Proof of Theorem sge0p1
Step | Hyp | Ref
| Expression |
1 | | sge0p1.1 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
2 | | fzsuc 13302 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) |
4 | 3 | mpteq1d 5174 |
. . 3
⊢ (𝜑 → (𝑘 ∈ (𝑀...(𝑁 + 1)) ↦ 𝐴) = (𝑘 ∈ ((𝑀...𝑁) ∪ {(𝑁 + 1)}) ↦ 𝐴)) |
5 | 4 | fveq2d 6775 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ (𝑀...(𝑁 + 1)) ↦ 𝐴)) =
(Σ^‘(𝑘 ∈ ((𝑀...𝑁) ∪ {(𝑁 + 1)}) ↦ 𝐴))) |
6 | | nfv 1921 |
. . 3
⊢
Ⅎ𝑘𝜑 |
7 | | ovex 7304 |
. . . 4
⊢ (𝑀...𝑁) ∈ V |
8 | 7 | a1i 11 |
. . 3
⊢ (𝜑 → (𝑀...𝑁) ∈ V) |
9 | | snex 5358 |
. . . 4
⊢ {(𝑁 + 1)} ∈ V |
10 | 9 | a1i 11 |
. . 3
⊢ (𝜑 → {(𝑁 + 1)} ∈ V) |
11 | | fzp1disj 13314 |
. . . 4
⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ |
12 | 11 | a1i 11 |
. . 3
⊢ (𝜑 → ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅) |
13 | | 0xr 11023 |
. . . . 5
⊢ 0 ∈
ℝ* |
14 | 13 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ∈
ℝ*) |
15 | | pnfxr 11030 |
. . . . 5
⊢ +∞
∈ ℝ* |
16 | 15 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → +∞ ∈
ℝ*) |
17 | | iccssxr 13161 |
. . . . 5
⊢
(0[,]+∞) ⊆ ℝ* |
18 | | simpl 483 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝜑) |
19 | | fzelp1 13307 |
. . . . . . 7
⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
20 | 19 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
21 | | sge0p1.2 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ (0[,]+∞)) |
22 | 18, 20, 21 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ (0[,]+∞)) |
23 | 17, 22 | sselid 3924 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈
ℝ*) |
24 | | iccgelb 13134 |
. . . . 5
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧
𝐴 ∈ (0[,]+∞))
→ 0 ≤ 𝐴) |
25 | 14, 16, 22, 24 | syl3anc 1370 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ 𝐴) |
26 | | iccleub 13133 |
. . . . 5
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧
𝐴 ∈ (0[,]+∞))
→ 𝐴 ≤
+∞) |
27 | 14, 16, 22, 26 | syl3anc 1370 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ≤ +∞) |
28 | 14, 16, 23, 25, 27 | eliccxrd 43036 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ (0[,]+∞)) |
29 | | simpl 483 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ {(𝑁 + 1)}) → 𝜑) |
30 | | elsni 4584 |
. . . . . 6
⊢ (𝑘 ∈ {(𝑁 + 1)} → 𝑘 = (𝑁 + 1)) |
31 | 30 | adantl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ {(𝑁 + 1)}) → 𝑘 = (𝑁 + 1)) |
32 | | simpr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 = (𝑁 + 1)) → 𝑘 = (𝑁 + 1)) |
33 | | peano2uz 12640 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
34 | | eluzfz2 13263 |
. . . . . . . 8
⊢ ((𝑁 + 1) ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
35 | 1, 33, 34 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
36 | 35 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 = (𝑁 + 1)) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
37 | 32, 36 | eqeltrd 2841 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 = (𝑁 + 1)) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
38 | 29, 31, 37 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ {(𝑁 + 1)}) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
39 | 29, 38, 21 | syl2anc 584 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ {(𝑁 + 1)}) → 𝐴 ∈ (0[,]+∞)) |
40 | 6, 8, 10, 12, 28, 39 | sge0splitmpt 43920 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ ((𝑀...𝑁) ∪ {(𝑁 + 1)}) ↦ 𝐴)) =
((Σ^‘(𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)) +𝑒
(Σ^‘(𝑘 ∈ {(𝑁 + 1)} ↦ 𝐴)))) |
41 | | ovex 7304 |
. . . . 5
⊢ (𝑁 + 1) ∈ V |
42 | 41 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝑁 + 1) ∈ V) |
43 | | id 22 |
. . . . 5
⊢ (𝜑 → 𝜑) |
44 | | eleq1 2828 |
. . . . . . . . 9
⊢ (𝑘 = (𝑁 + 1) → (𝑘 ∈ (𝑀...(𝑁 + 1)) ↔ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1)))) |
45 | 44 | anbi2d 629 |
. . . . . . . 8
⊢ (𝑘 = (𝑁 + 1) → ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) ↔ (𝜑 ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))))) |
46 | | sge0p1.3 |
. . . . . . . . 9
⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵) |
47 | 46 | eleq1d 2825 |
. . . . . . . 8
⊢ (𝑘 = (𝑁 + 1) → (𝐴 ∈ (0[,]+∞) ↔ 𝐵 ∈
(0[,]+∞))) |
48 | 45, 47 | imbi12d 345 |
. . . . . . 7
⊢ (𝑘 = (𝑁 + 1) → (((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → 𝐵 ∈ (0[,]+∞)))) |
49 | 48, 21 | vtoclg 3504 |
. . . . . 6
⊢ ((𝑁 + 1) ∈ V → ((𝜑 ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → 𝐵 ∈ (0[,]+∞))) |
50 | 41, 49 | ax-mp 5 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) → 𝐵 ∈ (0[,]+∞)) |
51 | 43, 35, 50 | syl2anc 584 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
52 | 42, 51, 46 | sge0snmpt 43892 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ {(𝑁 + 1)} ↦ 𝐴)) = 𝐵) |
53 | 52 | oveq2d 7287 |
. 2
⊢ (𝜑 →
((Σ^‘(𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)) +𝑒
(Σ^‘(𝑘 ∈ {(𝑁 + 1)} ↦ 𝐴))) =
((Σ^‘(𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)) +𝑒 𝐵)) |
54 | 5, 40, 53 | 3eqtrd 2784 |
1
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ (𝑀...(𝑁 + 1)) ↦ 𝐴)) =
((Σ^‘(𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)) +𝑒 𝐵)) |