Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sge0reuz Structured version   Visualization version   GIF version

Theorem sge0reuz 40982
Description: Value of the generalized sum of nonnegative reals, when the domain is a set of upper integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
sge0reuz.k 𝑘𝜑
sge0reuz.m (𝜑𝑀 ∈ ℤ)
sge0reuz.z 𝑍 = (ℤ𝑀)
sge0reuz.b ((𝜑𝑘𝑍) → 𝐵 ∈ (0[,)+∞))
Assertion
Ref Expression
sge0reuz (𝜑 → (Σ^‘(𝑘𝑍𝐵)) = sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
Distinct variable groups:   𝐵,𝑛   𝑘,𝑀,𝑛   𝑘,𝑍,𝑛   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑘)   𝐵(𝑘)

Proof of Theorem sge0reuz
Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sge0reuz.k . . 3 𝑘𝜑
2 sge0reuz.z . . . . 5 𝑍 = (ℤ𝑀)
32a1i 11 . . . 4 (𝜑𝑍 = (ℤ𝑀))
4 fvex 6239 . . . 4 (ℤ𝑀) ∈ V
53, 4syl6eqel 2738 . . 3 (𝜑𝑍 ∈ V)
6 sge0reuz.b . . 3 ((𝜑𝑘𝑍) → 𝐵 ∈ (0[,)+∞))
71, 5, 6sge0revalmpt 40913 . 2 (𝜑 → (Σ^‘(𝑘𝑍𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ))
8 nfv 1883 . . . . 5 𝑥𝜑
9 eqid 2651 . . . . 5 (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) = (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵)
10 nfv 1883 . . . . . . . 8 𝑘 𝑥 ∈ (𝒫 𝑍 ∩ Fin)
111, 10nfan 1868 . . . . . . 7 𝑘(𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin))
12 elinel2 3833 . . . . . . . 8 (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → 𝑥 ∈ Fin)
1312adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑥 ∈ Fin)
14 rge0ssre 12318 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
15 simpll 805 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘𝑥) → 𝜑)
16 elpwinss 39530 . . . . . . . . . . . 12 (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → 𝑥𝑍)
1716adantr 480 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘𝑥) → 𝑥𝑍)
18 simpr 476 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘𝑥) → 𝑘𝑥)
1917, 18sseldd 3637 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘𝑥) → 𝑘𝑍)
2019adantll 750 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘𝑥) → 𝑘𝑍)
2115, 20, 6syl2anc 694 . . . . . . . 8 (((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘𝑥) → 𝐵 ∈ (0[,)+∞))
2214, 21sseldi 3634 . . . . . . 7 (((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘𝑥) → 𝐵 ∈ ℝ)
2311, 13, 22fsumreclf 40126 . . . . . 6 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘𝑥 𝐵 ∈ ℝ)
2423rexrd 10127 . . . . 5 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘𝑥 𝐵 ∈ ℝ*)
258, 9, 24rnmptssd 39699 . . . 4 (𝜑 → ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) ⊆ ℝ*)
26 supxrcl 12183 . . . 4 (ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) ⊆ ℝ* → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ) ∈ ℝ*)
2725, 26syl 17 . . 3 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ) ∈ ℝ*)
28 nfv 1883 . . . . 5 𝑛𝜑
29 eqid 2651 . . . . 5 (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) = (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
30 nfv 1883 . . . . . . . 8 𝑘 𝑛𝑍
311, 30nfan 1868 . . . . . . 7 𝑘(𝜑𝑛𝑍)
32 fzfid 12812 . . . . . . 7 ((𝜑𝑛𝑍) → (𝑀...𝑛) ∈ Fin)
33 elfzuz 12376 . . . . . . . . . . 11 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
3433, 2syl6eleqr 2741 . . . . . . . . . 10 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
3534adantl 481 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...𝑛)) → 𝑘𝑍)
3614, 6sseldi 3634 . . . . . . . . 9 ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)
3735, 36syldan 486 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
3837adantlr 751 . . . . . . 7 (((𝜑𝑛𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
3931, 32, 38fsumreclf 40126 . . . . . 6 ((𝜑𝑛𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ)
4039rexrd 10127 . . . . 5 ((𝜑𝑛𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ*)
4128, 29, 40rnmptssd 39699 . . . 4 (𝜑 → ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ℝ*)
42 supxrcl 12183 . . . 4 (ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ℝ* → sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) ∈ ℝ*)
4341, 42syl 17 . . 3 (𝜑 → sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) ∈ ℝ*)
44 vex 3234 . . . . . . . 8 𝑦 ∈ V
459elrnmpt 5404 . . . . . . . 8 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) ↔ ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵))
4644, 45ax-mp 5 . . . . . . 7 (𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) ↔ ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵)
4746biimpi 206 . . . . . 6 (𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵)
