Step | Hyp | Ref
| Expression |
1 | | gsumesum.0 |
. . 3
⊢
Ⅎ𝑘𝜑 |
2 | | nfcv 2751 |
. . 3
⊢
Ⅎ𝑘𝐴 |
3 | | gsumesum.1 |
. . 3
⊢ (𝜑 → 𝐴 ∈ Fin) |
4 | | gsumesum.2 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
5 | | eqidd 2611 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) |
6 | 1, 2, 3, 4, 5 | esumval 29435 |
. 2
⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))), ℝ*, <
)) |
7 | | xrltso 11850 |
. . . 4
⊢ < Or
ℝ* |
8 | 7 | a1i 11 |
. . 3
⊢ (𝜑 → < Or
ℝ*) |
9 | | iccssxr 12127 |
. . . 4
⊢
(0[,]+∞) ⊆ ℝ* |
10 | | xrge0base 29016 |
. . . . 5
⊢
(0[,]+∞) = (Base‘(ℝ*𝑠
↾s (0[,]+∞))) |
11 | | xrge0cmn 19607 |
. . . . . 6
⊢
(ℝ*𝑠 ↾s
(0[,]+∞)) ∈ CMnd |
12 | 11 | a1i 11 |
. . . . 5
⊢ (𝜑 →
(ℝ*𝑠 ↾s (0[,]+∞))
∈ CMnd) |
13 | 4 | ex 449 |
. . . . . 6
⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
14 | 1, 13 | ralrimi 2940 |
. . . . 5
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
15 | 10, 12, 3, 14 | gsummptcl 18189 |
. . . 4
⊢ (𝜑 →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ (0[,]+∞)) |
16 | 9, 15 | sseldi 3566 |
. . 3
⊢ (𝜑 →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈
ℝ*) |
17 | | pwidg 4121 |
. . . . . . 7
⊢ (𝐴 ∈ Fin → 𝐴 ∈ 𝒫 𝐴) |
18 | 3, 17 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐴) |
19 | 18, 3 | elind 3760 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ (𝒫 𝐴 ∩ Fin)) |
20 | | eqidd 2611 |
. . . . 5
⊢ (𝜑 →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
21 | | mpteq1 4665 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑘 ∈ 𝑥 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)) |
22 | 21 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = 𝐴 →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
23 | 22 | eqeq2d 2620 |
. . . . . 6
⊢ (𝑥 = 𝐴 →
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) ↔
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
24 | 23 | rspcev 3282 |
. . . . 5
⊢ ((𝐴 ∈ (𝒫 𝐴 ∩ Fin) ∧
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) → ∃𝑥 ∈ (𝒫 𝐴 ∩
Fin)((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) |
25 | 19, 20, 24 | syl2anc 691 |
. . . 4
⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩
Fin)((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) |
26 | | eqid 2610 |
. . . . 5
⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) |
27 | | ovex 6577 |
. . . . 5
⊢
((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) ∈ V |
28 | 26, 27 | elrnmpti 5297 |
. . . 4
⊢
(((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩
Fin)((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) |
29 | 25, 28 | sylibr 223 |
. . 3
⊢ (𝜑 →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) |
30 | | simpr 476 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) → 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) |
31 | | mpteq1 4665 |
. . . . . . . . 9
⊢ (𝑥 = 𝑎 → (𝑘 ∈ 𝑥 ↦ 𝐵) = (𝑘 ∈ 𝑎 ↦ 𝐵)) |
32 | 31 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑥 = 𝑎 →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵))) |
33 | 32 | cbvmptv 4678 |
. . . . . . 7
⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) = (𝑎 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵))) |
34 | | ovex 6577 |
. . . . . . 7
⊢
((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∈ V |
35 | 33, 34 | elrnmpti 5297 |
. . . . . 6
⊢ (𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) ↔ ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ((ℝ*𝑠
↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑎 ↦ 𝐵))) |
36 | 30, 35 | sylib 207 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ((ℝ*𝑠
↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑎 ↦ 𝐵))) |
37 | 11 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) →
(ℝ*𝑠 ↾s (0[,]+∞))
∈ CMnd) |
38 | | inss2 3796 |
. . . . . . . . . . . 12
⊢
(𝒫 𝐴 ∩
Fin) ⊆ Fin |
39 | | simpr 476 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) |
40 | 38, 39 | sseldi 3566 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ∈ Fin) |
41 | | nfv 1830 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘 𝑎 ∈ (𝒫 𝐴 ∩ Fin) |
