Step | Hyp | Ref
| Expression |
1 | | eqeq2 2621 |
. 2
⊢
((#‘𝐷) =
if(𝐴 = 1 , (#‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (#‘𝐷) ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (#‘𝐷), 0))) |
2 | | eqeq2 2621 |
. 2
⊢ (0 =
if(𝐴 = 1 , (#‘𝐷), 0) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0 ↔ Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (#‘𝐷), 0))) |
3 | | fveq2 6103 |
. . . . . 6
⊢ (𝐴 = 1 → (𝑥‘𝐴) = (𝑥‘ 1 )) |
4 | | sumdchr.g |
. . . . . . . . 9
⊢ 𝐺 = (DChr‘𝑁) |
5 | | sumdchr2.z |
. . . . . . . . 9
⊢ 𝑍 =
(ℤ/nℤ‘𝑁) |
6 | | sumdchr.d |
. . . . . . . . 9
⊢ 𝐷 = (Base‘𝐺) |
7 | 4, 5, 6 | dchrmhm 24766 |
. . . . . . . 8
⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom
(mulGrp‘ℂfld)) |
8 | | simpr 476 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) |
9 | 7, 8 | sseldi 3566 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ((mulGrp‘𝑍) MndHom
(mulGrp‘ℂfld))) |
10 | | eqid 2610 |
. . . . . . . . 9
⊢
(mulGrp‘𝑍) =
(mulGrp‘𝑍) |
11 | | sumdchr2.1 |
. . . . . . . . 9
⊢ 1 =
(1r‘𝑍) |
12 | 10, 11 | ringidval 18326 |
. . . . . . . 8
⊢ 1 =
(0g‘(mulGrp‘𝑍)) |
13 | | eqid 2610 |
. . . . . . . . 9
⊢
(mulGrp‘ℂfld) =
(mulGrp‘ℂfld) |
14 | | cnfld1 19590 |
. . . . . . . . 9
⊢ 1 =
(1r‘ℂfld) |
15 | 13, 14 | ringidval 18326 |
. . . . . . . 8
⊢ 1 =
(0g‘(mulGrp‘ℂfld)) |
16 | 12, 15 | mhm0 17166 |
. . . . . . 7
⊢ (𝑥 ∈ ((mulGrp‘𝑍) MndHom
(mulGrp‘ℂfld)) → (𝑥‘ 1 ) = 1) |
17 | 9, 16 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥‘ 1 ) = 1) |
18 | 3, 17 | sylan9eqr 2666 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝐴 = 1 ) → (𝑥‘𝐴) = 1) |
19 | 18 | an32s 842 |
. . . 4
⊢ (((𝜑 ∧ 𝐴 = 1 ) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) = 1) |
20 | 19 | sumeq2dv 14281 |
. . 3
⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = Σ𝑥 ∈ 𝐷 1) |
21 | | sumdchr2.n |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℕ) |
22 | 4, 6 | dchrfi 24780 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin) |
23 | 21, 22 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ Fin) |
24 | | ax-1cn 9873 |
. . . . . 6
⊢ 1 ∈
ℂ |
25 | | fsumconst 14364 |
. . . . . 6
⊢ ((𝐷 ∈ Fin ∧ 1 ∈
ℂ) → Σ𝑥
∈ 𝐷 1 =
((#‘𝐷) ·
1)) |
26 | 23, 24, 25 | sylancl 693 |
. . . . 5
⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = ((#‘𝐷) · 1)) |
27 | | hashcl 13009 |
. . . . . . . 8
⊢ (𝐷 ∈ Fin →
(#‘𝐷) ∈
ℕ0) |
28 | 21, 22, 27 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (#‘𝐷) ∈
ℕ0) |
29 | 28 | nn0cnd 11230 |
. . . . . 6
⊢ (𝜑 → (#‘𝐷) ∈ ℂ) |
30 | 29 | mulid1d 9936 |
. . . . 5
⊢ (𝜑 → ((#‘𝐷) · 1) = (#‘𝐷)) |
31 | 26, 30 | eqtrd 2644 |
. . . 4
⊢ (𝜑 → Σ𝑥 ∈ 𝐷 1 = (#‘𝐷)) |
32 | 31 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 1 = (#‘𝐷)) |
33 | 20, 32 | eqtrd 2644 |
. 2
⊢ ((𝜑 ∧ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = (#‘𝐷)) |
34 | | df-ne 2782 |
. . 3
⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1 ) |
35 | | sumdchr2.b |
. . . . 5
⊢ 𝐵 = (Base‘𝑍) |
36 | 21 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝑁 ∈ ℕ) |
37 | | simpr 476 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝐴 ≠ 1 ) |
38 | | sumdchr2.x |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝐵) |
39 | 38 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → 𝐴 ∈ 𝐵) |
40 | 4, 5, 6, 35, 11, 36, 37, 39 | dchrpt 24792 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → ∃𝑦 ∈ 𝐷 (𝑦‘𝐴) ≠ 1) |
41 | 36 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑁 ∈ ℕ) |
