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Theorem itg2i1fseq 25280
Description: Subject to the conditions coming from mbfi1fseq 25246, 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 6891 . . . . . . . . 9 (𝑛 = π‘š β†’ (π‘ƒβ€˜π‘›) = (π‘ƒβ€˜π‘š))
21fveq1d 6893 . . . . . . . 8 (𝑛 = π‘š β†’ ((π‘ƒβ€˜π‘›)β€˜π‘₯) = ((π‘ƒβ€˜π‘š)β€˜π‘₯))
32cbvmptv 5261 . . . . . . 7 (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) = (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘₯))
4 fveq2 6891 . . . . . . . 8 (π‘₯ = 𝑦 β†’ ((π‘ƒβ€˜π‘š)β€˜π‘₯) = ((π‘ƒβ€˜π‘š)β€˜π‘¦))
54mpteq2dv 5250 . . . . . . 7 (π‘₯ = 𝑦 β†’ (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘₯)) = (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)))
63, 5eqtrid 2784 . . . . . 6 (π‘₯ = 𝑦 β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) = (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)))
76rneqd 5937 . . . . 5 (π‘₯ = 𝑦 β†’ ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) = ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)))
87supeq1d 9443 . . . 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 5261 . . 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 7086 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š) ∈ dom ∫1)
12 i1fmbf 25199 . . . 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 25200 . . . . 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 5160 . . . . . . . 8 (𝑛 = π‘š β†’ (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ↔ 0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š)))
18 fvoveq1 7434 . . . . . . . . 9 (𝑛 = π‘š β†’ (π‘ƒβ€˜(𝑛 + 1)) = (π‘ƒβ€˜(π‘š + 1)))
191, 18breq12d 5161 . . . . . . . 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 631 . . . . . . 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 3611 . . . . . 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 580 . . . . 5 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š) ∧ (π‘ƒβ€˜π‘š) ∘r ≀ (π‘ƒβ€˜(π‘š + 1))))
2322simpld 495 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š))
24 0plef 25196 . . . 4 ((π‘ƒβ€˜π‘š):β„βŸΆ(0[,)+∞) ↔ ((π‘ƒβ€˜π‘š):β„βŸΆβ„ ∧ 0𝑝 ∘r ≀ (π‘ƒβ€˜π‘š)))
2515, 23, 24sylanbrc 583 . . 3 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š):β„βŸΆ(0[,)+∞))
2622simprd 496 . . 3 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š) ∘r ≀ (π‘ƒβ€˜(π‘š + 1)))
27 rge0ssre 13435 . . . . 5 (0[,)+∞) βŠ† ℝ
28 itg2i1fseq.2 . . . . . 6 (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))
2928ffvelcdmda 7086 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (πΉβ€˜π‘¦) ∈ (0[,)+∞))
3027, 29sselid 3980 . . . 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 25279 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š) ∘r ≀ 𝐹)
3415ffnd 6718 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜π‘š) Fn ℝ)
3528ffnd 6718 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 Fn ℝ)
3635adantr 481 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 𝐹 Fn ℝ)
37 reex 11203 . . . . . . . . . 10 ℝ ∈ V
3837a1i 11 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ℝ ∈ V)
39 inidm 4218 . . . . . . . . 9 (ℝ ∩ ℝ) = ℝ
40 eqidd 2733 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) = ((π‘ƒβ€˜π‘š)β€˜π‘¦))
41 eqidd 2733 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘¦))
4234, 36, 38, 38, 39, 40, 41ofrfval 7682 . . . . . . . 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 3248 . . . . . 6 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ (πΉβ€˜π‘¦))
4544an32s 650 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ ℝ) ∧ π‘š ∈ β„•) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ (πΉβ€˜π‘¦))
4645ralrimiva 3146 . . . 4 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ βˆ€π‘š ∈ β„• ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ (πΉβ€˜π‘¦))
47 brralrspcev 5208 . . . 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 584 . . 3 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ βˆƒπ‘§ ∈ ℝ βˆ€π‘š ∈ β„• ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ 𝑧)
491fveq2d 6895 . . . . . 6 (𝑛 = π‘š β†’ (∫2β€˜(π‘ƒβ€˜π‘›)) = (∫2β€˜(π‘ƒβ€˜π‘š)))
5049cbvmptv 5261 . . . . 5 (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))) = (π‘š ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘š)))
5150rneqi 5936 . . . 4 ran (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))) = ran (π‘š ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘š)))
5251supeq1i 9444 . . 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 25278 . 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 6960 . . . . 5 (πœ‘ β†’ 𝐹 = (𝑦 ∈ ℝ ↦ (πΉβ€˜π‘¦)))
551fveq1d 6893 . . . . . . . . . 10 (𝑛 = π‘š β†’ ((π‘ƒβ€˜π‘›)β€˜π‘¦) = ((π‘ƒβ€˜π‘š)β€˜π‘¦))
5655cbvmptv 5261 . . . . . . . . 9 (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) = (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦))
5756rneqi 5936 . . . . . . . 8 ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) = ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦))
5857supeq1i 9444 . . . . . . 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 12867 . . . . . . . . 9 β„• = (β„€β‰₯β€˜1)
60 1zzd 12595 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ 1 ∈ β„€)
6115ffvelcdmda 7086 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ∈ ℝ)
6261an32s 650 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ ℝ) ∧ π‘š ∈ β„•) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ∈ ℝ)
