈ β„• ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ π‘š ∈ β„•)
528527adantlr 714 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑑 ∈ β„• ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ π‘š ∈ β„•)
529528adantlll 717 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ π‘š ∈ β„•)
530170ffvelcdmda 7084 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ (π‘“β€˜π‘š) ∈ (β„‚ Γ— β„‚))
531 ffvelcdm 7081 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚ ∧ (π‘“β€˜π‘š) ∈ (β„‚ Γ— β„‚)) β†’ ( βˆ’ β€˜(π‘“β€˜π‘š)) ∈ β„‚)
532137, 530, 531sylancr 588 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ ( βˆ’ β€˜(π‘“β€˜π‘š)) ∈ β„‚)
533532absge0d 15388 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ 0 ≀ (absβ€˜( βˆ’ β€˜(π‘“β€˜π‘š))))
534 fvco3 6988 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚ ∧ (π‘“β€˜π‘š) ∈ (β„‚ Γ— β„‚)) β†’ ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘š)) = (absβ€˜( βˆ’ β€˜(π‘“β€˜π‘š))))
535137, 530, 534sylancr 588 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘š)) = (absβ€˜( βˆ’ β€˜(π‘“β€˜π‘š))))
536505, 535eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š) = (absβ€˜( βˆ’ β€˜(π‘“β€˜π‘š))))
537533, 536breqtrrd 5176 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ π‘š ∈ β„•) β†’ 0 ≀ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š))
538537ad5ant15 758 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ β„•) β†’ 0 ≀ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š))
539529, 538syldan 592 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) ∧ π‘š ∈ ((𝑑 + 1)...𝑛)) β†’ 0 ≀ (((abs ∘ βˆ’ ) ∘ 𝑓)β€˜π‘š))
540471, 512, 513, 539sermono 13997 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘‘) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
541508, 540eqbrtrd 5170 . . . . . . . . . . . . . . . . . . 19 ((((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ≀ 𝑛) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
542402ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) β†’ 𝑛 ∈ ℝ)
543478adantl 483 . . . . . . . . . . . . . . . . . . 19 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) β†’ 𝑑 ∈ ℝ)
544468, 541, 542, 543ltlecasei 11319 . . . . . . . . . . . . . . . . . 18 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ 𝑑 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
545544ralrimiva 3147 . . . . . . . . . . . . . . . . 17 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘‘ ∈ β„• (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
546 breq1 5151 . . . . . . . . . . . . . . . . . . . 20 (π‘š = (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) β†’ (π‘š ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ↔ (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›)))
547546ralrn 7087 . . . . . . . . . . . . . . . . . . 19 (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))) Fn β„• β†’ (βˆ€π‘š ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))π‘š ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ↔ βˆ€π‘‘ ∈ β„• (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›)))
548355, 547syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ (βˆ€π‘š ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))π‘š ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ↔ βˆ€π‘‘ ∈ β„• (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›)))
549548adantr 482 . . . . . . . . . . . . . . . . 17 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (βˆ€π‘š ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))π‘š ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ↔ βˆ€π‘‘ ∈ β„• (seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))β€˜π‘‘) ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›)))
550545, 549mpbird 257 . . . . . . . . . . . . . . . 16 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘š ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))π‘š ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
551550r19.21bi 3249 . . . . . . . . . . . . . . 15 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ π‘š ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))) β†’ π‘š ≀ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
552361, 362, 551lensymd 11362 . . . . . . . . . . . . . 14 (((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) ∧ π‘š ∈ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩))))) β†’ Β¬ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) < π‘š)
553311, 322, 358, 552supmax 9459 . . . . . . . . . . . . 13 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))), ℝ*, < ) = (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
