Step | Hyp | Ref
| Expression |
1 | | eqeq2 2749 |
. 2
⊢
((ϕ‘𝑁) =
if(𝑋 = 1 , (ϕ‘𝑁), 0) → (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = (ϕ‘𝑁) ↔ Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0))) |
2 | | eqeq2 2749 |
. 2
⊢ (0 =
if(𝑋 = 1 , (ϕ‘𝑁), 0) → (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0 ↔ Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0))) |
3 | | fveq1 6716 |
. . . . . 6
⊢ (𝑋 = 1 → (𝑋‘𝑎) = ( 1 ‘𝑎)) |
4 | | dchrsum.g |
. . . . . . 7
⊢ 𝐺 = (DChr‘𝑁) |
5 | | dchrsum.z |
. . . . . . 7
⊢ 𝑍 =
(ℤ/nℤ‘𝑁) |
6 | | dchrsum.1 |
. . . . . . 7
⊢ 1 =
(0g‘𝐺) |
7 | | dchrsum2.u |
. . . . . . 7
⊢ 𝑈 = (Unit‘𝑍) |
8 | | dchrsum.x |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ 𝐷) |
9 | | dchrsum.d |
. . . . . . . . . 10
⊢ 𝐷 = (Base‘𝐺) |
10 | 4, 9 | dchrrcl 26121 |
. . . . . . . . 9
⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
11 | 8, 10 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℕ) |
12 | 11 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → 𝑁 ∈ ℕ) |
13 | | simpr 488 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → 𝑎 ∈ 𝑈) |
14 | 4, 5, 6, 7, 12, 13 | dchr1 26138 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → ( 1 ‘𝑎) = 1) |
15 | 3, 14 | sylan9eqr 2800 |
. . . . 5
⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑋 = 1 ) → (𝑋‘𝑎) = 1) |
16 | 15 | an32s 652 |
. . . 4
⊢ (((𝜑 ∧ 𝑋 = 1 ) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) = 1) |
17 | 16 | sumeq2dv 15267 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑎 ∈ 𝑈 1) |
18 | 5, 7 | znunithash 20529 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ →
(♯‘𝑈) =
(ϕ‘𝑁)) |
19 | 11, 18 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (♯‘𝑈) = (ϕ‘𝑁)) |
20 | 11 | phicld 16325 |
. . . . . . . . 9
⊢ (𝜑 → (ϕ‘𝑁) ∈
ℕ) |
21 | 20 | nnnn0d 12150 |
. . . . . . . 8
⊢ (𝜑 → (ϕ‘𝑁) ∈
ℕ0) |
22 | 19, 21 | eqeltrd 2838 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝑈) ∈
ℕ0) |
23 | 7 | fvexi 6731 |
. . . . . . . 8
⊢ 𝑈 ∈ V |
24 | | hashclb 13925 |
. . . . . . . 8
⊢ (𝑈 ∈ V → (𝑈 ∈ Fin ↔
(♯‘𝑈) ∈
ℕ0)) |
25 | 23, 24 | ax-mp 5 |
. . . . . . 7
⊢ (𝑈 ∈ Fin ↔
(♯‘𝑈) ∈
ℕ0) |
26 | 22, 25 | sylibr 237 |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ Fin) |
27 | | ax-1cn 10787 |
. . . . . 6
⊢ 1 ∈
ℂ |
28 | | fsumconst 15354 |
. . . . . 6
⊢ ((𝑈 ∈ Fin ∧ 1 ∈
ℂ) → Σ𝑎
∈ 𝑈 1 =
((♯‘𝑈) ·
1)) |
29 | 26, 27, 28 | sylancl 589 |
. . . . 5
⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = ((♯‘𝑈) · 1)) |
30 | 19 | oveq1d 7228 |
. . . . 5
⊢ (𝜑 → ((♯‘𝑈) · 1) =
((ϕ‘𝑁) ·
1)) |
31 | 20 | nncnd 11846 |
. . . . . 6
⊢ (𝜑 → (ϕ‘𝑁) ∈
ℂ) |
32 | 31 | mulid1d 10850 |
. . . . 5
⊢ (𝜑 → ((ϕ‘𝑁) · 1) =
(ϕ‘𝑁)) |
33 | 29, 30, 32 | 3eqtrd 2781 |
. . . 4
⊢ (𝜑 → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁)) |
34 | 33 | adantr 484 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 1 = (ϕ‘𝑁)) |
35 | 17, 34 | eqtrd 2777 |
. 2
⊢ ((𝜑 ∧ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = (ϕ‘𝑁)) |
36 | 4 | dchrabl 26135 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
37 | | ablgrp 19175 |
. . . . . . . 8