42 | 1, 41 | nfan 1816 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘(𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) |
43 | | simpll 786 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑎) → 𝜑) |
44 | | inss1 3795 |
. . . . . . . . . . . . . . . . . 18
⊢
(𝒫 𝐴 ∩
Fin) ⊆ 𝒫 𝐴 |
45 | 44 | sseli 3564 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ∈ 𝒫 𝐴) |
46 | 45 | elpwid 4118 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ (𝒫 𝐴 ∩ Fin) → 𝑎 ⊆ 𝐴) |
47 | 46 | ad2antlr 759 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑎) → 𝑎 ⊆ 𝐴) |
48 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑎) → 𝑘 ∈ 𝑎) |
49 | 47, 48 | sseldd 3569 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑎) → 𝑘 ∈ 𝐴) |
50 | 43, 49, 4 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑎) → 𝐵 ∈ (0[,]+∞)) |
51 | 50 | ex 449 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑘 ∈ 𝑎 → 𝐵 ∈ (0[,]+∞))) |
52 | 42, 51 | ralrimi 2940 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∀𝑘 ∈ 𝑎 𝐵 ∈ (0[,]+∞)) |
53 | 10, 37, 40, 52 | gsummptcl 18189 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∈ (0[,]+∞)) |
54 | 9, 53 | sseldi 3566 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∈
ℝ*) |
55 | | diffi 8077 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝑎) ∈ Fin) |
56 | 3, 55 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 ∖ 𝑎) ∈ Fin) |
57 | 56 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐴 ∖ 𝑎) ∈ Fin) |
58 | | simpll 786 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ (𝐴 ∖ 𝑎)) → 𝜑) |
59 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ (𝐴 ∖ 𝑎)) → 𝑘 ∈ (𝐴 ∖ 𝑎)) |
60 | 59 | eldifad 3552 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ (𝐴 ∖ 𝑎)) → 𝑘 ∈ 𝐴) |
61 | 58, 60, 4 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ (𝐴 ∖ 𝑎)) → 𝐵 ∈ (0[,]+∞)) |
62 | 61 | ex 449 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑘 ∈ (𝐴 ∖ 𝑎) → 𝐵 ∈ (0[,]+∞))) |
63 | 42, 62 | ralrimi 2940 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → ∀𝑘 ∈ (𝐴 ∖ 𝑎)𝐵 ∈ (0[,]+∞)) |
64 | 10, 37, 57, 63 | gsummptcl 18189 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) ∈ (0[,]+∞)) |
65 | 9, 64 | sseldi 3566 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) ∈
ℝ*) |
66 | | elxrge0 12152 |
. . . . . . . . . . 11
⊢
(((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) ∈ (0[,]+∞) ↔
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) ∈ ℝ* ∧ 0 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)))) |
67 | 66 | simprbi 479 |
. . . . . . . . . 10
⊢
(((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) ∈ (0[,]+∞) → 0 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵))) |
68 | 64, 67 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 0 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵))) |
69 | | xraddge02 28911 |
. . . . . . . . . 10
⊢
((((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∈ ℝ* ∧
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) ∈ ℝ*) → (0
≤ ((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) +𝑒
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵))))) |
70 | 69 | imp 444 |
. . . . . . . . 9
⊢
(((((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∈ ℝ* ∧
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) ∈ ℝ*) ∧ 0 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵))) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) +𝑒
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)))) |
71 | 54, 65, 68, 70 | syl21anc 1317 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) +𝑒
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)))) |
72 | 71 | adantlr 747 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) +𝑒
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)))) |
73 | | simpll 786 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝜑) |
74 | 46 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ⊆ 𝐴) |
75 | | xrge00 29017 |
. . . . . . . . . 10
⊢ 0 =
(0g‘(ℝ*𝑠
↾s (0[,]+∞))) |
76 | | xrge0plusg 29018 |
. . . . . . . . . 10
⊢
+𝑒 =
(+g‘(ℝ*𝑠
↾s (0[,]+∞))) |
77 | 11 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) →
(ℝ*𝑠 ↾s (0[,]+∞))
∈ CMnd) |
