Step | Hyp | Ref
| Expression |
1 | | simpl 472 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝜑) |
2 | | mdetuni.n |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ Fin) |
3 | | enrefg 7873 |
. . . . . . . . 9
⊢ (𝑁 ∈ Fin → 𝑁 ≈ 𝑁) |
4 | 2, 3 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ≈ 𝑁) |
5 | | f1finf1o 8072 |
. . . . . . . 8
⊢ ((𝑁 ≈ 𝑁 ∧ 𝑁 ∈ Fin) → (𝐸:𝑁–1-1→𝑁 ↔ 𝐸:𝑁–1-1-onto→𝑁)) |
6 | 4, 2, 5 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → (𝐸:𝑁–1-1→𝑁 ↔ 𝐸:𝑁–1-1-onto→𝑁)) |
7 | 6 | biimpa 500 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝐸:𝑁–1-1-onto→𝑁) |
8 | | mdetuni.r |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | | mdetuni.a |
. . . . . . . . . 10
⊢ 𝐴 = (𝑁 Mat 𝑅) |
10 | 9 | matring 20068 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
11 | 2, 8, 10 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ Ring) |
12 | | mdetuni.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝐴) |
13 | | eqid 2610 |
. . . . . . . . 9
⊢
(1r‘𝐴) = (1r‘𝐴) |
14 | 12, 13 | ringidcl 18391 |
. . . . . . . 8
⊢ (𝐴 ∈ Ring →
(1r‘𝐴)
∈ 𝐵) |
15 | 11, 14 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (1r‘𝐴) ∈ 𝐵) |
16 | 15 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (1r‘𝐴) ∈ 𝐵) |
17 | | mdetuni.k |
. . . . . . 7
⊢ 𝐾 = (Base‘𝑅) |
18 | | mdetuni.0g |
. . . . . . 7
⊢ 0 =
(0g‘𝑅) |
19 | | mdetuni.1r |
. . . . . . 7
⊢ 1 =
(1r‘𝑅) |
20 | | mdetuni.pg |
. . . . . . 7
⊢ + =
(+g‘𝑅) |
21 | | mdetuni.tg |
. . . . . . 7
⊢ · =
(.r‘𝑅) |
22 | | mdetuni.ff |
. . . . . . 7
⊢ (𝜑 → 𝐷:𝐵⟶𝐾) |
23 | | mdetuni.al |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 )) |
24 | | mdetuni.li |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) |
25 | | mdetuni.sc |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)))) |
26 | 9, 12, 17, 18, 19, 20, 21, 2, 8, 22, 23, 24, 25 | mdetunilem7 20243 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧
(1r‘𝐴)
∈ 𝐵) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)(1r‘𝐴)𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴)))) |
27 | 1, 7, 16, 26 | syl3anc 1318 |
. . . . 5
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)(1r‘𝐴)𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴)))) |
28 | 2 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝑁 ∈ Fin) |
29 | 28 | 3ad2ant1 1075 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑁 ∈ Fin) |
30 | 8 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝑅 ∈ Ring) |
31 | 30 | 3ad2ant1 1075 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ Ring) |
32 | | simp1r 1079 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐸:𝑁–1-1→𝑁) |
33 | | f1f 6014 |
. . . . . . . . . 10
⊢ (𝐸:𝑁–1-1→𝑁 → 𝐸:𝑁⟶𝑁) |
34 | 32, 33 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐸:𝑁⟶𝑁) |
35 | | simp2 1055 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁) |
36 | 34, 35 | ffvelrnd 6268 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐸‘𝑎) ∈ 𝑁) |
37 | | simp3 1056 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁) |
38 | 9, 19, 18, 29, 31, 36, 37, 13 | mat1ov 20073 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝐸‘𝑎)(1r‘𝐴)𝑏) = if((𝐸‘𝑎) = 𝑏, 1 , 0 )) |
39 | 38 | mpt2eq3dva 6617 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)(1r‘𝐴)𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) |
40 | 39 | fveq2d 6107 |
. . . . 5
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)(1r‘𝐴)𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 )))) |
41 | | mdetunilem8.id |
. . . . . . . 8
⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 ) |
42 | 41 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐷‘(1r‘𝐴)) = 0 ) |
43 | 42 | oveq2d 6565 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴))) =
((((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))‘𝐸) · 0 )) |
44 | | zrhpsgnmhm 19749 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) →
((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))
∈ ((SymGrp‘𝑁)
MndHom (mulGrp‘𝑅))) |
45 | 8, 2, 44 | syl2anc 691 |
. . . . . . . . . 10
⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅))) |
46 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁)) |
47 | | eqid 2610 |
. . . . . . . . . . . 12
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) |
48 | 47, 17 | mgpbas 18318 |
. . . . . . . . . . 11
⊢ 𝐾 =
(Base‘(mulGrp‘𝑅)) |