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
38 | 9, 6 | grpidcl 18395 |
. . . . . . . 8
⊢ (𝐺 ∈ Grp → 1 ∈ 𝐷) |
39 | 11, 36, 37, 38 | 4syl 19 |
. . . . . . 7
⊢ (𝜑 → 1 ∈ 𝐷) |
40 | 4, 5, 9, 7, 8, 39 | dchreq 26139 |
. . . . . 6
⊢ (𝜑 → (𝑋 = 1 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘))) |
41 | 40 | notbid 321 |
. . . . 5
⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘))) |
42 | | rexnal 3160 |
. . . . 5
⊢
(∃𝑘 ∈
𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) ↔ ¬ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = ( 1 ‘𝑘)) |
43 | 41, 42 | bitr4di 292 |
. . . 4
⊢ (𝜑 → (¬ 𝑋 = 1 ↔ ∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘))) |
44 | | df-ne 2941 |
. . . . . 6
⊢ ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) ↔ ¬ (𝑋‘𝑘) = ( 1 ‘𝑘)) |
45 | 11 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑁 ∈ ℕ) |
46 | | simpr 488 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝑈) |
47 | 4, 5, 6, 7, 45, 46 | dchr1 26138 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ( 1 ‘𝑘) = 1) |
48 | 47 | neeq2d 3001 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) ↔ (𝑋‘𝑘) ≠ 1)) |
49 | 26 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑈 ∈ Fin) |
50 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(Base‘𝑍) =
(Base‘𝑍) |
51 | 4, 5, 9, 50, 8 | dchrf 26123 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
52 | 50, 7 | unitss 19678 |
. . . . . . . . . . . . 13
⊢ 𝑈 ⊆ (Base‘𝑍) |
53 | 52 | sseli 3896 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ 𝑈 → 𝑎 ∈ (Base‘𝑍)) |
54 | | ffvelrn 6902 |
. . . . . . . . . . . 12
⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑋‘𝑎) ∈ ℂ) |
55 | 51, 53, 54 | syl2an 599 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ) |
56 | 55 | adantlr 715 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘𝑎) ∈ ℂ) |
57 | 49, 56 | fsumcl 15297 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) ∈ ℂ) |
58 | | 0cnd 10826 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 0 ∈
ℂ) |
59 | 51 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑋:(Base‘𝑍)⟶ℂ) |
60 | | simprl 771 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ 𝑈) |
61 | 52, 60 | sseldi 3899 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 𝑘 ∈ (Base‘𝑍)) |
62 | 59, 61 | ffvelrnd 6905 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ∈ ℂ) |
63 | | subcl 11077 |
. . . . . . . . . 10
⊢ (((𝑋‘𝑘) ∈ ℂ ∧ 1 ∈ ℂ)
→ ((𝑋‘𝑘) − 1) ∈
ℂ) |
64 | 62, 27, 63 | sylancl 589 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ∈ ℂ) |
65 | | simprr 773 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (𝑋‘𝑘) ≠ 1) |
66 | | subeq0 11104 |
. . . . . . . . . . . 12
⊢ (((𝑋‘𝑘) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((𝑋‘𝑘) − 1) = 0 ↔ (𝑋‘𝑘) = 1)) |
67 | 62, 27, 66 | sylancl 589 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) = 0 ↔ (𝑋‘𝑘) = 1)) |
68 | 67 | necon3bid 2985 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) ≠ 0 ↔ (𝑋‘𝑘) ≠ 1)) |
69 | 65, 68 | mpbird 260 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) − 1) ≠ 0) |
