MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itg2mono Structured version   Visualization version   GIF version

Theorem itg2mono 25670
Description: The Monotone Convergence Theorem for nonnegative functions. If {(πΉβ€˜π‘›):𝑛 ∈ β„•} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫2β€˜πΊ) is the limit of the sequence {(∫2β€˜(πΉβ€˜π‘›)):𝑛 ∈ β„•}. (Contributed by Mario Carneiro, 16-Aug-2014.)
Hypotheses
Ref Expression
itg2mono.1 𝐺 = (π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
itg2mono.2 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ MblFn)
itg2mono.3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞))
itg2mono.4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∘r ≀ (πΉβ€˜(𝑛 + 1)))
itg2mono.5 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦)
itg2mono.6 𝑆 = sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < )
Assertion
Ref Expression
itg2mono (πœ‘ β†’ (∫2β€˜πΊ) = 𝑆)
Distinct variable groups:   π‘₯,𝑛,𝑦,𝐺   𝑛,𝐹,π‘₯,𝑦   πœ‘,𝑛,π‘₯,𝑦   𝑆,𝑛,π‘₯,𝑦

Proof of Theorem itg2mono
Dummy variables 𝑓 π‘š 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itg2mono.3 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞))
2 rge0ssre 13457 . . . . . . . . . . . 12 (0[,)+∞) βŠ† ℝ
3 fss 6733 . . . . . . . . . . . 12 (((πΉβ€˜π‘›):β„βŸΆ(0[,)+∞) ∧ (0[,)+∞) βŠ† ℝ) β†’ (πΉβ€˜π‘›):β„βŸΆβ„)
41, 2, 3sylancl 585 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆβ„)
54ffvelcdmda 7088 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜π‘›)β€˜π‘₯) ∈ ℝ)
65an32s 651 . . . . . . . . 9 (((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›)β€˜π‘₯) ∈ ℝ)
76fmpttd 7119 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)):β„•βŸΆβ„)
87frnd 6724 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) βŠ† ℝ)
9 1nn 12245 . . . . . . . . . 10 1 ∈ β„•
10 eqid 2727 . . . . . . . . . . 11 (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) = (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))
1110, 6dmmptd 6694 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ dom (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) = β„•)
129, 11eleqtrrid 2835 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ 1 ∈ dom (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)))
1312ne0d 4331 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ dom (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) β‰  βˆ…)
14 dm0rn0 5921 . . . . . . . . 9 (dom (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) = βˆ… ↔ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) = βˆ…)
1514necon3bii 2988 . . . . . . . 8 (dom (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) β‰  βˆ… ↔ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) β‰  βˆ…)
1613, 15sylib 217 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) β‰  βˆ…)
17 itg2mono.5 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦)
187ffnd 6717 . . . . . . . . . . 11 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) Fn β„•)
19 breq1 5145 . . . . . . . . . . . 12 (𝑧 = ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) β†’ (𝑧 ≀ 𝑦 ↔ ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) ≀ 𝑦))