49 | 46, 48 | mhmf 17163 |
. . . . . . . . . 10
⊢
(((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾) |
50 | 45, 49 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾) |
51 | 50 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾) |
52 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(SymGrp‘𝑁) =
(SymGrp‘𝑁) |
53 | 52, 46 | elsymgbas 17625 |
. . . . . . . . . 10
⊢ (𝑁 ∈ Fin → (𝐸 ∈
(Base‘(SymGrp‘𝑁)) ↔ 𝐸:𝑁–1-1-onto→𝑁)) |
54 | 28, 53 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐸 ∈ (Base‘(SymGrp‘𝑁)) ↔ 𝐸:𝑁–1-1-onto→𝑁)) |
55 | 7, 54 | mpbird 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝐸 ∈ (Base‘(SymGrp‘𝑁))) |
56 | 51, 55 | ffvelrnd 6268 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) ∈ 𝐾) |
57 | 17, 21, 18 | ringrz 18411 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧
(((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))‘𝐸) ∈ 𝐾) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · 0 ) = 0 ) |
58 | 30, 56, 57 | syl2anc 691 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · 0 ) = 0 ) |
59 | 43, 58 | eqtrd 2644 |
. . . . 5
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴))) = 0 ) |
60 | 27, 40, 59 | 3eqtr3d 2652 |
. . . 4
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ) |
61 | 60 | ex 449 |
. . 3
⊢ (𝜑 → (𝐸:𝑁–1-1→𝑁 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )) |
62 | 61 | adantr 480 |
. 2
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐸:𝑁–1-1→𝑁 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )) |
63 | | ibar 524 |
. . . . . . 7
⊢ (𝐸:𝑁⟶𝑁 → (∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ (𝐸:𝑁⟶𝑁 ∧ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)))) |
64 | 63 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ (𝐸:𝑁⟶𝑁 ∧ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)))) |
65 | | dff13 6416 |
. . . . . 6
⊢ (𝐸:𝑁–1-1→𝑁 ↔ (𝐸:𝑁⟶𝑁 ∧ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))) |
66 | 64, 65 | syl6rbbr 278 |
. . . . 5
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐸:𝑁–1-1→𝑁 ↔ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))) |
67 | 66 | notbid 307 |
. . . 4
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (¬ 𝐸:𝑁–1-1→𝑁 ↔ ¬ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))) |
68 | | rexnal 2978 |
. . . . 5
⊢
(∃𝑐 ∈
𝑁 ¬ ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ¬ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)) |
69 | | rexnal 2978 |
. . . . . . 7
⊢
(∃𝑑 ∈
𝑁 ¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ¬ ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)) |
70 | | df-ne 2782 |
. . . . . . . . . 10
⊢ (𝑐 ≠ 𝑑 ↔ ¬ 𝑐 = 𝑑) |
71 | 70 | anbi2i 726 |
. . . . . . . . 9
⊢ (((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑) ↔ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ ¬ 𝑐 = 𝑑)) |
72 | | annim 440 |
. . . . . . . . 9
⊢ (((𝐸‘𝑐) = (𝐸‘𝑑) ∧ ¬ 𝑐 = 𝑑) ↔ ¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)) |
73 | 71, 72 | bitr2i 264 |
. . . . . . . 8
⊢ (¬
((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)) |
74 | 73 | rexbii 3023 |
. . . . . . 7
⊢
(∃𝑑 ∈
𝑁 ¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)) |
75 | 69, 74 | bitr3i 265 |
. . . . . 6
⊢ (¬
∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)) |
76 | 75 | rexbii 3023 |
. . . . 5
⊢
(∃𝑐 ∈
𝑁 ¬ ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)) |
77 | 68, 76 | bitr3i 265 |
. . . 4
⊢ (¬
∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)) |
78 | 67, 77 | syl6bb 275 |
. . 3
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (¬ 𝐸:𝑁–1-1→𝑁 ↔ ∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) |
79 | | simprrl 800 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝐸‘𝑐) = (𝐸‘𝑑)) |
80 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑐 → (𝐸‘𝑎) = (𝐸‘𝑐)) |
81 | 80 | eqeq1d 2612 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑐 → ((𝐸‘𝑎) = 𝑏 ↔ (𝐸‘𝑐) = 𝑏)) |
82 | 81 | ifbid 4058 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑐 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if((𝐸‘𝑐) = 𝑏, 1 , 0 )) |
83 | | iftrue 4042 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑐 → if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = if((𝐸‘𝑐) = 𝑏, 1 , 0 )) |
84 | 82, 83 | eqtr4d 2647 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑐 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))) |