4847adantl 481 . . . . 5 ((𝜑𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵)) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵)
49 sge0reuz.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
50493ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → 𝑀 ∈ ℤ)
51163ad2ant2 1103 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → 𝑥𝑍)
52133adant3 1101 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → 𝑥 ∈ Fin)
5350, 2, 51, 52uzfissfz 39855 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → ∃𝑛𝑍 𝑥 ⊆ (𝑀...𝑛))
54 nfv 1883 . . . . . . . . . 10 𝑛(𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵)
55 nfmpt1 4780 . . . . . . . . . . . 12 𝑛(𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
5655nfrn 5400 . . . . . . . . . . 11 𝑛ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
57 nfv 1883 . . . . . . . . . . 11 𝑛 𝑦𝑤
5856, 57nfrex 3036 . . . . . . . . . 10 𝑛𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤
59 id 22 . . . . . . . . . . . . . . 15 (𝑛𝑍𝑛𝑍)
60 sumex 14462 . . . . . . . . . . . . . . . 16 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V
6160a1i 11 . . . . . . . . . . . . . . 15 (𝑛𝑍 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V)
6229elrnmpt1 5406 . . . . . . . . . . . . . . 15 ((𝑛𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
6359, 61, 62syl2anc 694 . . . . . . . . . . . . . 14 (𝑛𝑍 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
64633ad2ant2 1103 . . . . . . . . . . . . 13 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑛𝑍𝑥 ⊆ (𝑀...𝑛)) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
65 simplr 807 . . . . . . . . . . . . . . 15 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 = Σ𝑘𝑥 𝐵)
66 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑘𝑦
67 nfcv 2793 . . . . . . . . . . . . . . . . . . . 20 𝑘𝑥
6867nfsum1 14464 . . . . . . . . . . . . . . . . . . 19 𝑘Σ𝑘𝑥 𝐵
6966, 68nfeq 2805 . . . . . . . . . . . . . . . . . 18 𝑘 𝑦 = Σ𝑘𝑥 𝐵
701, 69nfan 1868 . . . . . . . . . . . . . . . . 17 𝑘(𝜑𝑦 = Σ𝑘𝑥 𝐵)
71 nfv 1883 . . . . . . . . . . . . . . . . 17 𝑘 𝑥 ⊆ (𝑀...𝑛)
7270, 71nfan 1868 . . . . . . . . . . . . . . . 16 𝑘((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛))
73 fzfid 12812 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → (𝑀...𝑛) ∈ Fin)
7437ad4ant14 1317 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
75 simplll 813 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝜑)
7634adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘𝑍)
77 0xr 10124 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℝ*
7877a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑍) → 0 ∈ ℝ*)
79 pnfxr 10130 . . . . . . . . . . . . . . . . . . 19 +∞ ∈ ℝ*
8079a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑍) → +∞ ∈ ℝ*)
81 icogelb 12263 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝐵 ∈ (0[,)+∞)) → 0 ≤ 𝐵)
8278, 80, 6, 81syl3anc 1366 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑍) → 0 ≤ 𝐵)
8375, 76, 82syl2anc 694 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 0 ≤ 𝐵)
84 simpr 476 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑥 ⊆ (𝑀...𝑛))
8572, 73, 74, 83, 84fsumlessf 40127 . . . . . . . . . . . . . . 15 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → Σ𝑘𝑥 𝐵 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
8665, 85eqbrtrd 4707 . . . . . . . . . . . . . 14 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
87863adant2 1100 . . . . . . . . . . . . 13 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑛𝑍𝑥 ⊆ (𝑀...𝑛)) → 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
88 breq2 4689 . . . . . . . . . . . . . 14 (𝑤 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → (𝑦𝑤𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
8988rspcev 3340 . . . . . . . . . . . . 13 ((Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ∧ 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)
9064, 87, 89syl2anc 694 . . . . . . . . . . . 12 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑛𝑍𝑥 ⊆ (𝑀...𝑛)) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)
91903exp 1283 . . . . . . . . . . 11 ((𝜑𝑦 = Σ𝑘𝑥 𝐵) → (𝑛𝑍 → (𝑥 ⊆ (𝑀...𝑛) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)))
92913adant2 1100 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → (𝑛𝑍 → (𝑥 ⊆ (𝑀...𝑛) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)))
9354, 58, 92rexlimd 3055 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → (∃𝑛𝑍 𝑥 ⊆ (𝑀...𝑛) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤))