2019ralrn 7092 . . . . . . . . . . 11 ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) Fn β„• β†’ (βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))𝑧 ≀ 𝑦 ↔ βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) ≀ 𝑦))
2118, 20syl 17 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))𝑧 ≀ 𝑦 ↔ βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) ≀ 𝑦))
22 fveq2 6891 . . . . . . . . . . . . . . 15 (𝑛 = π‘š β†’ (πΉβ€˜π‘›) = (πΉβ€˜π‘š))
2322fveq1d 6893 . . . . . . . . . . . . . 14 (𝑛 = π‘š β†’ ((πΉβ€˜π‘›)β€˜π‘₯) = ((πΉβ€˜π‘š)β€˜π‘₯))
24 fvex 6904 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘š)β€˜π‘₯) ∈ V
2523, 10, 24fvmpt 6999 . . . . . . . . . . . . 13 (π‘š ∈ β„• β†’ ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) = ((πΉβ€˜π‘š)β€˜π‘₯))
2625breq1d 5152 . . . . . . . . . . . 12 (π‘š ∈ β„• β†’ (((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) ≀ 𝑦 ↔ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ 𝑦))
2726ralbiia 3086 . . . . . . . . . . 11 (βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) ≀ 𝑦 ↔ βˆ€π‘š ∈ β„• ((πΉβ€˜π‘š)β€˜π‘₯) ≀ 𝑦)
2823breq1d 5152 . . . . . . . . . . . 12 (𝑛 = π‘š β†’ (((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦 ↔ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ 𝑦))
2928cbvralvw 3229 . . . . . . . . . . 11 (βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦 ↔ βˆ€π‘š ∈ β„• ((πΉβ€˜π‘š)β€˜π‘₯) ≀ 𝑦)
3027, 29bitr4i 278 . . . . . . . . . 10 (βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) ≀ 𝑦 ↔ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦)
3121, 30bitrdi 287 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))𝑧 ≀ 𝑦 ↔ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦))
3231rexbidv 3173 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))𝑧 ≀ 𝑦 ↔ βˆƒπ‘¦ ∈ ℝ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦))
3317, 32mpbird 257 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))𝑧 ≀ 𝑦)
348, 16, 33suprcld 12199 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ ℝ)
3534rexrd 11286 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ ℝ*)
36 0red 11239 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ 0 ∈ ℝ)
37 fveq2 6891 . . . . . . . . . . 11 (𝑛 = 1 β†’ (πΉβ€˜π‘›) = (πΉβ€˜1))
3837feq1d 6701 . . . . . . . . . 10 (𝑛 = 1 β†’ ((πΉβ€˜π‘›):β„βŸΆ(0[,)+∞) ↔ (πΉβ€˜1):β„βŸΆ(0[,)+∞)))
391ralrimiva 3141 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘› ∈ β„• (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞))
409a1i 11 . . . . . . . . . 10 (πœ‘ β†’ 1 ∈ β„•)
4138, 39, 40rspcdva 3608 . . . . . . . . 9 (πœ‘ β†’ (πΉβ€˜1):β„βŸΆ(0[,)+∞))
4241ffvelcdmda 7088 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜1)β€˜π‘₯) ∈ (0[,)+∞))
43 elrege0 13455 . . . . . . . 8 (((πΉβ€˜1)β€˜π‘₯) ∈ (0[,)+∞) ↔ (((πΉβ€˜1)β€˜π‘₯) ∈ ℝ ∧ 0 ≀ ((πΉβ€˜1)β€˜π‘₯)))
4442, 43sylib 217 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (((πΉβ€˜1)β€˜π‘₯) ∈ ℝ ∧ 0 ≀ ((πΉβ€˜1)β€˜π‘₯)))
4544simpld 494 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜1)β€˜π‘₯) ∈ ℝ)
4644simprd 495 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ 0 ≀ ((πΉβ€˜1)β€˜π‘₯))
4737fveq1d 6893 . . . . . . . . . 10 (𝑛 = 1 β†’ ((πΉβ€˜π‘›)β€˜π‘₯) = ((πΉβ€˜1)β€˜π‘₯))
48 fvex 6904 . . . . . . . . . 10 ((πΉβ€˜1)β€˜π‘₯) ∈ V
4947, 10, 48fvmpt 6999 . . . . . . . . 9 (1 ∈ β„• β†’ ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜1) = ((πΉβ€˜1)β€˜π‘₯))