85 | | iffalse 4045 |
. . . . . . . . . . . 12
⊢ (¬
𝑎 = 𝑐 → if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) |
86 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑑 → (𝐸‘𝑎) = (𝐸‘𝑑)) |
87 | 86 | eqeq1d 2612 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑑 → ((𝐸‘𝑎) = 𝑏 ↔ (𝐸‘𝑑) = 𝑏)) |
88 | 87 | ifbid 4058 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if((𝐸‘𝑑) = 𝑏, 1 , 0 )) |
89 | | iftrue 4042 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑑 → if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = if((𝐸‘𝑑) = 𝑏, 1 , 0 )) |
90 | 88, 89 | eqtr4d 2647 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) |
91 | | iffalse 4045 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑎 = 𝑑 → if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = if((𝐸‘𝑎) = 𝑏, 1 , 0 )) |
92 | 91 | eqcomd 2616 |
. . . . . . . . . . . . 13
⊢ (¬
𝑎 = 𝑑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) |
93 | 90, 92 | pm2.61i 175 |
. . . . . . . . . . . 12
⊢ if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) |
94 | 85, 93 | syl6reqr 2663 |
. . . . . . . . . . 11
⊢ (¬
𝑎 = 𝑐 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))) |
95 | 84, 94 | pm2.61i 175 |
. . . . . . . . . 10
⊢ if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) |
96 | | eqeq1 2614 |
. . . . . . . . . . . . . 14
⊢ ((𝐸‘𝑑) = (𝐸‘𝑐) → ((𝐸‘𝑑) = 𝑏 ↔ (𝐸‘𝑐) = 𝑏)) |
97 | 96 | eqcoms 2618 |
. . . . . . . . . . . . 13
⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → ((𝐸‘𝑑) = 𝑏 ↔ (𝐸‘𝑐) = 𝑏)) |
98 | 97 | ifbid 4058 |
. . . . . . . . . . . 12
⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → if((𝐸‘𝑑) = 𝑏, 1 , 0 ) = if((𝐸‘𝑐) = 𝑏, 1 , 0 )) |
99 | 98 | ifeq1d 4054 |
. . . . . . . . . . 11
⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) |
100 | 99 | ifeq2d 4055 |
. . . . . . . . . 10
⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))) |
101 | 95, 100 | syl5eq 2656 |
. . . . . . . . 9
⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))) |
102 | 101 | mpt2eq3dv 6619 |
. . . . . . . 8
⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))) |
103 | 102 | fveq2d 6107 |
. . . . . . 7
⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))))) |
104 | 79, 103 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))))) |
105 | | simpll 786 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝜑) |
106 | | simprll 798 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝑐 ∈ 𝑁) |
107 | | simprlr 799 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝑑 ∈ 𝑁) |
108 | | simprrr 801 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝑐 ≠ 𝑑) |
109 | 106, 107,
108 | 3jca 1235 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ≠ 𝑑)) |
110 | 17, 19 | ringidcl 18391 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Ring → 1 ∈ 𝐾) |
111 | 8, 110 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈ 𝐾) |
112 | 17, 18 | ring0cl 18392 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
113 | 8, 112 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 0 ∈ 𝐾) |
114 | 111, 113 | ifcld 4081 |
. . . . . . . 8
⊢ (𝜑 → if((𝐸‘𝑐) = 𝑏, 1 , 0 ) ∈ 𝐾) |
115 | 114 | ad3antrrr 762 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) ∧ 𝑏 ∈ 𝑁) → if((𝐸‘𝑐) = 𝑏, 1 , 0 ) ∈ 𝐾) |
116 | | simp1ll 1117 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝜑) |
117 | 111, 113 | ifcld 4081 |
. . . . . . . 8
⊢ (𝜑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) ∈ 𝐾) |
118 | 116, 117 | syl 17 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) ∈ 𝐾) |
119 | 9, 12, 17, 18, 19, 20, 21, 2, 8, 22, 23, 24, 25, 105, 109, 115, 118 | mdetunilem2 20238 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))) = 0 ) |
120 | 104, 119 | eqtrd 2644 |
. . . . 5
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ) |
121 | 120 | expr 641 |
. . . 4
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ (𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁)) → (((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )) |
122 | 121 | rexlimdvva 3020 |
. . 3
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )) |
123 | 78, 122 | sylbid 229 |
. 2
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (¬ 𝐸:𝑁–1-1→𝑁 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )) |
124 | 62, 123 | pm2.61d 169 |
1
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ) |