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Theorem sge0reuz 39992
 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 6163 . . . 4 (ℤ𝑀) ∈ V
53, 4syl6eqel 2706 . . 3 (𝜑𝑍 ∈ V)
6 sge0reuz.b . . 3 ((𝜑𝑘𝑍) → 𝐵 ∈ (0[,)+∞))
71, 5, 6sge0revalmpt 39923 . 2 (𝜑 → (Σ^‘(𝑘𝑍𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ))
8 nfv 1840 . . . . 5 𝑥𝜑
9 eqid 2621 . . . . 5 (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) = (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵)
10 nfv 1840 . . . . . . . 8 𝑘 𝑥 ∈ (𝒫 𝑍 ∩ Fin)
111, 10nfan 1825 . . . . . . 7 𝑘(𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin))
12 elinel2 3783 . . . . . . . 8 (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → 𝑥 ∈ Fin)
1312adantl 482 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑥 ∈ Fin)
14 rge0ssre 12229 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
15 simpll 789 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘𝑥) → 𝜑)
16 elpwinss 38726 . . . . . . . . . . . 12 (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → 𝑥𝑍)
1716adantr 481 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘𝑥) → 𝑥𝑍)
18 simpr 477 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘𝑥) → 𝑘𝑥)
1917, 18sseldd 3588 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘𝑥) → 𝑘𝑍)
2019adantll 749 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘𝑥) → 𝑘𝑍)
2115, 20, 6syl2anc 692 . . . . . . . 8 (((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘𝑥) → 𝐵 ∈ (0[,)+∞))
2214, 21sseldi 3585 . . . . . . 7 (((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘𝑥) → 𝐵 ∈ ℝ)
2311, 13, 22fsumreclf 39235 . . . . . 6 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘𝑥 𝐵 ∈ ℝ)
2423rexrd 10040 . . . . 5 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘𝑥 𝐵 ∈ ℝ*)
258, 9, 24rnmptssd 38882 . . . 4 (𝜑 → ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵) ⊆ ℝ*)
26 supxrcl 12095 . . . 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 1840 . . . . 5 𝑛𝜑
29 eqid 2621 . . . . 5 (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) = (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
30 nfv 1840 . . . . . . . 8 𝑘 𝑛𝑍
311, 30nfan 1825 . . . . . . 7 𝑘(𝜑𝑛𝑍)
32 fzfid 12719 . . . . . . 7 ((𝜑𝑛𝑍) → (𝑀...𝑛) ∈ Fin)
33 elfzuz 12287 . . . . . . . . . . 11 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
3433, 2syl6eleqr 2709 . . . . . . . . . 10 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
3534adantl 482 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...𝑛)) → 𝑘𝑍)
3614, 6sseldi 3585 . . . . . . . . 9 ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)
3735, 36syldan 487 . . . . . . . 8 ((𝜑𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
3837adantlr 750 . . . . . . 7 (((𝜑𝑛𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
3931, 32, 38fsumreclf 39235 . . . . . 6 ((𝜑𝑛𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ)
4039rexrd 10040 . . . . 5 ((𝜑𝑛𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ*)
4128, 29, 40rnmptssd 38882 . . . 4 (𝜑 → ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ℝ*)
42 supxrcl 12095 . . . 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 3192 . . . . . . . 8 𝑦 ∈ V
459elrnmpt 5337 . . . . . . . 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 482 . . . . 5 ((𝜑𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵)) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵)
49 sge0reuz.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
50493ad2ant1 1080 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → 𝑀 ∈ ℤ)
51163ad2ant2 1081 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → 𝑥𝑍)
52133adant3 1079 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → 𝑥 ∈ Fin)
5350, 2, 51, 52uzfissfz 39029 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → ∃𝑛𝑍 𝑥 ⊆ (𝑀...𝑛))
54 nfv 1840 . . . . . . . . . 10 𝑛(𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵)
55 nfmpt1 4712 . . . . . . . . . . . 12 𝑛(𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
5655nfrn 5333 . . . . . . . . . . 11 𝑛ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
57 nfv 1840 . . . . . . . . . . 11 𝑛 𝑦𝑤
5856, 57nfrex 3002 . . . . . . . . . 10 𝑛𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤
59 id 22 . . . . . . . . . . . . . . 15 (𝑛𝑍𝑛𝑍)
60 sumex 14359 . . . . . . . . . . . . . . . 16 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V
6160a1i 11 . . . . . . . . . . . . . . 15 (𝑛𝑍 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V)
6229elrnmpt1 5339 . . . . . . . . . . . . . . 15 ((𝑛𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
6359, 61, 62syl2anc 692 . . . . . . . . . . . . . 14 (𝑛𝑍 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
64633ad2ant2 1081 . . . . . . . . . . . . 13 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑛𝑍𝑥 ⊆ (𝑀...𝑛)) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
65 simplr 791 . . . . . . . . . . . . . . 15 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 = Σ𝑘𝑥 𝐵)
66 nfcv 2761 . . . . . . . . . . . . . . . . . . 19 𝑘𝑦
67 nfcv 2761 . . . . . . . . . . . . . . . . . . . 20 𝑘𝑥
6867nfsum1 14361 . . . . . . . . . . . . . . . . . . 19 𝑘Σ𝑘𝑥 𝐵
6966, 68nfeq 2772 . . . . . . . . . . . . . . . . . 18 𝑘 𝑦 = Σ𝑘𝑥 𝐵
701, 69nfan 1825 . . . . . . . . . . . . . . . . 17 𝑘(𝜑𝑦 = Σ𝑘𝑥 𝐵)
71 nfv 1840 . . . . . . . . . . . . . . . . 17 𝑘 𝑥 ⊆ (𝑀...𝑛)
7270, 71nfan 1825 . . . . . . . . . . . . . . . 16 𝑘((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛))
