Proof of Theorem mdetunilem5
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mdetunilem5.ph | . 2
⊢ (𝜓 → 𝜑) | 
| 2 |  | mdetuni.a | . . 3
⊢ 𝐴 = (𝑁 Mat 𝑅) | 
| 3 |  | mdetuni.k | . . 3
⊢ 𝐾 = (Base‘𝑅) | 
| 4 |  | mdetuni.b | . . 3
⊢ 𝐵 = (Base‘𝐴) | 
| 5 |  | mdetuni.n | . . . 4
⊢ (𝜑 → 𝑁 ∈ Fin) | 
| 6 | 1, 5 | syl 17 | . . 3
⊢ (𝜓 → 𝑁 ∈ Fin) | 
| 7 |  | mdetuni.r | . . . 4
⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 8 | 1, 7 | syl 17 | . . 3
⊢ (𝜓 → 𝑅 ∈ Ring) | 
| 9 | 8 | 3ad2ant1 1133 | . . . . 5
⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ Ring) | 
| 10 |  | mdetunilem5.fgh | . . . . . 6
⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾 ∧ 𝐻 ∈ 𝐾)) | 
| 11 | 10 | simp1d 1142 | . . . . 5
⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐹 ∈ 𝐾) | 
| 12 | 10 | simp2d 1143 | . . . . 5
⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐺 ∈ 𝐾) | 
| 13 |  | mdetuni.pg | . . . . . 6
⊢  + =
(+g‘𝑅) | 
| 14 | 3, 13 | ringacl 20276 | . . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾) → (𝐹 + 𝐺) ∈ 𝐾) | 
| 15 | 9, 11, 12, 14 | syl3anc 1372 | . . . 4
⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐹 + 𝐺) ∈ 𝐾) | 
| 16 | 10 | simp3d 1144 | . . . 4
⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾) | 
| 17 | 15, 16 | ifcld 4571 | . . 3
⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻) ∈ 𝐾) | 
| 18 | 2, 3, 4, 6, 8, 17 | matbas2d 22430 | . 2
⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ∈ 𝐵) | 
| 19 | 11, 16 | ifcld 4571 | . . 3
⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐹, 𝐻) ∈ 𝐾) | 
| 20 | 2, 3, 4, 6, 8, 19 | matbas2d 22430 | . 2
⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ∈ 𝐵) | 
| 21 | 12, 16 | ifcld 4571 | . . 3
⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐺, 𝐻) ∈ 𝐾) | 
| 22 | 2, 3, 4, 6, 8, 21 | matbas2d 22430 | . 2
⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ∈ 𝐵) | 
| 23 |  | mdetunilem5.e | . 2
⊢ (𝜓 → 𝐸 ∈ 𝑁) | 
| 24 |  | snex 5435 | . . . . . . 7
⊢ {𝐸} ∈ V | 
| 25 | 24 | a1i 11 | . . . . . 6
⊢ (𝜓 → {𝐸} ∈ V) | 
| 26 | 23 | snssd 4808 | . . . . . . . . 9
⊢ (𝜓 → {𝐸} ⊆ 𝑁) | 
| 27 | 26 | 3ad2ant1 1133 | . . . . . . . 8
⊢ ((𝜓 ∧ 𝑎 ∈ {𝐸} ∧ 𝑏 ∈ 𝑁) → {𝐸} ⊆ 𝑁) | 
| 28 |  | simp2 1137 | . . . . . . . 8
⊢ ((𝜓 ∧ 𝑎 ∈ {𝐸} ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ {𝐸}) | 
| 29 | 27, 28 | sseldd 3983 | . . . . . . 7
⊢ ((𝜓 ∧ 𝑎 ∈ {𝐸} ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁) | 
| 30 | 29, 11 | syld3an2 1412 | . . . . . 6
⊢ ((𝜓 ∧ 𝑎 ∈ {𝐸} ∧ 𝑏 ∈ 𝑁) → 𝐹 ∈ 𝐾) | 
| 31 | 29, 12 | syld3an2 1412 | . . . . . 6
⊢ ((𝜓 ∧ 𝑎 ∈ {𝐸} ∧ 𝑏 ∈ 𝑁) → 𝐺 ∈ 𝐾) | 
| 32 |  | eqidd 2737 | . . . . . 6
⊢ (𝜓 → (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐹) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐹)) | 
| 33 |  | eqidd 2737 | . . . . . 6
⊢ (𝜓 → (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐺) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐺)) | 
| 34 | 25, 6, 30, 31, 32, 33 | offval22 8114 | . . . . 5
⊢ (𝜓 → ((𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐹) ∘f + (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐺)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ (𝐹 + 𝐺))) | 
| 35 | 34 | eqcomd 2742 | . . . 4