70 | | oveq2 7221 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑎 → (𝑘(.r‘𝑍)𝑥) = (𝑘(.r‘𝑍)𝑎)) |
71 | 70 | fveq2d 6721 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑎 → (𝑋‘(𝑘(.r‘𝑍)𝑥)) = (𝑋‘(𝑘(.r‘𝑍)𝑎))) |
72 | 71 | cbvsumv 15260 |
. . . . . . . . . . . . . 14
⊢
Σ𝑥 ∈
𝑈 (𝑋‘(𝑘(.r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(.r‘𝑍)𝑎)) |
73 | 4, 5, 9 | dchrmhm 26122 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom
(mulGrp‘ℂfld)) |
74 | 73, 8 | sseldi 3899 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom
(mulGrp‘ℂfld))) |
75 | 74 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom
(mulGrp‘ℂfld))) |
76 | 61 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑘 ∈ (Base‘𝑍)) |
77 | 53 | adantl 485 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → 𝑎 ∈ (Base‘𝑍)) |
78 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
(mulGrp‘𝑍) =
(mulGrp‘𝑍) |
79 | 78, 50 | mgpbas 19510 |
. . . . . . . . . . . . . . . . 17
⊢
(Base‘𝑍) =
(Base‘(mulGrp‘𝑍)) |
80 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
(.r‘𝑍) = (.r‘𝑍) |
81 | 78, 80 | mgpplusg 19508 |
. . . . . . . . . . . . . . . . 17
⊢
(.r‘𝑍) = (+g‘(mulGrp‘𝑍)) |
82 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
(mulGrp‘ℂfld) =
(mulGrp‘ℂfld) |
83 | | cnfldmul 20369 |
. . . . . . . . . . . . . . . . . 18
⊢ ·
= (.r‘ℂfld) |
84 | 82, 83 | mgpplusg 19508 |
. . . . . . . . . . . . . . . . 17
⊢ ·
= (+g‘(mulGrp‘ℂfld)) |
85 | 79, 81, 84 | mhmlin 18225 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom
(mulGrp‘ℂfld)) ∧ 𝑘 ∈ (Base‘𝑍) ∧ 𝑎 ∈ (Base‘𝑍)) → (𝑋‘(𝑘(.r‘𝑍)𝑎)) = ((𝑋‘𝑘) · (𝑋‘𝑎))) |
86 | 75, 76, 77, 85 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑎 ∈ 𝑈) → (𝑋‘(𝑘(.r‘𝑍)𝑎)) = ((𝑋‘𝑘) · (𝑋‘𝑎))) |
87 | 86 | sumeq2dv 15267 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘(𝑘(.r‘𝑍)𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎))) |
88 | 72, 87 | syl5eq 2790 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(.r‘𝑍)𝑥)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎))) |
89 | | fveq2 6717 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = (𝑘(.r‘𝑍)𝑥) → (𝑋‘𝑎) = (𝑋‘(𝑘(.r‘𝑍)𝑥))) |
90 | 11 | nnnn0d 12150 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
91 | 5 | zncrng 20509 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℕ0
→ 𝑍 ∈
CRing) |
92 | | crngring 19574 |
. . . . . . . . . . . . . . . 16
⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) |
93 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢
((mulGrp‘𝑍)
↾s 𝑈) =
((mulGrp‘𝑍)
↾s 𝑈) |
94 | 7, 93 | unitgrp 19685 |
. . . . . . . . . . . . . . . 16
⊢ (𝑍 ∈ Ring →
((mulGrp‘𝑍)
↾s 𝑈)
∈ Grp) |
95 | 90, 91, 92, 94 | 4syl 19 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((mulGrp‘𝑍) ↾s 𝑈) ∈ Grp) |
96 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(.r‘𝑍)𝑐))) = (𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(.r‘𝑍)𝑐))) |
97 | 7, 93 | unitgrpbas 19684 |
. . . . . . . . . . . . . . . 16
⊢ 𝑈 =
(Base‘((mulGrp‘𝑍) ↾s 𝑈)) |
98 | 93, 81 | ressplusg 16834 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑈 ∈ V →
(.r‘𝑍) =
(+g‘((mulGrp‘𝑍) ↾s 𝑈))) |