55452, 553sylan 581 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑧 ∈ β„• ↦ if(𝑧 ∈ (1...𝑛), (π‘“β€˜π‘§), ⟨0, 0⟩)))), ℝ*, < ) = (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›))
555255, 309, 5543eqtr3rd 2782 . . . . . . . . . . 11 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) = (vol*β€˜(βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩))))
556 elfznn 13527 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (1...𝑛) β†’ 𝑧 ∈ β„•)
557164, 65sselid 3980 . . . . . . . . . . . . . . . . 17 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ (π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ))
558 1st2nd2 8011 . . . . . . . . . . . . . . . . . . . 20 ((π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ) β†’ (π‘“β€˜π‘§) = ⟨(1st β€˜(π‘“β€˜π‘§)), (2nd β€˜(π‘“β€˜π‘§))⟩)
559558fveq2d 6893 . . . . . . . . . . . . . . . . . . 19 ((π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ) β†’ ([,]β€˜(π‘“β€˜π‘§)) = ([,]β€˜βŸ¨(1st β€˜(π‘“β€˜π‘§)), (2nd β€˜(π‘“β€˜π‘§))⟩))
560 df-ov 7409 . . . . . . . . . . . . . . . . . . 19 ((1st β€˜(π‘“β€˜π‘§))[,](2nd β€˜(π‘“β€˜π‘§))) = ([,]β€˜βŸ¨(1st β€˜(π‘“β€˜π‘§)), (2nd β€˜(π‘“β€˜π‘§))⟩)
561559, 560eqtr4di 2791 . . . . . . . . . . . . . . . . . 18 ((π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ) β†’ ([,]β€˜(π‘“β€˜π‘§)) = ((1st β€˜(π‘“β€˜π‘§))[,](2nd β€˜(π‘“β€˜π‘§))))
562 xp1st 8004 . . . . . . . . . . . . . . . . . . 19 ((π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(π‘“β€˜π‘§)) ∈ ℝ)
563 xp2nd 8005 . . . . . . . . . . . . . . . . . . 19 ((π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(π‘“β€˜π‘§)) ∈ ℝ)
564 iccssre 13403 . . . . . . . . . . . . . . . . . . 19 (((1st β€˜(π‘“β€˜π‘§)) ∈ ℝ ∧ (2nd β€˜(π‘“β€˜π‘§)) ∈ ℝ) β†’ ((1st β€˜(π‘“β€˜π‘§))[,](2nd β€˜(π‘“β€˜π‘§))) βŠ† ℝ)
565562, 563, 564syl2anc 585 . . . . . . . . . . . . . . . . . 18 ((π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ) β†’ ((1st β€˜(π‘“β€˜π‘§))[,](2nd β€˜(π‘“β€˜π‘§))) βŠ† ℝ)
566561, 565eqsstrd 4020 . . . . . . . . . . . . . . . . 17 ((π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ) β†’ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ℝ)
567557, 566syl 17 . . . . . . . . . . . . . . . 16 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ℝ)
56852, 556, 567syl2an 597 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) β†’ ([,]β€˜(π‘“β€˜π‘§)) βŠ† ℝ)
569568ralrimiva 3147 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆ€π‘§ ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† ℝ)
570 iunss 5048 . . . . . . . . . . . . . 14 (βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† ℝ ↔ βˆ€π‘§ ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† ℝ)
571569, 570sylibr 233 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† ℝ)
572571adantr 482 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† ℝ)
573 uzid 12834 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ β„€ β†’ (𝑛 + 1) ∈ (β„€β‰₯β€˜(𝑛 + 1)))
574 ne0i 4334 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ (β„€β‰₯β€˜(𝑛 + 1)) β†’ (β„€β‰₯β€˜(𝑛 + 1)) β‰  βˆ…)
575 iunconst 5006 . . . . . . . . . . . . . . . 16 ((β„€β‰₯β€˜(𝑛 + 1)) β‰  βˆ… β†’ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩) = ([,]β€˜βŸ¨0, 0⟩))
576373, 573, 574, 5754syl 19 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩) = ([,]β€˜βŸ¨0, 0⟩))
577 iccid 13366 . . . . . . . . . . . . . . . . 17 (0 ∈ ℝ* β†’ (0[,]0) = {0})
578259, 577ax-mp 5 . . . . . . . . . . . . . . . 16 (0[,]0) = {0}
579 df-ov 7409 . . . . . . . . . . . . . . . 16 (0[,]0) = ([,]β€˜βŸ¨0, 0⟩)
580578, 579eqtr3i 2763 . . . . . . . . . . . . . . 15 {0} = ([,]β€˜βŸ¨0, 0⟩)
581576, 580eqtr4di 2791 . . . . . . . . . . . . . 14 (𝑛 ∈ β„• β†’ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩) = {0})
582 snssi 4811 . . . . . . . . . . . . . . 15 (0 ∈ ℝ β†’ {0} βŠ† ℝ)
583199, 582ax-mp 5 . . . . . . . . . . . . . 14 {0} βŠ† ℝ
584581, 583eqsstrdi 4036 . . . . . . . . . . . . 13 (𝑛 ∈ β„• β†’ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩) βŠ† ℝ)
585584adantl 483 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩) βŠ† ℝ)
586581fveq2d 6893 . . . . . . . . . . . . . 14 (𝑛 ∈ β„• β†’ (vol*β€˜βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩)) = (vol*β€˜{0}))
587586adantl 483 . . . . . . . . . . . . 13 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩)) = (vol*β€˜{0}))
588 ovolsn 25004 . . . . . . . . . . . . . 14 (0 ∈ ℝ β†’ (vol*β€˜{0}) = 0)
589199, 588ax-mp 5 . . . . . . . . . . . . 13 (vol*β€˜{0}) = 0
590587, 589eqtrdi 2789 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩)) = 0)
