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Theorem itg2mono 25503
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 13437 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆβ„)
54ffvelcdmda 7085 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜π‘›)β€˜π‘₯) ∈ ℝ)
65an32s 648 . . . . . . . . 9 (((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›)β€˜π‘₯) ∈ ℝ)
76fmpttd 7115 . . . . . . . 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 12227 . . . . . . . . . 10 1 ∈ β„•
10 eqid 2730 . . . . . . . . . . 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 2838 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ 1 ∈ dom (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)))
1312ne0d 4334 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ dom (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) β‰  βˆ…)
14 dm0rn0 5923 . . . . . . . . 9 (dom (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) = βˆ… ↔ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) = βˆ…)
1514necon3bii 2991 . . . . . . . 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 5150 . . . . . . . . . . . 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 7088 . . . . . . . . . . 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 6890 . . . . . . . . . . . . . . 15 (𝑛 = π‘š β†’ (πΉβ€˜π‘›) = (πΉβ€˜π‘š))
2322fveq1d 6892 . . . . . . . . . . . . . 14 (𝑛 = π‘š β†’ ((πΉβ€˜π‘›)β€˜π‘₯) = ((πΉβ€˜π‘š)β€˜π‘₯))
24 fvex 6903 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘š)β€˜π‘₯) ∈ V
2523, 10, 24fvmpt 6997 . . . . . . . . . . . . 13 (π‘š ∈ β„• β†’ ((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) = ((πΉβ€˜π‘š)β€˜π‘₯))
2625breq1d 5157 . . . . . . . . . . . 12 (π‘š ∈ β„• β†’ (((𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))β€˜π‘š) ≀ 𝑦 ↔ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ 𝑦))
2726ralbiia 3089 . . . . . . . . . . 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 5157 . . . . . . . . . . . 12 (𝑛 = π‘š β†’ (((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦 ↔ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ 𝑦))
2928cbvralvw 3232 . . . . . . . . . . 11 (βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦 ↔ βˆ€π‘š ∈ β„• ((πΉβ€˜π‘š)β€˜π‘₯) ≀ 𝑦)
3027, 29bitr4i 277 . . . . . . . . . 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 286 . . . . . . . . 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 3176 . . . . . . . 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 256 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯))𝑧 ≀ 𝑦)
348, 16, 33suprcld 12181 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ ℝ)
3534rexrd 11268 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ ℝ*)
36 0red 11221 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ 0 ∈ ℝ)
37 fveq2 6890 . . . . . . . . . . 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 3144 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘› ∈ β„• (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞))
409a1i 11 . . . . . . . . . 10 (πœ‘ β†’ 1 ∈ β„•)
4138, 39, 40rspcdva 3612 . . . . . . . . 9 (πœ‘ β†’ (πΉβ€˜1):β„βŸΆ(0[,)+∞))
4241ffvelcdmda 7085 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜1)β€˜π‘₯) ∈ (0[,)+∞))
43 elrege0 13435 . . . . . . . 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 493 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜1)β€˜π‘₯) ∈ ℝ)
4644simprd 494 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ 0 ≀ ((πΉβ€˜1)β€˜π‘₯))
4737fveq1d 6892 . . . . . . . . . 10 (𝑛 = 1 β†’ ((πΉβ€˜π‘›)β€˜π‘₯) = ((πΉβ€˜1)β€˜π‘₯))
48 fvex 6903 . . . . . . . . . 10 ((πΉβ€˜1)β€˜π‘₯) ∈ V
4947, 10, 48fvmpt 6997 . . . . . . . . 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 7081 . . . . . . . . 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 584 . . . . . . . 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 2836 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜1)β€˜π‘₯) ∈ ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)))
548, 16, 33, 53suprubd 12180 . . . . . 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 11375 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ 0 ≀ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
56 elxrge0 13438 . . . . 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 581 . . . 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 7114 . . 3 (πœ‘ β†’ 𝐺:β„βŸΆ(0[,]+∞))
60 itg2cl 25482 . . 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 13417 . . . . . . . 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 584 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,]+∞))
66 itg2cl 25482 . . . . . . 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 7115 . . . . 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 13298 . . . 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 2835 . 2 (πœ‘ β†’ 𝑆 ∈ ℝ*)
73 itg2mono.2 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ MblFn)
7473adantlr 711 . . . . . . . 8 (((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ MblFn)
751adantlr 711 . . . . . . . 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 711 . . . . . . . 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 711 . . . . . . . 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 775 . . . . . . . 8 ((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) β†’ 𝑓 ∈ dom ∫1)
80 simprlr 776 . . . . . . . 8 ((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) β†’ 𝑓 ∘r ≀ 𝐺)
81 simprr 769 . . . . . . . 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 25502 . . . . . . 7 ((πœ‘ ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≀ 𝐺) ∧ Β¬ (∫1β€˜π‘“) ≀ 𝑆)) β†’ (∫1β€˜π‘“) ≀ 𝑆)