73 fzfid 12719 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → (𝑀...𝑛) ∈ Fin)
7437ad4ant14 1290 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
75 simplll 797 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝜑)
7634adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘𝑍)
77 0xr 10037 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℝ*
7877a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑍) → 0 ∈ ℝ*)
79 pnfxr 10043 . . . . . . . . . . . . . . . . . . 19 +∞ ∈ ℝ*
8079a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑍) → +∞ ∈ ℝ*)
81 icogelb 12174 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*𝐵 ∈ (0[,)+∞)) → 0 ≤ 𝐵)
8278, 80, 6, 81syl3anc 1323 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑍) → 0 ≤ 𝐵)
8375, 76, 82syl2anc 692 . . . . . . . . . . . . . . . 16 ((((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 0 ≤ 𝐵)
84 simpr 477 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑥 ⊆ (𝑀...𝑛))
8572, 73, 74, 83, 84fsumlessf 39236 . . . . . . . . . . . . . . 15 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → Σ𝑘𝑥 𝐵 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
8665, 85eqbrtrd 4640 . . . . . . . . . . . . . 14 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
87863adant2 1078 . . . . . . . . . . . . 13 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑛𝑍𝑥 ⊆ (𝑀...𝑛)) → 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
88 breq2 4622 . . . . . . . . . . . . . 14 (𝑤 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → (𝑦𝑤𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
8988rspcev 3298 . . . . . . . . . . . . 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 692 . . . . . . . . . . . 12 (((𝜑𝑦 = Σ𝑘𝑥 𝐵) ∧ 𝑛𝑍𝑥 ⊆ (𝑀...𝑛)) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)
91903exp 1261 . . . . . . . . . . 11 ((𝜑𝑦 = Σ𝑘𝑥 𝐵) → (𝑛𝑍 → (𝑥 ⊆ (𝑀...𝑛) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)))
92913adant2 1078 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘𝑥 𝐵) → (𝑛𝑍 → (𝑥 ⊆ (𝑀...𝑛) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)))
9354, 58, 92rexlimd 3020 . . . . . . . . 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 1261 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → (𝑦 = Σ𝑘𝑥 𝐵 → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)))
9695rexlimdv 3024 . . . . . 6 (𝜑 → (∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵 → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤))
9796imp 445 . . . . 5 ((𝜑 ∧ ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)
9848, 97syldan 487 . . . 4 ((𝜑𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵)) → ∃𝑤 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦𝑤)
9925, 41, 98suplesup2 39079 . . 3 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ) ≤ sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
10029elrnmpt 5337 . . . . . . . . . 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 482 . . . . . . 7 ((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → ∃𝑛𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
10434ssriv 3591 . . . . . . . . . . . . . . 15 (𝑀...𝑛) ⊆ 𝑍
105 ovex 6638 . . . . . . . . . . . . . . . 16 (𝑀...𝑛) ∈ V
106105elpw 4141 . . . . . . . . . . . . . . 15 ((𝑀...𝑛) ∈ 𝒫 𝑍 ↔ (𝑀...𝑛) ⊆ 𝑍)
107104, 106mpbir 221 . . . . . . . . . . . . . 14 (𝑀...𝑛) ∈ 𝒫 𝑍
108 fzfi 12718 . . . . . . . . . . . . . 14 (𝑀...𝑛) ∈ Fin
109107, 108elini 3780 . . . . . . . . . . . . 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 14360 . . . . . . . . . . . . . 14 (𝑥 = (𝑀...𝑛) → Σ𝑘𝑥 𝐵 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
113112eqeq2d 2631 . . . . . . . . . . . . 13 (𝑥 = (𝑀...𝑛) → (𝑦 = Σ𝑘𝑥 𝐵𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵))
114113rspcev 3298 . . . . . . . . . . . 12 (((𝑀...𝑛) ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘𝑥 𝐵)
115110, 111, 114syl2anc 692 . . . . . . . . . . 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 38863 . . . . . . . . . 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 3024 . . . . . . . 8 (𝜑 → (∃𝑛𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵)))
120119adantr 481 . . . . . . 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 2961 . . . . 5 (𝜑 → ∀𝑦 ∈ ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵))
123 dfss3 3577 . . . . 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 12112 . . . 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 692 . . 3 (𝜑 → sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) ≤ sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ))
12727, 43, 99, 126xrletrid 11937 . 2 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘𝑥 𝐵), ℝ*, < ) = sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
1287, 127eqtrd 2655 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 384   ∧ w3a 1036   = wceq 1480  Ⅎwnf 1705   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  Vcvv 3189   ∩ cin 3558   ⊆ wss 3559  𝒫 cpw 4135   class class class wbr 4618   ↦ cmpt 4678  ran crn 5080  ‘cfv 5852  (class class class)co 6610  Fincfn 7906  supcsup 8297  ℝcr 9886  0cc0 9887  +∞cpnf 10022  ℝ*cxr 10024   < clt 10025   ≤ cle 10026  ℤcz 11328  ℤ≥cuz 11638  [,)cico 12126  ...cfz 12275  Σcsu 14357  Σ^csumge0 39907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-oi 8366  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-z 11329  df-uz 11639  df-rp 11784  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-sum 14358  df-sumge0 39908 This theorem is referenced by:  sge0reuzb  39993
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