Step | Hyp | Ref
| Expression |
1 | | zlmodzxz.z |
. . 3
⊢ 𝑍 = (ℤring
freeLMod {0, 1}) |
2 | | eqid 2733 |
. . 3
⊢
(Base‘𝑍) =
(Base‘𝑍) |
3 | | zringring 20888 |
. . . 4
⊢
ℤring ∈ Ring |
4 | 3 | a1i 11 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
ℤring ∈ Ring) |
5 | | prex 5390 |
. . . 4
⊢ {0, 1}
∈ V |
6 | 5 | a1i 11 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {0, 1}
∈ V) |
7 | | simpl 484 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈
ℤ) |
8 | | simpl 484 |
. . . 4
⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → 𝐶 ∈
ℤ) |
9 | 1 | zlmodzxzel 46517 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) →
{⟨0, 𝐴⟩, ⟨1,
𝐶⟩} ∈
(Base‘𝑍)) |
10 | 7, 8, 9 | syl2an 597 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{⟨0, 𝐴⟩, ⟨1,
𝐶⟩} ∈
(Base‘𝑍)) |
11 | | simpr 486 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈
ℤ) |
12 | | simpr 486 |
. . . 4
⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → 𝐷 ∈
ℤ) |
13 | 1 | zlmodzxzel 46517 |
. . . 4
⊢ ((𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) →
{⟨0, 𝐵⟩, ⟨1,
𝐷⟩} ∈
(Base‘𝑍)) |
14 | 11, 12, 13 | syl2an 597 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{⟨0, 𝐵⟩, ⟨1,
𝐷⟩} ∈
(Base‘𝑍)) |
15 | | eqid 2733 |
. . 3
⊢
(+g‘ℤring) =
(+g‘ℤring) |
16 | | zlmodzxzadd.p |
. . 3
⊢ + =
(+g‘𝑍) |
17 | 1, 2, 4, 6, 10, 14, 15, 16 | frlmplusgval 21186 |
. 2
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({⟨0, 𝐴⟩,
⟨1, 𝐶⟩} + {⟨0,
𝐵⟩, ⟨1, 𝐷⟩}) = ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} ∘f
(+g‘ℤring){⟨0, 𝐵⟩, ⟨1, 𝐷⟩})) |
18 | | c0ex 11154 |
. . . . . 6
⊢ 0 ∈
V |
19 | | 1ex 11156 |
. . . . . 6
⊢ 1 ∈
V |
20 | 18, 19 | pm3.2i 472 |
. . . . 5
⊢ (0 ∈
V ∧ 1 ∈ V) |
21 | 20 | a1i 11 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (0
∈ V ∧ 1 ∈ V)) |
22 | 7, 8 | anim12i 614 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐴 ∈ ℤ ∧ 𝐶 ∈
ℤ)) |
23 | | 0ne1 12229 |
. . . . 5
⊢ 0 ≠
1 |
24 | 23 | a1i 11 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 0 ≠
1) |
25 | | fnprg 6561 |
. . . 4
⊢ (((0
∈ V ∧ 1 ∈ V) ∧ (𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 0 ≠ 1) →
{⟨0, 𝐴⟩, ⟨1,
𝐶⟩} Fn {0,
1}) |
26 | 21, 22, 24, 25 | syl3anc 1372 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{⟨0, 𝐴⟩, ⟨1,
𝐶⟩} Fn {0,
1}) |
27 | 11, 12 | anim12i 614 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐵 ∈ ℤ ∧ 𝐷 ∈
ℤ)) |
28 | | fnprg 6561 |
. . . 4
⊢ (((0
∈ V ∧ 1 ∈ V) ∧ (𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ 0 ≠ 1) →
{⟨0, 𝐵⟩, ⟨1,
𝐷⟩} Fn {0,
1}) |
29 | 21, 27, 24, 28 | syl3anc 1372 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{⟨0, 𝐵⟩, ⟨1,
𝐷⟩} Fn {0,
1}) |
30 | 6, 26, 29 | offvalfv 46504 |
. 2
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({⟨0, 𝐴⟩,
⟨1, 𝐶⟩}
∘f (+g‘ℤring){⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = (𝑥 ∈ {0, 1} ↦ (({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘𝑥)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘𝑥)))) |
31 | 18 | a1i 11 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 0
∈ V) |
32 | 19 | a1i 11 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 1
∈ V) |
33 | | ovexd 7393 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐴(+g‘ℤring)𝐵) ∈ V) |
34 | | ovexd 7393 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐶(+g‘ℤring)𝐷) ∈ V) |
35 | | fveq2 6843 |
. . . . . 6
⊢ (𝑥 = 0 → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘𝑥) = ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘0)) |
36 | | fveq2 6843 |
. . . . . 6
⊢ (𝑥 = 0 → ({⟨0, 𝐵⟩, ⟨1, 𝐷⟩}‘𝑥) = ({⟨0, 𝐵⟩, ⟨1, 𝐷⟩}‘0)) |
37 | 35, 36 | oveq12d 7376 |
. . . . 5
⊢ (𝑥 = 0 → (({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘𝑥)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘𝑥)) = (({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘0)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘0))) |
