| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl 482 | . . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝜑) | 
| 2 |  | mdetuni.n | . . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ Fin) | 
| 3 |  | enrefg 9024 | . . . . . . . . 9
⊢ (𝑁 ∈ Fin → 𝑁 ≈ 𝑁) | 
| 4 | 2, 3 | syl 17 | . . . . . . . 8
⊢ (𝜑 → 𝑁 ≈ 𝑁) | 
| 5 |  | f1finf1o 9305 | . . . . . . . 8
⊢ ((𝑁 ≈ 𝑁 ∧ 𝑁 ∈ Fin) → (𝐸:𝑁–1-1→𝑁 ↔ 𝐸:𝑁–1-1-onto→𝑁)) | 
| 6 | 4, 2, 5 | syl2anc 584 | . . . . . . 7
⊢ (𝜑 → (𝐸:𝑁–1-1→𝑁 ↔ 𝐸:𝑁–1-1-onto→𝑁)) | 
| 7 | 6 | biimpa 476 | . . . . . 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 22449 | . . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) | 
| 11 | 2, 8, 10 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ Ring) | 
| 12 |  | mdetuni.b | . . . . . . . . 9
⊢ 𝐵 = (Base‘𝐴) | 
| 13 |  | eqid 2737 | . . . . . . . . 9
⊢
(1r‘𝐴) = (1r‘𝐴) | 
| 14 | 12, 13 | ringidcl 20262 | . . . . . . . 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 22624 | . . . . . 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 1373 | . . . . 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 1134 | . . . . . . . 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 1134 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ Ring) | 
| 32 |  | simp1r 1199 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐸:𝑁–1-1→𝑁) | 
| 33 |  | f1f 6804 | . . . . . . . . . 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 1138 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁) | 
| 36 | 34, 35 | ffvelcdmd 7105 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐸‘𝑎) ∈ 𝑁) | 
| 37 |  | simp3 1139 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁) | 
| 38 | 9, 19, 18, 29, 31, 36, 37, 13 | mat1ov 22454 | . . . . . . 7
⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝐸‘𝑎)(1r‘𝐴)𝑏) = if((𝐸‘𝑎) = 𝑏, 1 , 0 )) | 
| 39 | 38 | mpoeq3dva 7510 | . . . . . 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 6910 | . . . . 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 7447 | . . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴))) =
((((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))‘𝐸) · 0 )) | 
| 44 |  | zrhpsgnmhm 21602 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) →
((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))
∈ ((SymGrp‘𝑁)
MndHom (mulGrp‘𝑅))) | 
| 45 | 8, 2, 44 | syl2anc 584 | . . . . . . . . . 10
⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅))) | 
| 46 |  | eqid 2737 | . . . . . . . . . . 11
⊢
(Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁)) | 
| 47 |  | eqid 2737 | . . . . . . . . . . . 12
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) | 
| 48 | 47, 17 | mgpbas 20142 | . . . . . . . . . . 11
⊢ 𝐾 =
(Base‘(mulGrp‘𝑅)) | 
| 49 | 46, 48 | mhmf 18802 | . . . . . . . . . 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 2737 | . . . . . . . . . . 11
⊢
(SymGrp‘𝑁) =
(SymGrp‘𝑁) | 
| 53 | 52, 46 | elsymgbas 19391 | . . . . . . . . . 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 257 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝐸 ∈ (Base‘(SymGrp‘𝑁))) | 
| 56 | 51, 55 | ffvelcdmd 7105 | . . . . . . 7
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) ∈ 𝐾) | 
| 57 | 17, 21, 18 | ringrz 20291 | . . . . . . 7
⊢ ((𝑅 ∈ Ring ∧
(((ℤRHom‘𝑅)
∘ (pmSgn‘𝑁))‘𝐸) ∈ 𝐾) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · 0 ) = 0 ) | 
| 58 | 30, 56, 57 | syl2anc 584 | . . . . . 6
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · 0 ) = 0 ) | 
| 59 | 43, 58 | eqtrd 2777 | . . . . 5