591 ovolunnul 25009 . . . . . . . . . . . 12 ((βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† ℝ ∧ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩) βŠ† ℝ ∧ (vol*β€˜βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩)) = 0) β†’ (vol*β€˜(βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩))) = (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))
592572, 585, 590, 591syl3anc 1372 . . . . . . . . . . 11 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜(βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βˆͺ βˆͺ 𝑧 ∈ (β„€β‰₯β€˜(𝑛 + 1))([,]β€˜βŸ¨0, 0⟩))) = (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))
593555, 592eqtrd 2773 . . . . . . . . . 10 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) = (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))
594593breq2d 5160 . . . . . . . . 9 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (𝑀 < (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) ↔ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)))))
595594biimpd 228 . . . . . . . 8 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑛 ∈ β„•) β†’ (𝑀 < (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) β†’ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)))))
596595reximdva 3169 . . . . . . 7 (𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ (βˆƒπ‘› ∈ β„• 𝑀 < (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) β†’ βˆƒπ‘› ∈ β„• 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)))))
597596adantl 483 . . . . . 6 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ (βˆƒπ‘› ∈ β„• 𝑀 < (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))β€˜π‘›) β†’ βˆƒπ‘› ∈ β„• 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)))))
598194, 597mpd 15 . . . . 5 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆƒπ‘› ∈ β„• 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))
599 fzfi 13934 . . . . . . . . . 10 (1...𝑛) ∈ Fin
600 icccld 24275 . . . . . . . . . . . . . . 15 (((1st β€˜(π‘“β€˜π‘§)) ∈ ℝ ∧ (2nd β€˜(π‘“β€˜π‘§)) ∈ ℝ) β†’ ((1st β€˜(π‘“β€˜π‘§))[,](2nd β€˜(π‘“β€˜π‘§))) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
601562, 563, 600syl2anc 585 . . . . . . . . . . . . . 14 ((π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ) β†’ ((1st β€˜(π‘“β€˜π‘§))[,](2nd β€˜(π‘“β€˜π‘§))) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
602561, 601eqeltrd 2834 . . . . . . . . . . . . 13 ((π‘“β€˜π‘§) ∈ (ℝ Γ— ℝ) β†’ ([,]β€˜(π‘“β€˜π‘§)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
603557, 602syl 17 . . . . . . . . . . . 12 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ ([,]β€˜(π‘“β€˜π‘§)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
604556, 603sylan2 594 . . . . . . . . . . 11 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) β†’ ([,]β€˜(π‘“β€˜π‘§)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
605604ralrimiva 3147 . . . . . . . . . 10 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆ€π‘§ ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
606 uniretop 24271 . . . . . . . . . . 11 ℝ = βˆͺ (topGenβ€˜ran (,))
607606iuncld 22541 . . . . . . . . . 10 (((topGenβ€˜ran (,)) ∈ Top ∧ (1...𝑛) ∈ Fin ∧ βˆ€π‘§ ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) ∈ (Clsdβ€˜(topGenβ€˜ran (,)))) β†’ βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
6081, 599, 605, 607mp3an12i 1466 . . . . . . . . 9 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
609608adantr 482 . . . . . . . 8 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (𝑛 ∈ β„• ∧ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))) β†’ βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
610 fveq2 6889 . . . . . . . . . . . . . . . 16 (𝑏 = (π‘“β€˜π‘§) β†’ ([,]β€˜π‘) = ([,]β€˜(π‘“β€˜π‘§)))
611610sseq1d 4013 . . . . . . . . . . . . . . 15 (𝑏 = (π‘“β€˜π‘§) β†’ (([,]β€˜π‘) βŠ† 𝐴 ↔ ([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴))
612611elrab 3683 . . . . . . . . . . . . . 14 ((π‘“β€˜π‘§) ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ↔ ((π‘“β€˜π‘§) ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∧ ([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴))
613612simprbi 498 . . . . . . . . . . . . 13 ((π‘“β€˜π‘§) ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} β†’ ([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴)
61465, 73, 6133syl 18 . . . . . . . . . . . 12 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ β„•) β†’ ([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴)
615556, 614sylan2 594 . . . . . . . . . . 11 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) β†’ ([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴)
616615ralrimiva 3147 . . . . . . . . . 10 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆ€π‘§ ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴)
