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Theorem itg2i1fseq 25143
Description: Subject to the conditions coming from mbfi1fseq 25109, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
Hypotheses
Ref Expression
itg2i1fseq.1 (πœ‘ β†’ 𝐹 ∈ MblFn)
itg2i1fseq.2 (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))
itg2i1fseq.3 (πœ‘ β†’ 𝑃:β„•βŸΆdom ∫1)
itg2i1fseq.4 (πœ‘ β†’ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ∧ (π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1))))
itg2i1fseq.5 (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯))
itg2i1fseq.6 𝑆 = (π‘š ∈ β„• ↦ (∫1β€˜(π‘ƒβ€˜π‘š)))
Assertion
Ref Expression
itg2i1fseq (πœ‘ β†’ (∫2β€˜πΉ) = sup(ran 𝑆, ℝ*, < ))
Distinct variable groups:   π‘š,𝑛,π‘₯,𝐹   𝑃,π‘š,𝑛,π‘₯   πœ‘,π‘š
Allowed substitution hints:   πœ‘(π‘₯,𝑛)   𝑆(π‘₯,π‘š,𝑛)

Proof of Theorem itg2i1fseq
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6846 . . . . . . . . 9 (𝑛 = π‘š β†’ (π‘ƒβ€˜π‘›) = (π‘ƒβ€˜π‘š))
21fveq1d 6848 . . . . . . . 8 (𝑛 = π‘š β†’ ((π‘ƒβ€˜π‘›)β€˜π‘₯) = ((π‘ƒβ€˜π‘š)β€˜π‘₯))
32cbvmptv 5222 . . . . . . 7 (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) = (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘₯))
4 fveq2 6846 . . . . . . . 8 (π‘₯ = 𝑦 β†’ ((π‘ƒβ€˜π‘š)β€˜π‘₯) = ((π‘ƒβ€˜π‘š)β€˜π‘¦))
54mpteq2dv 5211 . . . . . . 7 (π‘₯ = 𝑦 β†’ (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘₯)) = (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)))
63, 5eqtrid 2785 . . . . . 6 (π‘₯ = 𝑦 β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) = (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)))
76rneqd 5897 . . . . 5 (π‘₯ = 𝑦 β†’ ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) = ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)))
87supeq1d 9390 . . . 4 (π‘₯ = 𝑦 β†’ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)), ℝ, < ) = sup(ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)), ℝ, < ))
98cbvmptv 5222 . . 3 (π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)), ℝ, < )) = (𝑦 ∈ ℝ ↦ sup(ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)), ℝ, < ))
10 itg2i1fseq.3 . . . . 5 (πœ‘ β†’ 𝑃:β„•βŸΆdom ∫1)
1110ffvelcdmda 7039 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š) ∈ dom ∫1)
12 i1fmbf 25062 . . . 4 ((π‘ƒβ€˜π‘š) ∈ dom ∫1 β†’ (π‘ƒβ€˜π‘š) ∈ MblFn)
1311, 12syl 17 . . 3 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š) ∈ MblFn)
14 i1ff 25063 . . . . 5 ((π‘ƒβ€˜π‘š) ∈ dom ∫1 β†’ (π‘ƒβ€˜π‘š):β„βŸΆβ„)
1511, 14syl 17 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š):β„βŸΆβ„)
16 itg2i1fseq.4 . . . . . 6 (πœ‘ β†’ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ∧ (π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1))))
171breq2d 5121 . . . . . . . 8 (𝑛 = π‘š β†’ (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ↔ 0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š)))
18 fvoveq1 7384 . . . . . . . . 9 (𝑛 = π‘š β†’ (π‘ƒβ€˜(𝑛 + 1)) = (π‘ƒβ€˜(π‘š + 1)))
191, 18breq12d 5122 . . . . . . . 8 (𝑛 = π‘š β†’ ((π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1)) ↔ (π‘ƒβ€˜π‘š) ∘r ≀ (π‘ƒβ€˜(π‘š + 1))))
2017, 19anbi12d 632 . . . . . . 7 (𝑛 = π‘š β†’ ((0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ∧ (π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1))) ↔ (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š) ∧ (π‘ƒβ€˜π‘š) ∘r ≀ (π‘ƒβ€˜(π‘š + 1)))))
2120rspccva 3582 . . . . . 6 ((βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ∧ (π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1))) ∧ π‘š ∈ β„•) β†’ (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š) ∧ (π‘ƒβ€˜π‘š) ∘r ≀ (π‘ƒβ€˜(π‘š + 1))))
2216, 21sylan 581 . . . . 5 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š) ∧ (π‘ƒβ€˜π‘š) ∘r ≀ (π‘ƒβ€˜(π‘š + 1))))
2322simpld 496 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š))
24 0plef 25059 . . . 4 ((π‘ƒβ€˜π‘š):β„βŸΆ(0[,)+∞) ↔ ((π‘ƒβ€˜π‘š):β„βŸΆβ„ ∧ 0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š)))
2515, 23, 24sylanbrc 584 . . 3 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š):β„βŸΆ(0[,)+∞))