99 | 23, 98 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
(.r‘𝑍) =
(+g‘((mulGrp‘𝑍) ↾s 𝑈)) |
100 | 96, 97, 99 | grplactf1o 18467 |
. . . . . . . . . . . . . . 15
⊢
((((mulGrp‘𝑍)
↾s 𝑈)
∈ Grp ∧ 𝑘 ∈
𝑈) → ((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(.r‘𝑍)𝑐)))‘𝑘):𝑈–1-1-onto→𝑈) |
101 | 95, 60, 100 | syl2an2r 685 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(.r‘𝑍)𝑐)))‘𝑘):𝑈–1-1-onto→𝑈) |
102 | 96, 97 | grplactval 18465 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) → (((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(.r‘𝑍)𝑐)))‘𝑘)‘𝑥) = (𝑘(.r‘𝑍)𝑥)) |
103 | 60, 102 | sylan 583 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) ∧ 𝑥 ∈ 𝑈) → (((𝑏 ∈ 𝑈 ↦ (𝑐 ∈ 𝑈 ↦ (𝑏(.r‘𝑍)𝑐)))‘𝑘)‘𝑥) = (𝑘(.r‘𝑍)𝑥)) |
104 | 89, 49, 101, 103, 56 | fsumf1o 15287 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = Σ𝑥 ∈ 𝑈 (𝑋‘(𝑘(.r‘𝑍)𝑥))) |
105 | 49, 62, 56 | fsummulc2 15348 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 ((𝑋‘𝑘) · (𝑋‘𝑎))) |
106 | 88, 104, 105 | 3eqtr4rd 2788 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → ((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) |
107 | 57 | mulid2d 10851 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) |
108 | 106, 107 | oveq12d 7231 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))) = (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) − Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))) |
109 | 57 | subidd 11177 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) − Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = 0) |
110 | 108, 109 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎))) = 0) |
111 | | 1cnd 10828 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → 1 ∈
ℂ) |
112 | 62, 111, 57 | subdird 11289 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = (((𝑋‘𝑘) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) − (1 · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)))) |
113 | 64 | mul01d 11031 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · 0) =
0) |
114 | 110, 112,
113 | 3eqtr4d 2787 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → (((𝑋‘𝑘) − 1) · Σ𝑎 ∈ 𝑈 (𝑋‘𝑎)) = (((𝑋‘𝑘) − 1) · 0)) |
115 | 57, 58, 64, 69, 114 | mulcanad 11467 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑘 ∈ 𝑈 ∧ (𝑋‘𝑘) ≠ 1)) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0) |
116 | 115 | expr 460 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ 1 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)) |
117 | 48, 116 | sylbid 243 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)) |
118 | 44, 117 | syl5bir 246 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)) |
119 | 118 | rexlimdva 3203 |
. . . 4
⊢ (𝜑 → (∃𝑘 ∈ 𝑈 ¬ (𝑋‘𝑘) = ( 1 ‘𝑘) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)) |
120 | 43, 119 | sylbid 243 |
. . 3
⊢ (𝜑 → (¬ 𝑋 = 1 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0)) |
121 | 120 | imp 410 |
. 2
⊢ ((𝜑 ∧ ¬ 𝑋 = 1 ) → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = 0) |
122 | 1, 2, 35, 121 | ifbothda 4477 |
1
⊢ (𝜑 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)) |