42 | 41, 22 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐷 ∈ Fin) |
43 | | simpr 476 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) |
44 | 4, 5, 6, 35, 43 | dchrf 24767 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐵⟶ℂ) |
45 | 39 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐴 ∈ 𝐵) |
46 | 45 | adantr 480 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝐵) |
47 | 44, 46 | ffvelrnd 6268 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑥‘𝐴) ∈ ℂ) |
48 | 42, 47 | fsumcl 14311 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) ∈ ℂ) |
49 | | 0cnd 9912 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 0 ∈
ℂ) |
50 | | simprl 790 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦 ∈ 𝐷) |
51 | 4, 5, 6, 35, 50 | dchrf 24767 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝑦:𝐵⟶ℂ) |
52 | 51, 45 | ffvelrnd 6268 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ∈ ℂ) |
53 | | subcl 10159 |
. . . . . 6
⊢ (((𝑦‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ)
→ ((𝑦‘𝐴) − 1) ∈
ℂ) |
54 | 52, 24, 53 | sylancl 693 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) − 1) ∈
ℂ) |
55 | | simprr 792 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (𝑦‘𝐴) ≠ 1) |
56 | | subeq0 10186 |
. . . . . . . 8
⊢ (((𝑦‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((𝑦‘𝐴) − 1) = 0 ↔ (𝑦‘𝐴) = 1)) |
57 | 52, 24, 56 | sylancl 693 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) = 0 ↔ (𝑦‘𝐴) = 1)) |
58 | 57 | necon3bid 2826 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) ≠ 0 ↔ (𝑦‘𝐴) ≠ 1)) |
59 | 55, 58 | mpbird 246 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) − 1) ≠ 0) |
60 | | oveq2 6557 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑥 → (𝑦(+g‘𝐺)𝑧) = (𝑦(+g‘𝐺)𝑥)) |
61 | 60 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑥 → ((𝑦(+g‘𝐺)𝑧)‘𝐴) = ((𝑦(+g‘𝐺)𝑥)‘𝐴)) |
62 | 61 | cbvsumv 14274 |
. . . . . . . . . 10
⊢
Σ𝑧 ∈
𝐷 ((𝑦(+g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦(+g‘𝐺)𝑥)‘𝐴) |
63 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢
(+g‘𝐺) = (+g‘𝐺) |
64 | 50 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 ∈ 𝐷) |
65 | 4, 5, 6, 63, 64, 43 | dchrmul 24773 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → (𝑦(+g‘𝐺)𝑥) = (𝑦 ∘𝑓 · 𝑥)) |
66 | 65 | fveq1d 6105 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+g‘𝐺)𝑥)‘𝐴) = ((𝑦 ∘𝑓 · 𝑥)‘𝐴)) |
67 | 51 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦:𝐵⟶ℂ) |
68 | | ffn 5958 |
. . . . . . . . . . . . . 14
⊢ (𝑦:𝐵⟶ℂ → 𝑦 Fn 𝐵) |
69 | 67, 68 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑦 Fn 𝐵) |
70 | | ffn 5958 |
. . . . . . . . . . . . . 14
⊢ (𝑥:𝐵⟶ℂ → 𝑥 Fn 𝐵) |
71 | 44, 70 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝑥 Fn 𝐵) |
72 | | fvex 6113 |
. . . . . . . . . . . . . . 15
⊢
(Base‘𝑍)
∈ V |
73 | 35, 72 | eqeltri 2684 |
. . . . . . . . . . . . . 14
⊢ 𝐵 ∈ V |
74 | 73 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ V) |
75 | | fnfvof 6809 |
. . . . . . . . . . . . 13
⊢ (((𝑦 Fn 𝐵 ∧ 𝑥 Fn 𝐵) ∧ (𝐵 ∈ V ∧ 𝐴 ∈ 𝐵)) → ((𝑦 ∘𝑓 · 𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴))) |
76 | 69, 71, 74, 46, 75 | syl22anc 1319 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦 ∘𝑓 · 𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴))) |
77 | 66, 76 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑥 ∈ 𝐷) → ((𝑦(+g‘𝐺)𝑥)‘𝐴) = ((𝑦‘𝐴) · (𝑥‘𝐴))) |