2622simprd 497 . . 3 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š) ∘r ≀ (π‘ƒβ€˜(π‘š + 1)))
27 rge0ssre 13382 . . . . 5 (0[,)+∞) βŠ† ℝ
28 itg2i1fseq.2 . . . . . 6 (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))
2928ffvelcdmda 7039 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (πΉβ€˜π‘¦) ∈ (0[,)+∞))
3027, 29sselid 3946 . . . 4 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
31 itg2i1fseq.1 . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ MblFn)
32 itg2i1fseq.5 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯))
3331, 28, 10, 16, 32itg2i1fseqle 25142 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š) ∘r ≀ 𝐹)
3415ffnd 6673 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š) Fn ℝ)
3528ffnd 6673 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 Fn ℝ)
3635adantr 482 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝐹 Fn ℝ)
37 reex 11150 . . . . . . . . . 10 ℝ ∈ V
3837a1i 11 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ℝ ∈ V)
39 inidm 4182 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
40 eqidd 2734 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) = ((π‘ƒβ€˜π‘š)β€˜π‘¦))
41 eqidd 2734 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘¦))
4234, 36, 38, 38, 39, 40, 41ofrfval 7631 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((π‘ƒβ€˜π‘š) ∘r ≀ 𝐹 ↔ βˆ€π‘¦ ∈ ℝ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ (πΉβ€˜π‘¦)))
4333, 42mpbid 231 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ β„•) β†’ βˆ€π‘¦ ∈ ℝ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ (πΉβ€˜π‘¦))
4443r19.21bi 3233 . . . . . 6 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ (πΉβ€˜π‘¦))
4544an32s 651 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ ℝ) ∧ π‘š ∈ β„•) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ (πΉβ€˜π‘¦))
4645ralrimiva 3140 . . . 4 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ βˆ€π‘š ∈ β„• ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ (πΉβ€˜π‘¦))
47 brralrspcev 5169 . . . 4 (((πΉβ€˜π‘¦) ∈ ℝ ∧ βˆ€π‘š ∈ β„• ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ (πΉβ€˜π‘¦)) β†’ βˆƒπ‘§ ∈ ℝ βˆ€π‘š ∈ β„• ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ 𝑧)
4830, 46, 47syl2anc 585 . . 3 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ βˆƒπ‘§ ∈ ℝ βˆ€π‘š ∈ β„• ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ 𝑧)
491fveq2d 6850 . . . . . 6 (𝑛 = π‘š β†’ (∫2β€˜(π‘ƒβ€˜π‘›)) = (∫2β€˜(π‘ƒβ€˜π‘š)))
5049cbvmptv 5222 . . . . 5 (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))) = (π‘š ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘š)))
5150rneqi 5896 . . . 4 ran (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))) = ran (π‘š ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘š)))
5251supeq1i 9391 . . 3 sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))), ℝ*, < ) = sup(ran (π‘š ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘š))), ℝ*, < )
539, 13, 25, 26, 48, 52itg2mono 25141 . 2 (πœ‘ β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)), ℝ, < ))) = sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))), ℝ*, < ))
5428feqmptd 6914 . . . . 5 (πœ‘ β†’ 𝐹 = (𝑦 ∈ ℝ ↦ (πΉβ€˜π‘¦)))
551fveq1d 6848 . . . . . . . . . 10 (𝑛 = π‘š β†’ ((π‘ƒβ€˜π‘›)β€˜π‘¦) = ((π‘ƒβ€˜π‘š)β€˜π‘¦))
5655cbvmptv 5222 . . . . . . . . 9 (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) = (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦))
5756rneqi 5896 . . . . . . . 8 ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) = ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦))
5857supeq1i 9391 . . . . . . 7 sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)), ℝ, < ) = sup(ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)), ℝ, < )
59 nnuz 12814 . . . . . . . . 9 β„• = (β„€β‰₯β€˜1)
60 1zzd 12542 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ 1 ∈ β„€)
6115ffvelcdmda 7039 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ∈ ℝ)
6261an32s 651 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ ℝ) ∧ π‘š ∈ β„•) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ∈ ℝ)
6362, 56fmptd 7066 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)):β„•βŸΆβ„)
