Proof of Theorem telfsumo
Step | Hyp | Ref
| Expression |
1 | | telfsumo.3 |
. . . . . . . 8
⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) |
2 | 1 | eleq1d 2891 |
. . . . . . 7
⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐷 ∈ ℂ)) |
3 | | telfsumo.6 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
4 | 3 | ralrimiva 3175 |
. . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
5 | | telfsumo.5 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
6 | | eluzfz1 12641 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
7 | 5, 6 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
8 | 2, 4, 7 | rspcdva 3532 |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ ℂ) |
9 | 8 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → 𝐷 ∈ ℂ) |
10 | 9 | subidd 10701 |
. . . 4
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (𝐷 − 𝐷) = 0) |
11 | | sum0 14829 |
. . . 4
⊢
Σ𝑗 ∈
∅ (𝐵 − 𝐶) = 0 |
12 | 10, 11 | syl6reqr 2880 |
. . 3
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → Σ𝑗 ∈ ∅ (𝐵 − 𝐶) = (𝐷 − 𝐷)) |
13 | | oveq2 6913 |
. . . . . 6
⊢ (𝑁 = 𝑀 → (𝑀..^𝑁) = (𝑀..^𝑀)) |
14 | 13 | adantl 475 |
. . . . 5
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (𝑀..^𝑁) = (𝑀..^𝑀)) |
15 | | fzo0 12787 |
. . . . 5
⊢ (𝑀..^𝑀) = ∅ |
16 | 14, 15 | syl6eq 2877 |
. . . 4
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (𝑀..^𝑁) = ∅) |
17 | 16 | sumeq1d 14808 |
. . 3
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 − 𝐶) = Σ𝑗 ∈ ∅ (𝐵 − 𝐶)) |
18 | | eqeq1 2829 |
. . . . . . . 8
⊢ (𝑘 = 𝑁 → (𝑘 = 𝑀 ↔ 𝑁 = 𝑀)) |
19 | | telfsumo.4 |
. . . . . . . . 9
⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸) |
20 | 19 | eqeq1d 2827 |
. . . . . . . 8
⊢ (𝑘 = 𝑁 → (𝐴 = 𝐷 ↔ 𝐸 = 𝐷)) |
21 | 18, 20 | imbi12d 336 |
. . . . . . 7
⊢ (𝑘 = 𝑁 → ((𝑘 = 𝑀 → 𝐴 = 𝐷) ↔ (𝑁 = 𝑀 → 𝐸 = 𝐷))) |
22 | 21, 1 | vtoclg 3482 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑁 = 𝑀 → 𝐸 = 𝐷)) |
23 | 22 | imp 397 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ 𝑁 = 𝑀) → 𝐸 = 𝐷) |
24 | 5, 23 | sylan 575 |
. . . 4
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → 𝐸 = 𝐷) |
25 | 24 | oveq2d 6921 |
. . 3
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (𝐷 − 𝐸) = (𝐷 − 𝐷)) |
26 | 12, 17, 25 | 3eqtr4d 2871 |
. 2
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 − 𝐶) = (𝐷 − 𝐸)) |
27 | | fzofi 13068 |
. . . . . 6
⊢ (𝑀..^𝑁) ∈ Fin |
28 | 27 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin) |
29 | | telfsumo.1 |
. . . . . . 7
⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵) |
30 | 29 | eleq1d 2891 |
. . . . . 6
⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
31 | 4 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
32 | | elfzofz 12780 |
. . . . . . 7
⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁)) |
33 | 32 | adantl 475 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...𝑁)) |
34 | 30, 31, 33 | rspcdva 3532 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐵 ∈ ℂ) |
35 | | telfsumo.2 |
. . . . . . 7
⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶) |
36 | 35 | eleq1d 2891 |
. . . . . 6
⊢ (𝑘 = (𝑗 + 1) → (𝐴 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
37 | | fzofzp1 12860 |
. . . . . . 7
⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁)) |
38 | 37 | adantl 475 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (𝑀...𝑁)) |
39 | 36, 31, 38 | rspcdva 3532 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐶 ∈ ℂ) |
40 | 28, 34, 39 | fsumsub 14894 |
. . . 4
⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 − 𝐶) = (Σ𝑗 ∈ (𝑀..^𝑁)𝐵 − Σ𝑗 ∈ (𝑀..^𝑁)𝐶)) |