⊢ (𝜓 → (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ (𝐹 + 𝐺)) = ((𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐹) ∘f + (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐺))) | 
| 36 |  | mposnif 7550 | . . . 4
⊢ (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ (𝐹 + 𝐺)) | 
| 37 |  | mposnif 7550 | . . . . 5
⊢ (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐹) | 
| 38 |  | mposnif 7550 | . . . . 5
⊢ (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐺) | 
| 39 | 37, 38 | oveq12i 7444 | . . . 4
⊢ ((𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ∘f + (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻))) = ((𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐹) ∘f + (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ 𝐺)) | 
| 40 | 35, 36, 39 | 3eqtr4g 2801 | . . 3
⊢ (𝜓 → (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) = ((𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ∘f + (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)))) | 
| 41 |  | ssid 4005 | . . . 4
⊢ 𝑁 ⊆ 𝑁 | 
| 42 |  | resmpo 7554 | . . . 4
⊢ (({𝐸} ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ({𝐸} × 𝑁)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻))) | 
| 43 | 26, 41, 42 | sylancl 586 | . . 3
⊢ (𝜓 → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ({𝐸} × 𝑁)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻))) | 
| 44 |  | resmpo 7554 | . . . . 5
⊢ (({𝐸} ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ↾ ({𝐸} × 𝑁)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) | 
| 45 | 26, 41, 44 | sylancl 586 | . . . 4
⊢ (𝜓 → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ↾ ({𝐸} × 𝑁)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) | 
| 46 |  | resmpo 7554 | . . . . 5
⊢ (({𝐸} ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ↾ ({𝐸} × 𝑁)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻))) | 
| 47 | 26, 41, 46 | sylancl 586 | . . . 4
⊢ (𝜓 → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ↾ ({𝐸} × 𝑁)) = (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻))) | 
| 48 | 45, 47 | oveq12d 7450 | . . 3
⊢ (𝜓 → (((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ↾ ({𝐸} × 𝑁)) ∘f + ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ↾ ({𝐸} × 𝑁))) = ((𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ∘f + (𝑎 ∈ {𝐸}, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)))) | 
| 49 | 40, 43, 48 | 3eqtr4d 2786 | . 2
⊢ (𝜓 → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ({𝐸} × 𝑁)) = (((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ↾ ({𝐸} × 𝑁)) ∘f + ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ↾ ({𝐸} × 𝑁)))) | 
| 50 |  | eldifsni 4789 | . . . . . . 7
⊢ (𝑎 ∈ (𝑁 ∖ {𝐸}) → 𝑎 ≠ 𝐸) | 
| 51 | 50 | 3ad2ant2 1134 | . . . . . 6
⊢ ((𝜓 ∧ 𝑎 ∈ (𝑁 ∖ {𝐸}) ∧ 𝑏 ∈ 𝑁) → 𝑎 ≠ 𝐸) | 
| 52 | 51 | neneqd 2944 | . . . . 5
⊢ ((𝜓 ∧ 𝑎 ∈ (𝑁 ∖ {𝐸}) ∧ 𝑏 ∈ 𝑁) → ¬ 𝑎 = 𝐸) | 
| 53 |  | iffalse 4533 | . . . . . 6
⊢ (¬
𝑎 = 𝐸 → if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻) = 𝐻) | 
| 54 |  | iffalse 4533 | . . . . . 6
⊢ (¬
𝑎 = 𝐸 → if(𝑎 = 𝐸, 𝐹, 𝐻) = 𝐻) | 
| 55 | 53, 54 | eqtr4d 2779 | . . . . 5
⊢ (¬
𝑎 = 𝐸 → if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻) = if(𝑎 = 𝐸, 𝐹, 𝐻)) | 