64 peano2nn 12173 . . . . . . . . . . . . . . . . 17 (π‘š ∈ β„• β†’ (π‘š + 1) ∈ β„•)
65 ffvelcdm 7036 . . . . . . . . . . . . . . . . 17 ((𝑃:β„•βŸΆdom ∫1 ∧ (π‘š + 1) ∈ β„•) β†’ (π‘ƒβ€˜(π‘š + 1)) ∈ dom ∫1)
6610, 64, 65syl2an 597 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜(π‘š + 1)) ∈ dom ∫1)
67 i1ff 25063 . . . . . . . . . . . . . . . 16 ((π‘ƒβ€˜(π‘š + 1)) ∈ dom ∫1 β†’ (π‘ƒβ€˜(π‘š + 1)):β„βŸΆβ„)
6866, 67syl 17 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜(π‘š + 1)):β„βŸΆβ„)
6968ffnd 6673 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜(π‘š + 1)) Fn ℝ)
70 eqidd 2734 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦) = ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
7134, 69, 38, 38, 39, 40, 70ofrfval 7631 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((π‘ƒβ€˜π‘š) ∘r ≀ (π‘ƒβ€˜(π‘š + 1)) ↔ βˆ€π‘¦ ∈ ℝ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦)))
7226, 71mpbid 231 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ β„•) β†’ βˆ€π‘¦ ∈ ℝ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
7372r19.21bi 3233 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
7473an32s 651 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ ℝ) ∧ π‘š ∈ β„•) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
75 eqid 2733 . . . . . . . . . . . 12 (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) = (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))
76 fvex 6859 . . . . . . . . . . . 12 ((π‘ƒβ€˜π‘š)β€˜π‘¦) ∈ V
7755, 75, 76fvmpt 6952 . . . . . . . . . . 11 (π‘š ∈ β„• β†’ ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜π‘š) = ((π‘ƒβ€˜π‘š)β€˜π‘¦))
7877adantl 483 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ ℝ) ∧ π‘š ∈ β„•) β†’ ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜π‘š) = ((π‘ƒβ€˜π‘š)β€˜π‘¦))
79 fveq2 6846 . . . . . . . . . . . . . 14 (𝑛 = (π‘š + 1) β†’ (π‘ƒβ€˜π‘›) = (π‘ƒβ€˜(π‘š + 1)))
8079fveq1d 6848 . . . . . . . . . . . . 13 (𝑛 = (π‘š + 1) β†’ ((π‘ƒβ€˜π‘›)β€˜π‘¦) = ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
81 fvex 6859 . . . . . . . . . . . . 13 ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦) ∈ V
8280, 75, 81fvmpt 6952 . . . . . . . . . . . 12 ((π‘š + 1) ∈ β„• β†’ ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜(π‘š + 1)) = ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
8364, 82syl 17 . . . . . . . . . . 11 (π‘š ∈ β„• β†’ ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜(π‘š + 1)) = ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
8483adantl 483 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ ℝ) ∧ π‘š ∈ β„•) β†’ ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜(π‘š + 1)) = ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
8574, 78, 843brtr4d 5141 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ ℝ) ∧ π‘š ∈ β„•) β†’ ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜π‘š) ≀ ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜(π‘š + 1)))
8677breq1d 5119 . . . . . . . . . . . 12 (π‘š ∈ β„• β†’ (((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜π‘š) ≀ 𝑧 ↔ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ 𝑧))
8786ralbiia 3091 . . . . . . . . . . 11 (βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜π‘š) ≀ 𝑧 ↔ βˆ€π‘š ∈ β„• ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ 𝑧)
8887rexbii 3094 . . . . . . . . . 10 (βˆƒπ‘§ ∈ ℝ βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜π‘š) ≀ 𝑧 ↔ βˆƒπ‘§ ∈ ℝ βˆ€π‘š ∈ β„• ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ 𝑧)
8948, 88sylibr 233 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ βˆƒπ‘§ ∈ ℝ βˆ€π‘š ∈ β„• ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜π‘š) ≀ 𝑧)
9059, 60, 63, 85, 89climsup 15563 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) ⇝ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)), ℝ, < ))
91 fveq2 6846 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ ((π‘ƒβ€˜π‘›)β€˜π‘₯) = ((π‘ƒβ€˜π‘›)β€˜π‘¦))
9291mpteq2dv 5211 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) = (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)))