9453, 93mpd 15 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)
95943exp 1283 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → (𝑦 = Σ𝑘𝑥 𝐵 → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)))
9695rexlimdv 3059 . . . . . 6 (𝜑 → (∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵 → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤))
9796imp 444 . . . . 5 ((𝜑 ∧ ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)
9848, 97syldan 486 . . . 4 ((𝜑𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵)) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)
9925, 41, 98suplesup2 39905 . . 3 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ) ≤ sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
10029elrnmpt 5404 . . . . . . . . . 10 (𝑦 ∈ V → (𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ↔ ∃𝑛𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵))
10144, 100ax-mp 5 . . . . . . . . 9 (𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ↔ ∃𝑛𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
102101biimpi 206 . . . . . . . 8 (𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑛𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
103102adantl 481 . . . . . . 7 ((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → ∃𝑛𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
10434ssriv 3640 . . . . . . . . . . . . . . 15 (𝑀...𝑛) ⊆ 𝑍
105 ovex 6718 . . . . . . . . . . . . . . . 16 (𝑀...𝑛) ∈ V
106105elpw 4197 . . . . . . . . . . . . . . 15 ((𝑀...𝑛) ∈ 𝒫 𝑍 ↔ (𝑀...𝑛) ⊆ 𝑍)
107104, 106mpbir 221 . . . . . . . . . . . . . 14 (𝑀...𝑛) ∈ 𝒫 𝑍
108 fzfi 12811 . . . . . . . . . . . . . 14 (𝑀...𝑛) ∈ Fin
109107, 108elini 3830 . . . . . . . . . . . . 13 (𝑀...𝑛) ∈ (𝒫 𝑍 ∩ Fin)
110109a1i 11 . . . . . . . . . . . 12 (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → (𝑀...𝑛) ∈ (𝒫 𝑍 ∩ Fin))
111 id 22 . . . . . . . . . . . 12 (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
112 sumeq1 14463 . . . . . . . . . . . . . 14 (𝑥 = (𝑀...𝑛) → Σ𝑘𝑥 𝐵 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
113112eqeq2d 2661 . . . . . . . . . . . . 13 (𝑥 = (𝑀...𝑛) → (𝑦 = Σ𝑘𝑥 𝐵𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵))
114113rspcev 3340 . . . . . . . . . . . 12 (((𝑀...𝑛) ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵)
115110, 111, 114syl2anc 694 . . . . . . . . . . 11 (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵)
11644a1i 11 . . . . . . . . . . 11 (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵𝑦 ∈ V)
1179, 115, 116elrnmptd 39680 . . . . . . . . . 10 (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵))
1181172a1i 12 . . . . . . . . 9 (𝜑 → (𝑛𝑍 → (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵))))
119118rexlimdv 3059 . . . . . . . 8 (𝜑 → (∃𝑛𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵)))
120119adantr 480 . . . . . . 7 ((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → (∃𝑛𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵)))
121103, 120mpd 15 . . . . . 6 ((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵))
122121ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵))
123 dfss3 3625 . . . . 5 (ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) ↔ ∀𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵))
124122, 123sylibr 224 . . . 4 (𝜑 → ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵))
125 supxrss 12200 . . . 4 ((ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) ∧ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) ⊆ ℝ*) → sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) ≤ sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ))
126124, 25, 125syl2anc 694 . . 3 (𝜑 → sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) ≤ sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ))
12727, 43, 99, 126xrletrid 12024 . 2 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ) = sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
1287, 127eqtrd 2685 1 (𝜑 → (Σ^‘(𝑘𝑍𝐵)) = sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191   class class class wbr 4685  cmpt 4762  ran crn 5144  cfv 5926  (class class class)co 6690  Fincfn 7997  supcsup 8387  cr 9973  0cc0 9974  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cz 11415  cuz 11725  [,)cico 12215  ...cfz 12364  Σcsu 14460  Σ^csumge0 40897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-sumge0 40898
This theorem is referenced by:  sge0reuzb  40983
  Copyright terms: Public domain W3C validator