Step | Hyp | Ref
| Expression |
1 | | fsumge0.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ Fin) |
2 | 1 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝐴 ∈ Fin) |
3 | | fsumge0.2 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
4 | 3 | adantlr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
5 | | fsumge0.3 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) |
6 | 5 | adantlr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) |
7 | | snssi 3724 |
. . . . . . . . . 10
⊢ (𝑚 ∈ 𝐴 → {𝑚} ⊆ 𝐴) |
8 | 7 | adantl 275 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → {𝑚} ⊆ 𝐴) |
9 | | snfig 6792 |
. . . . . . . . . 10
⊢ (𝑚 ∈ 𝐴 → {𝑚} ∈ Fin) |
10 | 9 | adantl 275 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → {𝑚} ∈ Fin) |
11 | 2, 4, 6, 8, 10 | fsumlessfi 11423 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ {𝑚}𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
12 | 11 | adantlr 474 |
. . . . . . 7
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ {𝑚}𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
13 | | simpr 109 |
. . . . . . . 8
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐴) |
14 | 3, 5 | jca 304 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) |
15 | 14 | ralrimiva 2543 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) |
16 | 15 | adantr 274 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) → ∀𝑘 ∈ 𝐴 (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) |
17 | | nfcsb1v 3082 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 |
18 | 17 | nfel1 2323 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 ∈ ℝ |
19 | | nfcv 2312 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘0 |
20 | | nfcv 2312 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘
≤ |
21 | 19, 20, 17 | nfbr 4035 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘0 ≤
⦋𝑚 / 𝑘⦌𝐵 |
22 | 18, 21 | nfan 1558 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘(⦋𝑚 / 𝑘⦌𝐵 ∈ ℝ ∧ 0 ≤
⦋𝑚 / 𝑘⦌𝐵) |
23 | | csbeq1a 3058 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵) |
24 | 23 | eleq1d 2239 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → (𝐵 ∈ ℝ ↔ ⦋𝑚 / 𝑘⦌𝐵 ∈ ℝ)) |
25 | 23 | breq2d 4001 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑚 / 𝑘⦌𝐵)) |
26 | 24, 25 | anbi12d 470 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑚 → ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ↔ (⦋𝑚 / 𝑘⦌𝐵 ∈ ℝ ∧ 0 ≤
⦋𝑚 / 𝑘⦌𝐵))) |
27 | 22, 26 | rspc 2828 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) → (⦋𝑚 / 𝑘⦌𝐵 ∈ ℝ ∧ 0 ≤
⦋𝑚 / 𝑘⦌𝐵))) |
28 | 16, 27 | mpan9 279 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → (⦋𝑚 / 𝑘⦌𝐵 ∈ ℝ ∧ 0 ≤
⦋𝑚 / 𝑘⦌𝐵)) |
29 | 28 | simpld 111 |
. . . . . . . . 9
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐵 ∈ ℝ) |
30 | 29 | recnd 7948 |
. . . . . . . 8
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐵 ∈ ℂ) |
31 | | sumsns 11378 |
. . . . . . . 8
⊢ ((𝑚 ∈ 𝐴 ∧ ⦋𝑚 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑚}𝐵 = ⦋𝑚 / 𝑘⦌𝐵) |
32 | 13, 30, 31 | syl2anc 409 |
. . . . . . 7
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ {𝑚}𝐵 = ⦋𝑚 / 𝑘⦌𝐵) |
33 | | simplr 525 |
. . . . . . 7
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
34 | 12, 32, 33 | 3brtr3d 4020 |
. . . . . 6
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐵 ≤ 0) |
35 | 28 | simprd 113 |
. . . . . 6
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → 0 ≤ ⦋𝑚 / 𝑘⦌𝐵) |
36 | | 0re 7920 |
. . . . . . 7
⊢ 0 ∈
ℝ |
37 | | letri3 8000 |
. . . . . . 7
⊢
((⦋𝑚 /
𝑘⦌𝐵 ∈ ℝ ∧ 0 ∈
ℝ) → (⦋𝑚 / 𝑘⦌𝐵 = 0 ↔ (⦋𝑚 / 𝑘⦌𝐵 ≤ 0 ∧ 0 ≤ ⦋𝑚 / 𝑘⦌𝐵))) |
38 | 29, 36, 37 | sylancl 411 |
. . . . . 6
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → (⦋𝑚 / 𝑘⦌𝐵 = 0 ↔ (⦋𝑚 / 𝑘⦌𝐵 ≤ 0 ∧ 0 ≤ ⦋𝑚 / 𝑘⦌𝐵))) |
39 | 34, 35, 38 | mpbir2and 939 |
. . . . 5
⊢ (((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐵 = 0) |
40 | 39 | ralrimiva 2543 |
. . . 4
⊢ ((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) → ∀𝑚 ∈ 𝐴 ⦋𝑚 / 𝑘⦌𝐵 = 0) |
41 | | nfv 1521 |
. . . . 5
⊢
Ⅎ𝑚 𝐵 = 0 |
42 | 17 | nfeq1 2322 |
. . . . 5
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 = 0 |
43 | 23 | eqeq1d 2179 |
. . . . 5
⊢ (𝑘 = 𝑚 → (𝐵 = 0 ↔ ⦋𝑚 / 𝑘⦌𝐵 = 0)) |
44 | 41, 42, 43 | cbvral 2692 |
. . . 4
⊢
(∀𝑘 ∈
𝐴 𝐵 = 0 ↔ ∀𝑚 ∈ 𝐴 ⦋𝑚 / 𝑘⦌𝐵 = 0) |
45 | 40, 44 | sylibr 133 |
. . 3
⊢ ((𝜑 ∧ Σ𝑘 ∈ 𝐴 𝐵 = 0) → ∀𝑘 ∈ 𝐴 𝐵 = 0) |
46 | 45 | ex 114 |
. 2
⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 = 0 → ∀𝑘 ∈ 𝐴 𝐵 = 0)) |
47 | | isumz 11352 |
. . . . 5
⊢ (((0
∈ ℤ ∧ 𝐴
⊆ (ℤ≥‘0) ∧ ∀𝑥 ∈
(ℤ≥‘0)DECID 𝑥 ∈ 𝐴) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0) |
48 | 47 | olcs 731 |
. . . 4
⊢ (𝐴 ∈ Fin → Σ𝑘 ∈ 𝐴 0 = 0) |
49 | | sumeq2 11322 |
. . . . 5
⊢
(∀𝑘 ∈
𝐴 𝐵 = 0 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 0) |
50 | 49 | eqeq1d 2179 |
. . . 4
⊢
(∀𝑘 ∈
𝐴 𝐵 = 0 → (Σ𝑘 ∈ 𝐴 𝐵 = 0 ↔ Σ𝑘 ∈ 𝐴 0 = 0)) |
51 | 48, 50 | syl5ibrcom 156 |
. . 3
⊢ (𝐴 ∈ Fin →
(∀𝑘 ∈ 𝐴 𝐵 = 0 → Σ𝑘 ∈ 𝐴 𝐵 = 0)) |
52 | 1, 51 | syl 14 |
. 2
⊢ (𝜑 → (∀𝑘 ∈ 𝐴 𝐵 = 0 → Σ𝑘 ∈ 𝐴 𝐵 = 0)) |
53 | 46, 52 | impbid 128 |
1
⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 = 0 ↔ ∀𝑘 ∈ 𝐴 𝐵 = 0)) |