93 fveq2 6846 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
9492, 93breq12d 5122 . . . . . . . . . 10 (π‘₯ = 𝑦 β†’ ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯) ↔ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) ⇝ (πΉβ€˜π‘¦)))
9594rspccva 3582 . . . . . . . . 9 ((βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯) ∧ 𝑦 ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) ⇝ (πΉβ€˜π‘¦))
9632, 95sylan 581 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) ⇝ (πΉβ€˜π‘¦))
97 climuni 15443 . . . . . . . 8 (((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) ⇝ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)), ℝ, < ) ∧ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) ⇝ (πΉβ€˜π‘¦)) β†’ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)), ℝ, < ) = (πΉβ€˜π‘¦))
9890, 96, 97syl2anc 585 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)), ℝ, < ) = (πΉβ€˜π‘¦))
9958, 98eqtr3id 2787 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ sup(ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)), ℝ, < ) = (πΉβ€˜π‘¦))
10099mpteq2dva 5209 . . . . 5 (πœ‘ β†’ (𝑦 ∈ ℝ ↦ sup(ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)), ℝ, < )) = (𝑦 ∈ ℝ ↦ (πΉβ€˜π‘¦)))
10154, 100eqtr4d 2776 . . . 4 (πœ‘ β†’ 𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)), ℝ, < )))
102101, 9eqtr4di 2791 . . 3 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)), ℝ, < )))
103102fveq2d 6850 . 2 (πœ‘ β†’ (∫2β€˜πΉ) = (∫2β€˜(π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)), ℝ, < ))))
104 itg2i1fseq.6 . . . . . 6 𝑆 = (π‘š ∈ β„• ↦ (∫1β€˜(π‘ƒβ€˜π‘š)))
105 itg2itg1 25124 . . . . . . . 8 (((π‘ƒβ€˜π‘š) ∈ dom ∫1 ∧ 0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š)) β†’ (∫2β€˜(π‘ƒβ€˜π‘š)) = (∫1β€˜(π‘ƒβ€˜π‘š)))
10611, 23, 105syl2anc 585 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (∫2β€˜(π‘ƒβ€˜π‘š)) = (∫1β€˜(π‘ƒβ€˜π‘š)))
107106mpteq2dva 5209 . . . . . 6 (πœ‘ β†’ (π‘š ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘š))) = (π‘š ∈ β„• ↦ (∫1β€˜(π‘ƒβ€˜π‘š))))
108104, 107eqtr4id 2792 . . . . 5 (πœ‘ β†’ 𝑆 = (π‘š ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘š))))
109108, 50eqtr4di 2791 . . . 4 (πœ‘ β†’ 𝑆 = (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))))
110109rneqd 5897 . . 3 (πœ‘ β†’ ran 𝑆 = ran (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))))
111110supeq1d 9390 . 2 (πœ‘ β†’ sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))), ℝ*, < ))
11253, 103, 1113eqtr4d 2783 1 (πœ‘ β†’ (∫2β€˜πΉ) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5637  ran crn 5638   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∘r cofr 7620  supcsup 9384  β„cr 11058  0cc0 11059  1c1 11060   + caddc 11062  +∞cpnf 11194  β„*cxr 11196   < clt 11197   ≀ cle 11198  β„•cn 12161  [,)cico 13275   ⇝ cli 15375  MblFncmbf 25001  βˆ«1citg1 25002  βˆ«2citg2 25003  0𝑝c0p 25056
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-cc 10379  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-addf 11138
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-disj 5075  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-ofr 7622  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-oadd 8420  df-omul 8421  df-er 8654  df-map 8773  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fi 9355  df-sup 9386  df-inf 9387  df-oi 9454  df-dju 9845  df-card 9883  df-acn 9886  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-q 12882  df-rp 12924  df-xneg 13041  df-xadd 13042  df-xmul 13043  df-ioo 13277  df-ioc 13278  df-ico 13279  df-icc 13280  df-fz 13434  df-fzo 13577  df-fl 13706  df-seq 13916  df-exp 13977  df-hash 14240  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-clim 15379  df-rlim 15380  df-sum 15580  df-rest 17312  df-topgen 17333  df-psmet 20811  df-xmet 20812  df-met 20813  df-bl 20814  df-mopn 20815  df-top 22266  df-topon 22283  df-bases 22319  df-cmp 22761  df-ovol 24851  df-vol 24852  df-mbf 25006  df-itg1 25007  df-itg2 25008  df-0p 25057
This theorem is referenced by:  itg2i1fseq2  25144
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