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴))) = 0 ) | 
| 60 | 27, 40, 59 | 3eqtr3d 2785 | . . . 4
⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ) | 
| 61 | 60 | ex 412 | . . 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 |  | dff13 7275 | . . . . . 6
⊢ (𝐸:𝑁–1-1→𝑁 ↔ (𝐸:𝑁⟶𝑁 ∧ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))) | 
| 64 |  | ibar 528 | . . . . . . 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)))) | 
| 65 | 64 | 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)))) | 
| 66 | 63, 65 | bitr4id 290 | . . . . 5
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐸:𝑁–1-1→𝑁 ↔ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))) | 
| 67 | 66 | notbid 318 | . . . 4
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (¬ 𝐸:𝑁–1-1→𝑁 ↔ ¬ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))) | 
| 68 |  | rexnal 3100 | . . . . 5
⊢
(∃𝑐 ∈
𝑁 ¬ ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ¬ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)) | 
| 69 |  | rexnal 3100 | . . . . . . 7
⊢
(∃𝑑 ∈
𝑁 ¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ¬ ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)) | 
| 70 |  | df-ne 2941 | . . . . . . . . . 10
⊢ (𝑐 ≠ 𝑑 ↔ ¬ 𝑐 = 𝑑) | 
| 71 | 70 | anbi2i 623 | . . . . . . . . 9
⊢ (((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑) ↔ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ ¬ 𝑐 = 𝑑)) | 
| 72 |  | annim 403 | . . . . . . . . 9
⊢ (((𝐸‘𝑐) = (𝐸‘𝑑) ∧ ¬ 𝑐 = 𝑑) ↔ ¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)) | 
| 73 | 71, 72 | bitr2i 276 | . . . . . . . 8
⊢ (¬
((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)) | 
| 74 | 73 | rexbii 3094 | . . . . . . 7
⊢
(∃𝑑 ∈
𝑁 ¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)) | 
| 75 | 69, 74 | bitr3i 277 | . . . . . 6
⊢ (¬
∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)) | 
| 76 | 75 | rexbii 3094 | . . . . 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 277 | . . . 4
⊢ (¬
∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)) | 
| 78 | 67, 77 | bitrdi 287 | . . 3
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (¬ 𝐸:𝑁–1-1→𝑁 ↔ ∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) | 
| 79 |  | simprrl 781 | . . . . . . 7
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝐸‘𝑐) = (𝐸‘𝑑)) | 
| 80 |  | fveqeq2 6915 | . . . . . . . . . . . . 13
⊢ (𝑎 = 𝑐 → ((𝐸‘𝑎) = 𝑏 ↔ (𝐸‘𝑐) = 𝑏)) | 
| 81 | 80 | ifbid 4549 | . . . . . . . . . . . 12
⊢ (𝑎 = 𝑐 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if((𝐸‘𝑐) = 𝑏, 1 , 0 )) | 
| 82 |  | iftrue 4531 | . . . . . . . . . . . 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 )) | 
| 83 | 81, 82 | eqtr4d 2780 | . . . . . . . . . . 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 )))) | 
| 84 |  | fveqeq2 6915 | . . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑑 → ((𝐸‘𝑎) = 𝑏 ↔ (𝐸‘𝑑) = 𝑏)) | 
| 85 | 84 | ifbid 4549 | . . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if((𝐸‘𝑑) = 𝑏, 1 , 0 )) | 
| 86 |  | iftrue 4531 | . . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑑 → if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = if((𝐸‘𝑑) = 𝑏, 1 , 0 )) | 
| 87 | 85, 86 | eqtr4d 2780 | . . . . . . . . . . . . 13
⊢ (𝑎 = 𝑑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) | 
| 88 |  | iffalse 4534 | . . . . . . . . . . . . . 14
⊢ (¬
𝑎 = 𝑑 → if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = if((𝐸‘𝑎) = 𝑏, 1 , 0 )) | 
| 89 | 88 | eqcomd 2743 | . . . . . . . . . . . . 13
⊢ (¬