509, 49ax-mp 5 . . . . . . . 8 ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜1) = ((πΉβ€˜1)β€˜π‘₯)
51 fnfvelrn 7084 . . . . . . . . 9 (((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) Fn β„• ∧ 1 ∈ β„•) β†’ ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜1) ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)))
5218, 9, 51sylancl 585 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜1) ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)))
5350, 52eqeltrrid 2833 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜1)β€˜π‘₯) ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)))
548, 16, 33, 53suprubd 12198 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜1)β€˜π‘₯) ≀ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
5536, 45, 34, 46, 54letrd 11393 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ 0 ≀ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
56 elxrge0 13458 . . . . 5 (sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ (0[,]+∞) ↔ (sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ ℝ* ∧ 0 ≀ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < )))
5735, 55, 56sylanbrc 582 . . . 4 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ (0[,]+∞))
58 itg2mono.1 . . . 4 𝐺 = (π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
5957, 58fmptd 7118 . . 3 (πœ‘ β†’ 𝐺:β„βŸΆ(0[,]+∞))
60 itg2cl 25649 . . 3 (𝐺:β„βŸΆ(0[,]+∞) β†’ (∫2β€˜πΊ) ∈ ℝ*)
6159, 60syl 17 . 2 (πœ‘ β†’ (∫2β€˜πΊ) ∈ ℝ*)
62 itg2mono.6 . . 3 𝑆 = sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < )
63 icossicc 13437 . . . . . . . 8 (0[,)+∞) βŠ† (0[,]+∞)
64 fss 6733 . . . . . . . 8 (((πΉβ€˜π‘›):β„βŸΆ(0[,)+∞) ∧ (0[,)+∞) βŠ† (0[,]+∞)) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,]+∞))
651, 63, 64sylancl 585 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,]+∞))
66 itg2cl 25649 . . . . . . 7 ((πΉβ€˜π‘›):β„βŸΆ(0[,]+∞) β†’ (∫2β€˜(πΉβ€˜π‘›)) ∈ ℝ*)
6765, 66syl 17 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (∫2β€˜(πΉβ€˜π‘›)) ∈ ℝ*)
6867fmpttd 7119 . . . . 5 (πœ‘ β†’ (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))):β„•βŸΆβ„*)
6968frnd 6724 . . . 4 (πœ‘ β†’ ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))) βŠ† ℝ*)
70 supxrcl 13318 . . . 4 (ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))) βŠ† ℝ* β†’ sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < ) ∈ ℝ*)
7169, 70syl 17 . . 3 (πœ‘ β†’ sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < ) ∈ ℝ*)
7262, 71eqeltrid 2832 . 2 (πœ‘ β†’ 𝑆 ∈ ℝ*)
73 itg2mono.2 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ MblFn)
7473adantlr 714 . . . . . . . 8 (((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ MblFn)
751adantlr 714 . . . . . . . 8 (((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞))
76 itg2mono.4 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∘r ≀ (πΉβ€˜(𝑛 + 1)))
7776adantlr 714 . . . . . . . 8 (((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∘r ≀ (πΉβ€˜(𝑛 + 1)))
7817adantlr 714 . . . . . . . 8 (((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) ∧ π‘₯ ∈ ℝ) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦)
79 simprll 778 . . . . . . . 8 ((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) β†’ 𝑓 ∈ dom ∫1)