8382expr 455 . . . . . 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 455 . . . 4 ((πœ‘ ∧ 𝑓 ∈ dom ∫1) β†’ (𝑓 ∘r ≀ 𝐺 β†’ (∫1β€˜π‘“) ≀ 𝑆))
8685ralrimiva 3144 . . 3 (πœ‘ β†’ βˆ€π‘“ ∈ dom ∫1(𝑓 ∘r ≀ 𝐺 β†’ (∫1β€˜π‘“) ≀ 𝑆))
87 itg2leub 25484 . . . 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 582 . . 3 (πœ‘ β†’ ((∫2β€˜πΊ) ≀ 𝑆 ↔ βˆ€π‘“ ∈ dom ∫1(𝑓 ∘r ≀ 𝐺 β†’ (∫1β€˜π‘“) ≀ 𝑆)))
8986, 88mpbird 256 . 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 3232 . . . . . . . . . 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 3246 . . . . . . . 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 584 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΉβ€˜π‘š):β„βŸΆ(0[,]+∞))
9659adantr 479 . . . . . . 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 711 . . . . . . . . . . . 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 723 . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)) Fn β„•)
101 simplr 765 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ π‘š ∈ β„•)
102 fnfvelrn 7081 . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . 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 2832 . . . . . . . . . . . 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 12179 . . . . . . . . . . . 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 582 . . . . . . . . . . 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 483 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ π‘₯ ∈ ℝ)
108 ltso 11298 . . . . . . . . . . . . 13 < Or ℝ
109108supex 9460 . . . . . . . . . . . 12 sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ) ∈ V
11058fvmpt2 7008 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ (πΊβ€˜π‘₯) = sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))
112106, 111breqtrrd 5175 . . . . . . . . . 10 (((πœ‘ ∧ π‘š ∈ β„•) ∧ π‘₯ ∈ ℝ) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ (πΊβ€˜π‘₯))
113112ralrimiva 3144 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ βˆ€π‘₯ ∈ ℝ ((πΉβ€˜π‘š)β€˜π‘₯) ≀ (πΊβ€˜π‘₯))
114 fveq2 6890 . . . . . . . . . . 11 (π‘₯ = 𝑧 β†’ ((πΉβ€˜π‘š)β€˜π‘₯) = ((πΉβ€˜π‘š)β€˜π‘§))
115 fveq2 6890 . . . . . . . . . . 11 (π‘₯ = 𝑧 β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘§))
116114, 115breq12d 5160 . . . . . . . . . 10 (π‘₯ = 𝑧 β†’ (((πΉβ€˜π‘š)β€˜π‘₯) ≀ (πΊβ€˜π‘₯) ↔ ((πΉβ€˜π‘š)β€˜π‘§) ≀ (πΊβ€˜π‘§)))
117116cbvralvw 3232 . . . . . . . . 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 7114 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
121120ffnd 6717 . . . . . . . . . 10 (πœ‘ β†’ 𝐺 Fn ℝ)
122121adantr 479 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝐺 Fn ℝ)
123 reex 11203 . . . . . . . . . 10 ℝ ∈ V
124123a1i 11 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ℝ ∈ V)
125 inidm 4217 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
126 eqidd 2731 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑧 ∈ ℝ) β†’ ((πΉβ€˜π‘š)β€˜π‘§) = ((πΉβ€˜π‘š)β€˜π‘§))
127 eqidd 2731 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑧 ∈ ℝ) β†’ (πΊβ€˜π‘§) = (πΊβ€˜π‘§))
128119, 122, 124, 124, 125, 126, 127ofrfval 7682 . . . . . . . 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 256 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΉβ€˜π‘š) ∘r ≀ 𝐺)
130 itg2le 25489 . . . . . . 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 3144 . . . . 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 5150 . . . . . . . 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 7088 . . . . . . 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 6895 . . . . . . . . 9 (𝑛 = π‘š β†’ (∫2β€˜(πΉβ€˜π‘›)) = (∫2β€˜(πΉβ€˜π‘š)))
138 eqid 2730 . . . . . . . . 9 (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))) = (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))
139 fvex 6903 . . . . . . . . 9 (∫2β€˜(πΉβ€˜π‘š)) ∈ V
140137, 138, 139fvmpt 6997 . . . . . . . 8 (π‘š ∈ β„• β†’ ((𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))β€˜π‘š) = (∫2β€˜(πΉβ€˜π‘š)))
141140breq1d 5157 . . . . . . 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 3089 . . . . . 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 286 . . . . 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 256 . . . 4 (πœ‘ β†’ βˆ€π‘§ ∈ ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›)))𝑧 ≀ (∫2β€˜πΊ))
145 supxrleub 13309 . . . . 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 582 . . . 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 256 . . 3 (πœ‘ β†’ sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < ) ≀ (∫2β€˜πΊ))
14862, 147eqbrtrid 5182 . 2 (πœ‘ β†’ 𝑆 ≀ (∫2β€˜πΊ))
14961, 72, 89, 148xrletrid 13138 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675  ran crn 5676   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∘r cofr 7671  supcsup 9437  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115  +∞cpnf 11249  β„*cxr 11251   < clt 11252   ≀ cle 11253  β„•cn 12216  [,)cico 13330  [,]cicc 13331  MblFncmbf 25363  βˆ«1citg1 25364  βˆ«2citg2 25365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cc 10432  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-ofr 7673  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-omul 8473  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-acn 9939  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-ioc 13333  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-fl 13761  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-rlim 15437  df-sum 15637  df-rest 17372  df-topgen 17393  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-top 22616  df-topon 22633  df-bases 22669  df-cmp 23111  df-ovol 25213  df-vol 25214  df-mbf 25368  df-itg1 25369  df-itg2 25370
This theorem is referenced by:  itg2i1fseq  25505  itg2cnlem1  25511
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