78 | 3 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) → 𝐴 ∈ Fin) |
79 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) |
80 | 1, 4, 79 | fmptdf 6294 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
81 | 80 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
82 | 79 | fnmpt 5933 |
. . . . . . . . . . . . 13
⊢
(∀𝑘 ∈
𝐴 𝐵 ∈ (0[,]+∞) → (𝑘 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
83 | 14, 82 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
84 | | c0ex 9913 |
. . . . . . . . . . . . 13
⊢ 0 ∈
V |
85 | 84 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ∈
V) |
86 | 83, 3, 85 | fndmfifsupp 8171 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
87 | 86 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
88 | | disjdif 3992 |
. . . . . . . . . . 11
⊢ (𝑎 ∩ (𝐴 ∖ 𝑎)) = ∅ |
89 | 88 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑎 ∩ (𝐴 ∖ 𝑎)) = ∅) |
90 | | undif 4001 |
. . . . . . . . . . . . 13
⊢ (𝑎 ⊆ 𝐴 ↔ (𝑎 ∪ (𝐴 ∖ 𝑎)) = 𝐴) |
91 | 90 | biimpi 205 |
. . . . . . . . . . . 12
⊢ (𝑎 ⊆ 𝐴 → (𝑎 ∪ (𝐴 ∖ 𝑎)) = 𝐴) |
92 | 91 | eqcomd 2616 |
. . . . . . . . . . 11
⊢ (𝑎 ⊆ 𝐴 → 𝐴 = (𝑎 ∪ (𝐴 ∖ 𝑎))) |
93 | 92 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) → 𝐴 = (𝑎 ∪ (𝐴 ∖ 𝑎))) |
94 | 10, 75, 76, 77, 78, 81, 87, 89, 93 | gsumsplit 18151 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑎)) +𝑒
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ (𝐴 ∖ 𝑎))))) |
95 | | resmpt 5369 |
. . . . . . . . . . . 12
⊢ (𝑎 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑎) = (𝑘 ∈ 𝑎 ↦ 𝐵)) |
96 | 95 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ (𝑎 ⊆ 𝐴 →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑎)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵))) |
97 | 96 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑎)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵))) |
98 | | difss 3699 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∖ 𝑎) ⊆ 𝐴 |
99 | | resmpt 5369 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∖ 𝑎) ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ (𝐴 ∖ 𝑎)) = (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) |
100 | 98, 99 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ (𝐴 ∖ 𝑎)) = (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵) |
101 | 100 | oveq2i 6560 |
. . . . . . . . . . 11
⊢
((ℝ*𝑠 ↾s
(0[,]+∞)) Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ (𝐴 ∖ 𝑎))) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)) |
102 | 101 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ (𝐴 ∖ 𝑎))) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵))) |
103 | 97, 102 | oveq12d 6567 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) →
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑎)) +𝑒
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ (𝐴 ∖ 𝑎)))) =
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) +𝑒
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)))) |
104 | 94, 103 | eqtrd 2644 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐴) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) +𝑒
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)))) |
105 | 73, 74, 104 | syl2anc 691 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) =
(((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) +𝑒
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ (𝐴 ∖ 𝑎) ↦ 𝐵)))) |
106 | 72, 105 | breqtrrd 4611 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) ∧ 𝑎 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
107 | 106 | ralrimiva 2949 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) → ∀𝑎 ∈ (𝒫 𝐴 ∩
Fin)((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
108 | | r19.29r 3055 |
. . . . . 6
⊢
((∃𝑎 ∈
(𝒫 𝐴 ∩
Fin)𝑦 =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∧ ∀𝑎 ∈ (𝒫 𝐴 ∩
Fin)((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 = ((ℝ*𝑠
↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∧
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
109 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑦 =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) → (𝑦 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ↔