80 simprlr 779 . . . . . . . 8 ((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) β†’ 𝑓 ∘r ≀ 𝐺)
81 simprr 772 . . . . . . . 8 ((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) β†’ Β¬ (∫1β€˜π‘“) ≀ 𝑆)
8258, 74, 75, 77, 78, 62, 79, 80, 81itg2monolem3 25669 . . . . . . 7 ((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) β†’ (∫1β€˜π‘“) ≀ 𝑆)
8382expr 456 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺)) β†’ (Β¬ (∫1β€˜π‘“) ≀ 𝑆 β†’ (∫1β€˜π‘“) ≀ 𝑆))
8483pm2.18d 127 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺)) β†’ (∫1β€˜π‘“) ≀ 𝑆)
8584expr 456 . . . 4 ((πœ‘ ∧ 𝑓 ∈ dom ∫1) β†’ (𝑓 ∘r ≀ 𝐺 β†’ (∫1β€˜π‘“) ≀ 𝑆))
8685ralrimiva 3141 . . 3 (πœ‘ β†’ βˆ€π‘“ ∈ dom ∫1(𝑓 ∘r ≀ 𝐺 β†’ (∫1β€˜π‘“) ≀ 𝑆))
87 itg2leub 25651 . . . 4 ((𝐺:β„βŸΆ(0[,]+∞) ∧ 𝑆 ∈ ℝ*) β†’ ((∫2β€˜πΊ) ≀ 𝑆 ↔ βˆ€π‘“ ∈ dom ∫1(𝑓 ∘r ≀ 𝐺 β†’ (∫1β€˜π‘“) ≀ 𝑆)))
8859, 72, 87syl2anc 583 . . 3 (πœ‘ β†’ ((∫2β€˜πΊ) ≀ 𝑆 ↔ βˆ€π‘“ ∈ dom ∫1(𝑓 ∘r ≀ 𝐺 β†’ (∫1β€˜π‘“) ≀ 𝑆)))
8986, 88mpbird 257 . 2 (πœ‘ β†’ (∫2β€˜πΊ) ≀ 𝑆)
9022feq1d 6701 . . . . . . . . . . 11 (𝑛 = π‘š β†’ ((πΉβ€˜π‘›):β„βŸΆ(0[,)+∞) ↔ (πΉβ€˜π‘š):β„βŸΆ(0[,)+∞)))
9190cbvralvw 3229 . . . . . . . . . 10 (βˆ€π‘› ∈ β„• (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞) ↔ βˆ€π‘š ∈ β„• (πΉβ€˜π‘š):β„βŸΆ(0[,)+∞))
9239, 91sylib 217 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘š ∈ β„• (πΉβ€˜π‘š):β„βŸΆ(0[,)+∞))
9392r19.21bi 3243 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΉβ€˜π‘š):β„βŸΆ(0[,)+∞))
94 fss 6733 . . . . . . . 8 (((πΉβ€˜π‘š):β„βŸΆ(0[,)+∞) ∧ (0[,)+∞) βŠ† (0[,]+∞)) β†’ (πΉβ€˜π‘š):β„βŸΆ(0[,]+∞))
9593, 63, 94sylancl 585 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΉβ€˜π‘š):β„βŸΆ(0[,]+∞))
9659adantr 480 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝐺:β„βŸΆ(0[,]+∞))
978, 16, 333jca 1126 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) βŠ† ℝ ∧ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) β‰  βˆ… ∧ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))𝑧 ≀ 𝑦))
9897adantlr 714 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ (ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) βŠ† ℝ ∧ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) β‰  βˆ… ∧ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))𝑧 ≀ 𝑦))
9925ad2antlr 726 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) = ((πΉβ€˜π‘š)β€˜π‘₯))
10018adantlr 714 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) Fn β„•)
101 simplr 768 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ π‘š ∈ β„•)
102 fnfvelrn 7084 . . . . . . . . . . . . . 14 (((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) Fn β„• ∧ π‘š ∈ β„•) β†’ ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)))
103100, 101, 102syl2anc 583 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)))
10499, 103eqeltrrd 2829 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)))
105 suprub 12197 . . . . . . . . . . . 12 (((ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) βŠ† ℝ ∧ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) β‰  βˆ… ∧ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))𝑧 ≀ 𝑦) ∧ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
10698, 104, 105syl2anc 583 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
107 simpr 484 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ π‘₯ ∈ ℝ)