| 56 | 52, 55 | syl 17 | . . . 4
⊢ ((𝜓 ∧ 𝑎 ∈ (𝑁 ∖ {𝐸}) ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻) = if(𝑎 = 𝐸, 𝐹, 𝐻)) | 
| 57 | 56 | mpoeq3dva 7511 | . . 3
⊢ (𝜓 → (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) = (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) | 
| 58 |  | difss 4135 | . . . 4
⊢ (𝑁 ∖ {𝐸}) ⊆ 𝑁 | 
| 59 |  | resmpo 7554 | . . . 4
⊢ (((𝑁 ∖ {𝐸}) ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻))) | 
| 60 | 58, 41, 59 | mp2an 692 | . . 3
⊢ ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) | 
| 61 |  | resmpo 7554 | . . . 4
⊢ (((𝑁 ∖ {𝐸}) ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) | 
| 62 | 58, 41, 61 | mp2an 692 | . . 3
⊢ ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) | 
| 63 | 57, 60, 62 | 3eqtr4g 2801 | . 2
⊢ (𝜓 → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁))) | 
| 64 |  | iffalse 4533 | . . . . . 6
⊢ (¬
𝑎 = 𝐸 → if(𝑎 = 𝐸, 𝐺, 𝐻) = 𝐻) | 
| 65 | 53, 64 | eqtr4d 2779 | . . . . 5
⊢ (¬
𝑎 = 𝐸 → if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻) = if(𝑎 = 𝐸, 𝐺, 𝐻)) | 
| 66 | 52, 65 | syl 17 | . . . 4
⊢ ((𝜓 ∧ 𝑎 ∈ (𝑁 ∖ {𝐸}) ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻) = if(𝑎 = 𝐸, 𝐺, 𝐻)) | 
| 67 | 66 | mpoeq3dva 7511 | . . 3
⊢ (𝜓 → (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) = (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻))) | 
| 68 |  | resmpo 7554 | . . . 4
⊢ (((𝑁 ∖ {𝐸}) ⊆ 𝑁 ∧ 𝑁 ⊆ 𝑁) → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻))) | 
| 69 | 58, 41, 68 | mp2an 692 | . . 3
⊢ ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = (𝑎 ∈ (𝑁 ∖ {𝐸}), 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) | 
| 70 | 67, 60, 69 | 3eqtr4g 2801 | . 2
⊢ (𝜓 → ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁))) | 
| 71 |  | mdetuni.0g | . . 3
⊢  0 =
(0g‘𝑅) | 
| 72 |  | mdetuni.1r | . . 3
⊢  1 =
(1r‘𝑅) | 
| 73 |  | mdetuni.tg | . . 3
⊢  · =
(.r‘𝑅) | 
| 74 |  | mdetuni.ff | . . 3
⊢ (𝜑 → 𝐷:𝐵⟶𝐾) | 
| 75 |  | mdetuni.al | . . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 )) | 
| 76 |  | mdetuni.li | . . 3
⊢ (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)))) | 
| 77 |  | mdetuni.sc | . . 3
⊢ (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)))) | 
| 78 | 2, 4, 3, 71, 72, 13, 73, 5, 7, 74, 75, 76, 77 | mdetunilem3 22621 | . 2
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ∈ 𝐵 ∧ (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ∈ 𝐵) ∧ ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ∈ 𝐵 ∧ 𝐸 ∈ 𝑁 ∧ ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ({𝐸} × 𝑁)) = (((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ↾ ({𝐸} × 𝑁)) ∘f + ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ↾ ({𝐸} × 𝑁)))) ∧ (((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) ∧ ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)) = ((𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)) ↾ ((𝑁 ∖ {𝐸}) × 𝑁)))) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻))) = ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻))))) | 
| 79 | 1, 18, 20, 22, 23, 49, 63, 70, 78 | syl332anc 1402 | 1
⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻))) = ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻))))) |