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
110 | 109 | biimpar 501 |
. . . . . . 7
⊢ ((𝑦 =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∧
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) → 𝑦 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
111 | 110 | rexlimivw 3011 |
. . . . . 6
⊢
(∃𝑎 ∈
(𝒫 𝐴 ∩
Fin)(𝑦 =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∧
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) → 𝑦 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
112 | 108, 111 | syl 17 |
. . . . 5
⊢
((∃𝑎 ∈
(𝒫 𝐴 ∩
Fin)𝑦 =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ∧ ∀𝑎 ∈ (𝒫 𝐴 ∩
Fin)((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝑎 ↦ 𝐵)) ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) → 𝑦 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
113 | 36, 107, 112 | syl2anc 691 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) → 𝑦 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
114 | 16 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈
ℝ*) |
115 | 11 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) →
(ℝ*𝑠 ↾s (0[,]+∞))
∈ CMnd) |
116 | | simpr 476 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) |
117 | 38, 116 | sseldi 3566 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ Fin) |
118 | | nfv 1830 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘 𝑥 ∈ (𝒫 𝐴 ∩ Fin) |
119 | 1, 118 | nfan 1816 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) |
120 | | simpll 786 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝜑) |
121 | 44 | sseli 3564 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴) |
122 | 121 | ad2antlr 759 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ∈ 𝒫 𝐴) |
123 | 122 | elpwid 4118 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ⊆ 𝐴) |
124 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑥) |
125 | 123, 124 | sseldd 3569 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐴) |
126 | 120, 125,
4 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ (0[,]+∞)) |
127 | 126 | ex 449 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑘 ∈ 𝑥 → 𝐵 ∈ (0[,]+∞))) |
128 | 119, 127 | ralrimi 2940 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ∀𝑘 ∈ 𝑥 𝐵 ∈ (0[,]+∞)) |
129 | 10, 115, 117, 128 | gsummptcl 18189 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) ∈ (0[,]+∞)) |
130 | 9, 129 | sseldi 3566 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) ∈
ℝ*) |
131 | 130 | ralrimiva 2949 |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ (𝒫 𝐴 ∩
Fin)((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) ∈
ℝ*) |
132 | 26 | rnmptss 6299 |
. . . . . . 7
⊢
(∀𝑥 ∈
(𝒫 𝐴 ∩
Fin)((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) ∈ ℝ* → ran
(𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) ⊆
ℝ*) |
133 | 131, 132 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))) ⊆
ℝ*) |
134 | 133 | sselda 3568 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) → 𝑦 ∈ ℝ*) |
135 | | xrltnle 9984 |
. . . . . 6
⊢
((((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ* ∧ 𝑦 ∈ ℝ*)
→ (((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) < 𝑦 ↔ ¬ 𝑦 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
136 | 135 | con2bid 343 |
. . . . 5
⊢
((((ℝ*𝑠 ↾s
(0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ* ∧ 𝑦 ∈ ℝ*)
→ (𝑦 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ↔ ¬
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) < 𝑦)) |
137 | 114, 134,
136 | syl2anc 691 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) → (𝑦 ≤
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ↔ ¬
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) < 𝑦)) |
138 | 113, 137 | mpbid 221 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵)))) → ¬
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) < 𝑦) |
139 | 8, 16, 29, 138 | supmax 8256 |
. 2
⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑥 ↦ 𝐵))), ℝ*, < ) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
140 | 6, 139 | eqtr2d 2645 |
1
⊢ (𝜑 →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ*𝑘 ∈ 𝐴𝐵) |