𝑎 = 𝑑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) | 
| 90 | 87, 89 | pm2.61i 182 | . . . . . . . . . . . 12
⊢ if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) | 
| 91 |  | iffalse 4534 | . . . . . . . . . . . 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 ))) | 
| 92 | 90, 91 | eqtr4id 2796 | . . . . . . . . . . 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 )))) | 
| 93 | 83, 92 | pm2.61i 182 | . . . . . . . . . 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 ))) | 
| 94 |  | eqeq1 2741 | . . . . . . . . . . . . . 14
⊢ ((𝐸‘𝑑) = (𝐸‘𝑐) → ((𝐸‘𝑑) = 𝑏 ↔ (𝐸‘𝑐) = 𝑏)) | 
| 95 | 94 | eqcoms 2745 | . . . . . . . . . . . . 13
⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → ((𝐸‘𝑑) = 𝑏 ↔ (𝐸‘𝑐) = 𝑏)) | 
| 96 | 95 | ifbid 4549 | . . . . . . . . . . . 12
⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → if((𝐸‘𝑑) = 𝑏, 1 , 0 ) = if((𝐸‘𝑐) = 𝑏, 1 , 0 )) | 
| 97 | 96 | ifeq1d 4545 | . . . . . . . . . . 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 ))) | 
| 98 | 97 | ifeq2d 4546 | . . . . . . . . . 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 )))) | 
| 99 | 93, 98 | eqtrid 2789 | . . . . . . . . 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 )))) | 
| 100 | 99 | mpoeq3dv 7512 | . . . . . . . 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 ))))) | 
| 101 | 100 | fveq2d 6910 | . . . . . . 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 )))))) | 
| 102 | 79, 101 | 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 )))))) | 
| 103 |  | simpll 767 | . . . . . . 7
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝜑) | 
| 104 |  | simprll 779 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝑐 ∈ 𝑁) | 
| 105 |  | simprlr 780 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝑑 ∈ 𝑁) | 
| 106 |  | simprrr 782 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝑐 ≠ 𝑑) | 
| 107 | 104, 105,
106 | 3jca 1129 | . . . . . . 7
⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ≠ 𝑑)) | 
| 108 | 17, 19 | ringidcl 20262 | . . . . . . . . . 10
⊢ (𝑅 ∈ Ring → 1 ∈ 𝐾) | 
| 109 | 8, 108 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 1 ∈ 𝐾) | 
| 110 | 17, 18 | ring0cl 20264 | . . . . . . . . . 10
⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) | 
| 111 | 8, 110 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 0 ∈ 𝐾) | 
| 112 | 109, 111 | ifcld 4572 | . . . . . . . 8
⊢ (𝜑 → if((𝐸‘𝑐) = 𝑏, 1 , 0 ) ∈ 𝐾) | 
| 113 | 112 | ad3antrrr 730 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) ∧ 𝑏 ∈ 𝑁) → if((𝐸‘𝑐) = 𝑏, 1 , 0 ) ∈ 𝐾) | 
| 114 |  | simp1ll 1237 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝜑) | 
| 115 | 109, 111 | ifcld 4572 | . . . . . . . 8
⊢ (𝜑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) ∈ 𝐾) | 
| 116 | 114, 115 | 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) | 
| 117 | 9, 12, 17, 18, 19, 20, 21, 2, 8, 22, 23, 24, 25, 103, 107, 113, 116 | mdetunilem2 22619 | . . . . . 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 ) | 
| 118 | 102, 117 | eqtrd 2777 | . . . . 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 ) | 
| 119 | 118 | expr 456 | . . . 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 )) | 
| 120 | 119 | rexlimdvva 3213 | . . 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 )) | 
| 121 | 78, 120 | sylbid 240 | . 2
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (¬ 𝐸:𝑁–1-1→𝑁 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )) | 
| 122 | 62, 121 | pm2.61d 179 | 1
⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ) |