41 | 40 | adantr 474 |
. . 3
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 − 𝐶) = (Σ𝑗 ∈ (𝑀..^𝑁)𝐵 − Σ𝑗 ∈ (𝑀..^𝑁)𝐶)) |
42 | 29 | cbvsumv 14803 |
. . . . . 6
⊢
Σ𝑘 ∈
(𝑀..^𝑁)𝐴 = Σ𝑗 ∈ (𝑀..^𝑁)𝐵 |
43 | | eluzel2 11973 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
44 | 5, 43 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℤ) |
45 | | eluzp1m1 11992 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈
(ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈
(ℤ≥‘𝑀)) |
46 | 44, 45 | sylan 575 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈
(ℤ≥‘𝑀)) |
47 | | eluzelz 11978 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
48 | 5, 47 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℤ) |
49 | 48 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ ℤ) |
50 | | fzoval 12766 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
52 | | fzossfz 12783 |
. . . . . . . . . . 11
⊢ (𝑀..^𝑁) ⊆ (𝑀...𝑁) |
53 | 51, 52 | syl6eqssr 3881 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑀...(𝑁 − 1)) ⊆ (𝑀...𝑁)) |
54 | 53 | sselda 3827 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝑘 ∈ (𝑀...𝑁)) |
55 | 3 | adantlr 706 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
56 | 54, 55 | syldan 585 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝐴 ∈ ℂ) |
57 | 46, 56, 1 | fsum1p 14859 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 = (𝐷 + Σ𝑘 ∈ ((𝑀 + 1)...(𝑁 − 1))𝐴)) |
58 | 51 | sumeq1d 14808 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴) |
59 | | fzoval 12766 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℤ → ((𝑀 + 1)..^𝑁) = ((𝑀 + 1)...(𝑁 − 1))) |
60 | 49, 59 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑀 + 1)..^𝑁) = ((𝑀 + 1)...(𝑁 − 1))) |
61 | 60 | sumeq1d 14808 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 1)...(𝑁 − 1))𝐴) |
62 | 61 | oveq2d 6921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐷 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴) = (𝐷 + Σ𝑘 ∈ ((𝑀 + 1)...(𝑁 − 1))𝐴)) |
63 | 57, 58, 62 | 3eqtr4d 2871 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (𝐷 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴)) |
64 | 42, 63 | syl5eqr 2875 |
. . . . 5
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑗 ∈ (𝑀..^𝑁)𝐵 = (𝐷 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴)) |
65 | | simpr 479 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) |
66 | | fzp1ss 12685 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
67 | 44, 66 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) |
68 | 67 | sselda 3827 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁)) |
69 | 68, 3 | syldan 585 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ ℂ) |
70 | 69 | adantlr 706 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ ℂ) |
71 | 65, 70, 19 | fsumm1 14857 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 = (Σ𝑘 ∈ ((𝑀 + 1)...(𝑁 − 1))𝐴 + 𝐸)) |
72 | | 1zzd 11736 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈
ℤ) |
73 | 44 | peano2zd 11813 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
74 | 72, 73, 48, 69, 35 | fsumshftm 14887 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 = Σ𝑗 ∈ (((𝑀 + 1) − 1)...(𝑁 − 1))𝐶) |
75 | 44 | zcnd 11811 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℂ) |
76 | | ax-1cn 10310 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
77 | | pncan 10607 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑀 + 1)
− 1) = 𝑀) |
78 | 75, 76, 77 | sylancl 580 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀) |
79 | 78 | oveq1d 6920 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝑀 + 1) − 1)...(𝑁 − 1)) = (𝑀...(𝑁 − 1))) |
80 | 48, 50 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
81 | 79, 80 | eqtr4d 2864 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑀 + 1) − 1)...(𝑁 − 1)) = (𝑀..^𝑁)) |
82 | 81 | sumeq1d 14808 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑗 ∈ (((𝑀 + 1) − 1)...(𝑁 − 1))𝐶 = Σ𝑗 ∈ (𝑀..^𝑁)𝐶) |
83 | 74, 82 | eqtrd 2861 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 = Σ𝑗 ∈ (𝑀..^𝑁)𝐶) |
84 | 83 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 = Σ𝑗 ∈ (𝑀..^𝑁)𝐶) |
85 | 48, 59 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑀 + 1)..^𝑁) = ((𝑀 + 1)...(𝑁 − 1))) |
86 | 85 | sumeq1d 14808 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 1)...(𝑁 − 1))𝐴) |
87 | 86 | oveq1d 6920 |
. . . . . . . 8
⊢ (𝜑 → (Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴 + 𝐸) = (Σ𝑘 ∈ ((𝑀 + 1)...(𝑁 − 1))𝐴 + 𝐸)) |
88 | | fzofi 13068 |
. . . . . . . . . . 11
⊢ ((𝑀 + 1)..^𝑁) ∈ Fin |
89 | 88 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑀 + 1)..^𝑁) ∈ Fin) |
90 | | elfzofz 12780 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ((𝑀 + 1)..^𝑁) → 𝑘 ∈ ((𝑀 + 1)...𝑁)) |
91 | 90, 69 | sylan2 586 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)..^𝑁)) → 𝐴 ∈ ℂ) |
92 | 89, 91 | fsumcl 14841 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴 ∈ ℂ) |
93 | 19 | eleq1d 2891 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
94 | | eluzfz2 12642 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
95 | 5, 94 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
96 | 93, 4, 95 | rspcdva 3532 |
. . . . . . . . 9
⊢ (𝜑 → 𝐸 ∈ ℂ) |
97 | 92, 96 | addcomd 10557 |
. . . . . . . 8
⊢ (𝜑 → (Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴 + 𝐸) = (𝐸 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴)) |
98 | 87, 97 | eqtr3d 2863 |
. . . . . . 7
⊢ (𝜑 → (Σ𝑘 ∈ ((𝑀 + 1)...(𝑁 − 1))𝐴 + 𝐸) = (𝐸 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴)) |
99 | 98 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (Σ𝑘 ∈ ((𝑀 + 1)...(𝑁 − 1))𝐴 + 𝐸) = (𝐸 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴)) |
100 | 71, 84, 99 | 3eqtr3d 2869 |
. . . . 5
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑗 ∈ (𝑀..^𝑁)𝐶 = (𝐸 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴)) |
101 | 64, 100 | oveq12d 6923 |
. . . 4
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (Σ𝑗 ∈ (𝑀..^𝑁)𝐵 − Σ𝑗 ∈ (𝑀..^𝑁)𝐶) = ((𝐷 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴) − (𝐸 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴))) |
102 | 8, 96, 92 | pnpcan2d 10751 |
. . . . 5
⊢ (𝜑 → ((𝐷 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴) − (𝐸 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴)) = (𝐷 − 𝐸)) |
103 | 102 | adantr 474 |
. . . 4
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐷 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴) − (𝐸 + Σ𝑘 ∈ ((𝑀 + 1)..^𝑁)𝐴)) = (𝐷 − 𝐸)) |
104 | 101, 103 | eqtrd 2861 |
. . 3
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (Σ𝑗 ∈ (𝑀..^𝑁)𝐵 − Σ𝑗 ∈ (𝑀..^𝑁)𝐶) = (𝐷 − 𝐸)) |
105 | 41, 104 | eqtrd 2861 |
. 2
⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 − 𝐶) = (𝐷 − 𝐸)) |
106 | | uzp1 12003 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))) |
107 | 5, 106 | syl 17 |
. 2
⊢ (𝜑 → (𝑁 = 𝑀 ∨ 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))) |
108 | 26, 105, 107 | mpjaodan 986 |
1
⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 − 𝐶) = (𝐷 − 𝐸)) |