6362, 56fmptd 7115 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)):β„•βŸΆβ„)
64 peano2nn 12226 . . . . . . . . . . . . . . . . 17 (π‘š ∈ β„• β†’ (π‘š + 1) ∈ β„•)
65 ffvelcdm 7083 . . . . . . . . . . . . . . . . 17 ((𝑃:β„•βŸΆdom ∫1 ∧ (π‘š + 1) ∈ β„•) β†’ (π‘ƒβ€˜(π‘š + 1)) ∈ dom ∫1)
6610, 64, 65syl2an 596 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜(π‘š + 1)) ∈ dom ∫1)
67 i1ff 25200 . . . . . . . . . . . . . . . 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 6718 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘ƒβ€˜(π‘š + 1)) Fn ℝ)
70 eqidd 2733 . . . . . . . . . . . . . 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 7682 . . . . . . . . . . . . 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 3248 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ β„•) ∧ 𝑦 ∈ ℝ) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
7473an32s 650 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ ℝ) ∧ π‘š ∈ β„•) β†’ ((π‘ƒβ€˜π‘š)β€˜π‘¦) ≀ ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
75 eqid 2732 . . . . . . . . . . . 12 (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) = (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))
76 fvex 6904 . . . . . . . . . . . 12 ((π‘ƒβ€˜π‘š)β€˜π‘¦) ∈ V
7755, 75, 76fvmpt 6998 . . . . . . . . . . 11 (π‘š ∈ β„• β†’ ((𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦))β€˜π‘š) = ((π‘ƒβ€˜π‘š)β€˜π‘¦))
7877adantl 482 . . . . . . . . . 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 6891 . . . . . . . . . . . . . 14 (𝑛 = (π‘š + 1) β†’ (π‘ƒβ€˜π‘›) = (π‘ƒβ€˜(π‘š + 1)))
8079fveq1d 6893 . . . . . . . . . . . . 13 (𝑛 = (π‘š + 1) β†’ ((π‘ƒβ€˜π‘›)β€˜π‘¦) = ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦))
81 fvex 6904 . . . . . . . . . . . . 13 ((π‘ƒβ€˜(π‘š + 1))β€˜π‘¦) ∈ V
8280, 75, 81fvmpt 6998 . . . . . . . . . . . 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 482 . . . . . . . . . 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 5180 . . . . . . . . 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 5158 . . . . . . . . . . . 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 15618 . . . . . . . 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 6891 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ ((π‘ƒβ€˜π‘›)β€˜π‘₯) = ((π‘ƒβ€˜π‘›)β€˜π‘¦))
9291mpteq2dv 5250 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) = (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)))
93 fveq2 6891 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
9492, 93breq12d 5161 . . . . . . . . . 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 3611 . . . . . . . . 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 580 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)) ⇝ (πΉβ€˜π‘¦))
97 climuni 15498 . . . . . . . 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 584 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘¦)), ℝ, < ) = (πΉβ€˜π‘¦))
9958, 98eqtr3id 2786 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ sup(ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)), ℝ, < ) = (πΉβ€˜π‘¦))
10099mpteq2dva 5248 . . . . 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 2775 . . . 4 (πœ‘ β†’ 𝐹 = (𝑦 ∈ ℝ ↦ sup(ran (π‘š ∈ β„• ↦ ((π‘ƒβ€˜π‘š)β€˜π‘¦)), ℝ, < )))
102101, 9eqtr4di 2790 . . 3 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)), ℝ, < )))
103102fveq2d 6895 . 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 25261 . . . . . . . 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 584 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (∫2β€˜(π‘ƒβ€˜π‘š)) = (∫1β€˜(π‘ƒβ€˜π‘š)))
107106mpteq2dva 5248 . . . . . 6 (πœ‘ β†’ (π‘š ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘š))) = (π‘š ∈ β„• ↦ (∫1β€˜(π‘ƒβ€˜π‘š))))
108104, 107eqtr4id 2791 . . . . 5 (πœ‘ β†’ 𝑆 = (π‘š ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘š))))
109108, 50eqtr4di 2790 . . . 4 (πœ‘ β†’ 𝑆 = (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))))
110109rneqd 5937 . . 3 (πœ‘ β†’ ran 𝑆 = ran (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))))
111110supeq1d 9443 . 2 (πœ‘ β†’ sup(ran 𝑆, ℝ*, < ) = sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(π‘ƒβ€˜π‘›))), ℝ*, < ))
11253, 103, 1113eqtr4d 2782 1 (πœ‘ β†’ (∫2β€˜πΉ) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∘r cofr 7671  supcsup 9437  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115  +∞cpnf 11247  β„*cxr 11249   < clt 11250   ≀ cle 11251  β„•cn 12214  [,)cico 13328   ⇝ cli 15430  MblFncmbf 25138  βˆ«1citg1 25139  βˆ«2citg2 25140  0𝑝c0p 25193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  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  ax-addf 11191
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  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 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-ioo 13330  df-ioc 13331  df-ico 13332  df-icc 13333  df-fz 13487  df-fzo 13630  df-fl 13759  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-rlim 15435  df-sum 15635  df-rest 17370  df-topgen 17391  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-top 22403  df-topon 22420  df-bases 22456  df-cmp 22898  df-ovol 24988  df-vol 24989  df-mbf 25143  df-itg1 25144  df-itg2 25145  df-0p 25194
This theorem is referenced by:  itg2i1fseq2  25281
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