108 ltso 11316 . . . . . . . . . . . . 13 < Or ℝ
109108supex 9478 . . . . . . . . . . . 12 sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ V
11058fvmpt2 7010 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ V) β†’ (πΊβ€˜π‘₯) = sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
111107, 109, 110sylancl 585 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ (πΊβ€˜π‘₯) = sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
112106, 111breqtrrd 5170 . . . . . . . . . 10 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ (πΊβ€˜π‘₯))
113112ralrimiva 3141 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ βˆ€π‘₯ ∈ ℝ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ (πΊβ€˜π‘₯))
114 fveq2 6891 . . . . . . . . . . 11 (π‘₯ = 𝑧 β†’ ((πΉβ€˜π‘š)β€˜π‘₯) = ((πΉβ€˜π‘š)β€˜π‘§))
115 fveq2 6891 . . . . . . . . . . 11 (π‘₯ = 𝑧 β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘§))
116114, 115breq12d 5155 . . . . . . . . . 10 (π‘₯ = 𝑧 β†’ (((πΉβ€˜π‘š)β€˜π‘₯) ≀ (πΊβ€˜π‘₯) ↔ ((πΉβ€˜π‘š)β€˜π‘§) ≀ (πΊβ€˜π‘§)))
117116cbvralvw 3229 . . . . . . . . 9 (βˆ€π‘₯ ∈ ℝ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ (πΊβ€˜π‘₯) ↔ βˆ€π‘§ ∈ ℝ ((πΉβ€˜π‘š)β€˜π‘§) ≀ (πΊβ€˜π‘§))
118113, 117sylib 217 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ β„•) β†’ βˆ€π‘§ ∈ ℝ ((πΉβ€˜π‘š)β€˜π‘§) ≀ (πΊβ€˜π‘§))
11993ffnd 6717 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΉβ€˜π‘š) Fn ℝ)
12034, 58fmptd 7118 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
121120ffnd 6717 . . . . . . . . . 10 (πœ‘ β†’ 𝐺 Fn ℝ)
122121adantr 480 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝐺 Fn ℝ)
123 reex 11221 . . . . . . . . . 10 ℝ ∈ V
124123a1i 11 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ℝ ∈ V)
125 inidm 4214 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
126 eqidd 2728 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑧 ∈ ℝ) β†’ ((πΉβ€˜π‘š)β€˜π‘§) = ((πΉβ€˜π‘š)β€˜π‘§))
127 eqidd 2728 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑧 ∈ ℝ) β†’ (πΊβ€˜π‘§) = (πΊβ€˜π‘§))
128119, 122, 124, 124, 125, 126, 127ofrfval 7689 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((πΉβ€˜π‘š) ∘r ≀ 𝐺 ↔ βˆ€π‘§ ∈ ℝ ((πΉβ€˜π‘š)β€˜π‘§) ≀ (πΊβ€˜π‘§)))
129118, 128mpbird 257 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΉβ€˜π‘š) ∘r ≀ 𝐺)
130 itg2le 25656 . . . . . . 7 (((πΉβ€˜π‘š):β„βŸΆ(0[,]+∞) ∧ 𝐺:β„βŸΆ(0[,]+∞) ∧ (πΉβ€˜π‘š) ∘r ≀ 𝐺) β†’ (∫2β€˜(πΉβ€˜π‘š)) ≀ (∫2β€˜πΊ))
13195, 96, 129, 130syl3anc 1369 . . . . . 6 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (∫2β€˜(πΉβ€˜π‘š)) ≀ (∫2β€˜πΊ))
132131ralrimiva 3141 . . . . 5 (πœ‘ β†’ βˆ€π‘š ∈ β„• (∫2β€˜(πΉβ€˜π‘š)) ≀ (∫2β€˜πΊ))
13368ffnd 6717 . . . . . . 7 (πœ‘ β†’ (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))) Fn β„•)
134 breq1 5145 . . . . . . . 8 (𝑧 = ((𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))β€˜π‘š) β†’ (𝑧 ≀ (∫2β€˜πΊ) ↔ ((𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))β€˜π‘š) ≀ (∫2β€˜πΊ)))
135134ralrn 7092 . . . . . . 7 ((𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))) Fn β„• β†’ (βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))𝑧 ≀ (∫2β€˜πΊ) ↔ βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))β€˜π‘š) ≀ (∫2β€˜πΊ)))