617 iunss 5048 . . . . . . . . . 10 (βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴 ↔ βˆ€π‘§ ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴)
618616, 617sylibr 233 . . . . . . . . 9 (𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} β†’ βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴)
619618adantr 482 . . . . . . . 8 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (𝑛 ∈ β„• ∧ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))) β†’ βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴)
620 simprr 772 . . . . . . . 8 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (𝑛 ∈ β„• ∧ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))) β†’ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))
621 sseq1 4007 . . . . . . . . . 10 (𝑠 = βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) β†’ (𝑠 βŠ† 𝐴 ↔ βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴))
622 fveq2 6889 . . . . . . . . . . 11 (𝑠 = βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) β†’ (vol*β€˜π‘ ) = (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))
623622breq2d 5160 . . . . . . . . . 10 (𝑠 = βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) β†’ (𝑀 < (vol*β€˜π‘ ) ↔ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)))))
624621, 623anbi12d 632 . . . . . . . . 9 (𝑠 = βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) β†’ ((𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )) ↔ (βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))))
625624rspcev 3613 . . . . . . . 8 ((βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§)) βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
626609, 619, 620, 625syl12anc 836 . . . . . . 7 ((𝑓:β„•βŸΆ{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (𝑛 ∈ β„• ∧ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
62752, 626sylan 581 . . . . . 6 ((𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)} ∧ (𝑛 ∈ β„• ∧ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
628627adantll 713 . . . . 5 ((((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) ∧ (𝑛 ∈ β„• ∧ 𝑀 < (vol*β€˜βˆͺ 𝑧 ∈ (1...𝑛)([,]β€˜(π‘“β€˜π‘§))))) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
629598, 628rexlimddv 3162 . . . 4 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
630629adantlr 714 . . 3 ((((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝐴 β‰  βˆ…) ∧ 𝑓:ℕ–1-1-ontoβ†’{π‘Ž ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} ∣ βˆ€π‘ ∈ {𝑏 ∈ ran (π‘₯ ∈ β„€, 𝑦 ∈ β„•0 ↦ ⟨(π‘₯ / (2↑𝑦)), ((π‘₯ + 1) / (2↑𝑦))⟩) ∣ ([,]β€˜π‘) βŠ† 𝐴} (([,]β€˜π‘Ž) βŠ† ([,]β€˜π‘) β†’ π‘Ž = 𝑐)}) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
63117, 630exlimddv 1939 . 2 (((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) ∧ 𝐴 β‰  βˆ…) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
63215, 631pm2.61dane 3030 1 ((𝐴 ∈ (topGenβ€˜ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*β€˜π΄)) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† 𝐴 ∧ 𝑀 < (vol*β€˜π‘ )))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  ifcif 4528  π’« cpw 4602  {csn 4628  βŸ¨cop 4634  βˆͺ cuni 4908  βˆͺ ciun 4997  Disj wdisj 5113   class class class wbr 5148   ↦ cmpt 5231   Or wor 5587   Γ— cxp 5674  ran crn 5677   β€œ cima 5679   ∘ ccom 5680   Fn wfn 6536  βŸΆwf 6537  β€“1-1β†’wf1 6538  β€“ontoβ†’wfo 6539  β€“1-1-ontoβ†’wf1o 6540  β€˜cfv 6541  (class class class)co 7406   ∈ cmpo 7408  1st c1st 7970  2nd c2nd 7971  Fincfn 8936  supcsup 9432  β„‚cc 11105  β„cr 11106  0cc0 11107  1c1 11108   + caddc 11110  β„*cxr 11244   < clt 11245   ≀ cle 11246   βˆ’ cmin 11441   / cdiv 11868  β„•cn 12209  2c2 12264  β„•0cn0 12469  β„€cz 12555  β„€β‰₯cuz 12819  (,)cioo 13321  [,]cicc 13324  ...cfz 13481  ..^cfzo 13624  seqcseq 13963  β†‘cexp 14024  abscabs 15178  topGenctg 17380  Topctop 22387  Clsdccld 22512  vol*covol 24971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-oadd 8467  df-omul 8468  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9502  df-dju 9893  df-card 9931  df-acn 9934  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-ioo 13325  df-ico 13327  df-icc 13328  df-fz 13482  df-fzo 13625  df-fl 13754  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-rlim 15430  df-sum 15630  df-rest 17365  df-topgen 17386  df-psmet 20929  df-xmet 20930  df-met 20931  df-bl 20932  df-mopn 20933  df-top 22388  df-topon 22405  df-bases 22441  df-cld 22515  df-cmp 22883  df-conn 22908  df-ovol 24973  df-vol 24974
This theorem is referenced by:  mblfinlem4  36517  ismblfin  36518
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