78 | 77 | sumeq2dv 14281 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 ((𝑦(+g‘𝐺)𝑥)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴))) |
79 | 62, 78 | syl5eq 2656 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑧 ∈ 𝐷 ((𝑦(+g‘𝐺)𝑧)‘𝐴) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴))) |
80 | | fveq1 6102 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑦(+g‘𝐺)𝑧) → (𝑥‘𝐴) = ((𝑦(+g‘𝐺)𝑧)‘𝐴)) |
81 | 4 | dchrabl 24779 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
82 | | ablgrp 18021 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
83 | 41, 81, 82 | 3syl 18 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 𝐺 ∈ Grp) |
84 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+g‘𝐺)𝑏))) = (𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+g‘𝐺)𝑏))) |
85 | 84, 6, 63 | grplactf1o 17342 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐷) → ((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+g‘𝐺)𝑏)))‘𝑦):𝐷–1-1-onto→𝐷) |
86 | 83, 50, 85 | syl2anc 691 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+g‘𝐺)𝑏)))‘𝑦):𝐷–1-1-onto→𝐷) |
87 | 84, 6 | grplactval 17340 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+g‘𝐺)𝑏)))‘𝑦)‘𝑧) = (𝑦(+g‘𝐺)𝑧)) |
88 | 50, 87 | sylan 487 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) ∧ 𝑧 ∈ 𝐷) → (((𝑎 ∈ 𝐷 ↦ (𝑏 ∈ 𝐷 ↦ (𝑎(+g‘𝐺)𝑏)))‘𝑦)‘𝑧) = (𝑦(+g‘𝐺)𝑧)) |
89 | 80, 42, 86, 88, 47 | fsumf1o 14301 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = Σ𝑧 ∈ 𝐷 ((𝑦(+g‘𝐺)𝑧)‘𝐴)) |
90 | 42, 52, 47 | fsummulc2 14358 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 ((𝑦‘𝐴) · (𝑥‘𝐴))) |
91 | 79, 89, 90 | 3eqtr4rd 2655 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → ((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) |
92 | 48 | mulid2d 9937 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) |
93 | 91, 92 | oveq12d 6567 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) − (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))) = (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) − Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))) |
94 | 48 | subidd 10259 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) − Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = 0) |
95 | 93, 94 | eqtrd 2644 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) − (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴))) = 0) |
96 | 24 | a1i 11 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → 1 ∈
ℂ) |
97 | 52, 96, 48 | subdird 10366 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = (((𝑦‘𝐴) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) − (1 · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)))) |
98 | 54 | mul01d 10114 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · 0) =
0) |
99 | 95, 97, 98 | 3eqtr4d 2654 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → (((𝑦‘𝐴) − 1) · Σ𝑥 ∈ 𝐷 (𝑥‘𝐴)) = (((𝑦‘𝐴) − 1) · 0)) |
100 | 48, 49, 54, 59, 99 | mulcanad 10541 |
. . . 4
⊢ (((𝜑 ∧ 𝐴 ≠ 1 ) ∧ (𝑦 ∈ 𝐷 ∧ (𝑦‘𝐴) ≠ 1)) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0) |
101 | 40, 100 | rexlimddv 3017 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ≠ 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0) |
102 | 34, 101 | sylan2br 492 |
. 2
⊢ ((𝜑 ∧ ¬ 𝐴 = 1 ) → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = 0) |
103 | 1, 2, 33, 102 | ifbothda 4073 |
1
⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (#‘𝐷), 0)) |