38 | 7 | adantr 482 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐴 ∈
ℤ) |
39 | | fvpr1g 7137 |
. . . . . . 7
⊢ ((0
∈ V ∧ 𝐴 ∈
ℤ ∧ 0 ≠ 1) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘0) = 𝐴) |
40 | 31, 38, 24, 39 | syl3anc 1372 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({⟨0, 𝐴⟩,
⟨1, 𝐶⟩}‘0)
= 𝐴) |
41 | 11 | adantr 482 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐵 ∈
ℤ) |
42 | | fvpr1g 7137 |
. . . . . . 7
⊢ ((0
∈ V ∧ 𝐵 ∈
ℤ ∧ 0 ≠ 1) → ({⟨0, 𝐵⟩, ⟨1, 𝐷⟩}‘0) = 𝐵) |
43 | 31, 41, 24, 42 | syl3anc 1372 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({⟨0, 𝐵⟩,
⟨1, 𝐷⟩}‘0)
= 𝐵) |
44 | 40, 43 | oveq12d 7376 |
. . . . 5
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
(({⟨0, 𝐴⟩,
⟨1, 𝐶⟩}‘0)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘0)) = (𝐴(+g‘ℤring)𝐵)) |
45 | 37, 44 | sylan9eqr 2795 |
. . . 4
⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) ∧ 𝑥 = 0) → (({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘𝑥)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘𝑥)) = (𝐴(+g‘ℤring)𝐵)) |
46 | | fveq2 6843 |
. . . . . 6
⊢ (𝑥 = 1 → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘𝑥) = ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘1)) |
47 | | fveq2 6843 |
. . . . . 6
⊢ (𝑥 = 1 → ({⟨0, 𝐵⟩, ⟨1, 𝐷⟩}‘𝑥) = ({⟨0, 𝐵⟩, ⟨1, 𝐷⟩}‘1)) |
48 | 46, 47 | oveq12d 7376 |
. . . . 5
⊢ (𝑥 = 1 → (({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘𝑥)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘𝑥)) = (({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘1)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘1))) |
49 | 8 | adantl 483 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐶 ∈
ℤ) |
50 | | fvpr2g 7138 |
. . . . . . 7
⊢ ((1
∈ V ∧ 𝐶 ∈
ℤ ∧ 0 ≠ 1) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘1) = 𝐶) |
51 | 32, 49, 24, 50 | syl3anc 1372 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({⟨0, 𝐴⟩,
⟨1, 𝐶⟩}‘1)
= 𝐶) |
52 | 12 | adantl 483 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐷 ∈
ℤ) |
53 | | fvpr2g 7138 |
. . . . . . 7
⊢ ((1
∈ V ∧ 𝐷 ∈
ℤ ∧ 0 ≠ 1) → ({⟨0, 𝐵⟩, ⟨1, 𝐷⟩}‘1) = 𝐷) |
54 | 32, 52, 24, 53 | syl3anc 1372 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({⟨0, 𝐵⟩,
⟨1, 𝐷⟩}‘1)
= 𝐷) |
55 | 51, 54 | oveq12d 7376 |
. . . . 5
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
(({⟨0, 𝐴⟩,
⟨1, 𝐶⟩}‘1)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘1)) = (𝐶(+g‘ℤring)𝐷)) |
56 | 48, 55 | sylan9eqr 2795 |
. . . 4
⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) ∧ 𝑥 = 1) → (({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘𝑥)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘𝑥)) = (𝐶(+g‘ℤring)𝐷)) |
57 | 31, 32, 33, 34, 45, 56 | fmptpr 7119 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{⟨0, (𝐴(+g‘ℤring)𝐵)⟩, ⟨1, (𝐶(+g‘ℤring)𝐷)⟩} = (𝑥 ∈ {0, 1} ↦ (({⟨0, 𝐴⟩, ⟨1, 𝐶⟩}‘𝑥)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘𝑥)))) |
58 | | zringplusg 20892 |
. . . . . . 7
⊢ + =
(+g‘ℤring) |
59 | 58 | eqcomi 2742 |
. . . . . 6
⊢
(+g‘ℤring) = + |
60 | 59 | oveqi 7371 |
. . . . 5
⊢ (𝐴(+g‘ℤring)𝐵) = (𝐴 + 𝐵) |
61 | 60 | opeq2i 4835 |
. . . 4
⊢ ⟨0,
(𝐴(+g‘ℤring)𝐵)⟩ = ⟨0, (𝐴 + 𝐵)⟩ |
62 | 59 | oveqi 7371 |
. . . . 5
⊢ (𝐶(+g‘ℤring)𝐷) = (𝐶 + 𝐷) |
63 | 62 | opeq2i 4835 |
. . . 4
⊢ ⟨1,
(𝐶(+g‘ℤring)𝐷)⟩ = ⟨1, (𝐶 + 𝐷)⟩ |
64 | 61, 63 | preq12i 4700 |
. . 3
⊢ {⟨0,
(𝐴(+g‘ℤring)𝐵)⟩, ⟨1, (𝐶(+g‘ℤring)𝐷)⟩} = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩} |
65 | 57, 64 | eqtr3di 2788 |
. 2
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝑥 ∈ {0, 1} ↦
(({⟨0, 𝐴⟩,
⟨1, 𝐶⟩}‘𝑥)(+g‘ℤring)({⟨0,
𝐵⟩, ⟨1, 𝐷⟩}‘𝑥))) = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩}) |
66 | 17, 30, 65 | 3eqtrd 2777 |
1
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({⟨0, 𝐴⟩,
⟨1, 𝐶⟩} + {⟨0,
𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩}) |