136133, 135syl 17 . . . . . 6 (πœ‘ β†’ (βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))𝑧 ≀ (∫2β€˜πΊ) ↔ βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))β€˜π‘š) ≀ (∫2β€˜πΊ)))
137 2fveq3 6896 . . . . . . . . 9 (𝑛 = π‘š β†’ (∫2β€˜(πΉβ€˜π‘›)) = (∫2β€˜(πΉβ€˜π‘š)))
138 eqid 2727 . . . . . . . . 9 (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))) = (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))
139 fvex 6904 . . . . . . . . 9 (∫2β€˜(πΉβ€˜π‘š)) ∈ V
140137, 138, 139fvmpt 6999 . . . . . . . 8 (π‘š ∈ β„• β†’ ((𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))β€˜π‘š) = (∫2β€˜(πΉβ€˜π‘š)))
141140breq1d 5152 . . . . . . 7 (π‘š ∈ β„• β†’ (((𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))β€˜π‘š) ≀ (∫2β€˜πΊ) ↔ (∫2β€˜(πΉβ€˜π‘š)) ≀ (∫2β€˜πΊ)))
142141ralbiia 3086 . . . . . 6 (βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))β€˜π‘š) ≀ (∫2β€˜πΊ) ↔ βˆ€π‘š ∈ β„• (∫2β€˜(πΉβ€˜π‘š)) ≀ (∫2β€˜πΊ))
143136, 142bitrdi 287 . . . . 5 (πœ‘ β†’ (βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))𝑧 ≀ (∫2β€˜πΊ) ↔ βˆ€π‘š ∈ β„• (∫2β€˜(πΉβ€˜π‘š)) ≀ (∫2β€˜πΊ)))
144132, 143mpbird 257 . . . 4 (πœ‘ β†’ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))𝑧 ≀ (∫2β€˜πΊ))
145 supxrleub 13329 . . . . 5 ((ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))) βŠ† ℝ* ∧ (∫2β€˜πΊ) ∈ ℝ*) β†’ (sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < ) ≀ (∫2β€˜πΊ) ↔ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))𝑧 ≀ (∫2β€˜πΊ)))
14669, 61, 145syl2anc 583 . . . 4 (πœ‘ β†’ (sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < ) ≀ (∫2β€˜πΊ) ↔ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))𝑧 ≀ (∫2β€˜πΊ)))
147144, 146mpbird 257 . . 3 (πœ‘ β†’ sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < ) ≀ (∫2β€˜πΊ))
14862, 147eqbrtrid 5177 . 2 (πœ‘ β†’ 𝑆 ≀ (∫2β€˜πΊ))
14961, 72, 89, 148xrletrid 13158 1 (πœ‘ β†’ (∫2β€˜πΊ) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆ€wral 3056  βˆƒwrex 3065  Vcvv 3469   βŠ† wss 3944  βˆ…c0 4318   class class class wbr 5142   ↦ cmpt 5225  dom cdm 5672  ran crn 5673   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ∘r cofr 7678  supcsup 9455  β„cr 11129  0cc0 11130  1c1 11131   + caddc 11133  +∞cpnf 11267  β„*cxr 11269   < clt 11270   ≀ cle 11271  β„•cn 12234  [,)cico 13350  [,]cicc 13351  MblFncmbf 25530  βˆ«1citg1 25531  βˆ«2citg2 25532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cc 10450  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-ofr 7680  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fi 9426  df-sup 9457  df-inf 9458  df-oi 9525  df-dju 9916  df-card 9954  df-acn 9957  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-q 12955  df-rp 12999  df-xneg 13116  df-xadd 13117  df-xmul 13118  df-ioo 13352  df-ioc 13353  df-ico 13354  df-icc 13355  df-fz 13509  df-fzo 13652  df-fl 13781  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-rlim 15457  df-sum 15657  df-rest 17395  df-topgen 17416  df-psmet 21258  df-xmet 21259  df-met 21260  df-bl 21261  df-mopn 21262  df-top 22783  df-topon 22800  df-bases 22836  df-cmp 23278  df-ovol 25380  df-vol 25381  df-mbf 25535  df-itg1 25536  df-itg2 25537
This theorem is referenced by:  itg2i1fseq  25672  itg2cnlem1  25678
  Copyright terms: Public domain W3C validator