Step | Hyp | Ref
| Expression |
1 | | zlmodzxz.z |
. . 3
⊢ 𝑍 = (ℤring
freeLMod {0, 1}) |
2 | | eqid 2771 |
. . 3
⊢
(Base‘𝑍) =
(Base‘𝑍) |
3 | | zringring 20337 |
. . . 4
⊢
ℤring ∈ Ring |
4 | 3 | a1i 11 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
ℤring ∈ Ring) |
5 | | prex 5185 |
. . . 4
⊢ {0, 1}
∈ V |
6 | 5 | a1i 11 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {0, 1}
∈ V) |
7 | | simpl 475 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈
ℤ) |
8 | | simpl 475 |
. . . 4
⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → 𝐶 ∈
ℤ) |
9 | 1 | zlmodzxzel 43801 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) →
{〈0, 𝐴〉, 〈1,
𝐶〉} ∈
(Base‘𝑍)) |
10 | 7, 8, 9 | syl2an 587 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{〈0, 𝐴〉, 〈1,
𝐶〉} ∈
(Base‘𝑍)) |
11 | | simpr 477 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈
ℤ) |
12 | | simpr 477 |
. . . 4
⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → 𝐷 ∈
ℤ) |
13 | 1 | zlmodzxzel 43801 |
. . . 4
⊢ ((𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) →
{〈0, 𝐵〉, 〈1,
𝐷〉} ∈
(Base‘𝑍)) |
14 | 11, 12, 13 | syl2an 587 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{〈0, 𝐵〉, 〈1,
𝐷〉} ∈
(Base‘𝑍)) |
15 | | eqid 2771 |
. . 3
⊢
(+g‘ℤring) =
(+g‘ℤring) |
16 | | zlmodzxzadd.p |
. . 3
⊢ + =
(+g‘𝑍) |
17 | 1, 2, 4, 6, 10, 14, 15, 16 | frlmplusgval 20625 |
. 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 10431 |
. . . . . 6
⊢ 0 ∈
V |
19 | | 1ex 10433 |
. . . . . 6
⊢ 1 ∈
V |
20 | 18, 19 | pm3.2i 463 |
. . . . 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 604 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐴 ∈ ℤ ∧ 𝐶 ∈
ℤ)) |
23 | | 0ne1 11509 |
. . . . 5
⊢ 0 ≠
1 |
24 | 23 | a1i 11 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 0 ≠
1) |
25 | | fnprg 6243 |
. . . 4
⊢ (((0
∈ V ∧ 1 ∈ V) ∧ (𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 0 ≠ 1) →
{〈0, 𝐴〉, 〈1,
𝐶〉} Fn {0,
1}) |
26 | 21, 22, 24, 25 | syl3anc 1352 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{〈0, 𝐴〉, 〈1,
𝐶〉} Fn {0,
1}) |
27 | 11, 12 | anim12i 604 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐵 ∈ ℤ ∧ 𝐷 ∈
ℤ)) |
28 | | fnprg 6243 |
. . . 4
⊢ (((0
∈ V ∧ 1 ∈ V) ∧ (𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ 0 ≠ 1) →
{〈0, 𝐵〉, 〈1,
𝐷〉} Fn {0,
1}) |
29 | 21, 27, 24, 28 | syl3anc 1352 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{〈0, 𝐵〉, 〈1,
𝐷〉} Fn {0,
1}) |
30 | 6, 26, 29 | offvalfv 43789 |
. 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 | | zringplusg 20341 |
. . . . . . 7
⊢ + =
(+g‘ℤring) |
32 | 31 | eqcomi 2780 |
. . . . . 6
⊢
(+g‘ℤring) = + |
33 | 32 | oveqi 6987 |
. . . . 5
⊢ (𝐴(+g‘ℤring)𝐵) = (𝐴 + 𝐵) |
34 | 33 | opeq2i 4677 |
. . . 4
⊢ 〈0,
(𝐴(+g‘ℤring)𝐵)〉 = 〈0, (𝐴 + 𝐵)〉 |
35 | 32 | oveqi 6987 |
. . . . 5
⊢ (𝐶(+g‘ℤring)𝐷) = (𝐶 + 𝐷) |
36 | 35 | opeq2i 4677 |
. . . 4
⊢ 〈1,
(𝐶(+g‘ℤring)𝐷)〉 = 〈1, (𝐶 + 𝐷)〉 |
37 | 34, 36 | preq12i 4544 |
. . 3
⊢ {〈0,
(𝐴(+g‘ℤring)𝐵)〉, 〈1, (𝐶(+g‘ℤring)𝐷)〉} = {〈0, (𝐴 + 𝐵)〉, 〈1, (𝐶 + 𝐷)〉} |
38 | 18 | a1i 11 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 0
∈ V) |
39 | 19 | a1i 11 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 1
∈ V) |
40 | | ovexd 7008 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐴(+g‘ℤring)𝐵) ∈ V) |
41 | | ovexd 7008 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐶(+g‘ℤring)𝐷) ∈ V) |
42 | | fveq2 6496 |
. . . . . 6
⊢ (𝑥 = 0 → ({〈0, 𝐴〉, 〈1, 𝐶〉}‘𝑥) = ({〈0, 𝐴〉, 〈1, 𝐶〉}‘0)) |
43 | | fveq2 6496 |
. . . . . 6
⊢ (𝑥 = 0 → ({〈0, 𝐵〉, 〈1, 𝐷〉}‘𝑥) = ({〈0, 𝐵〉, 〈1, 𝐷〉}‘0)) |
44 | 42, 43 | oveq12d 6992 |
. . . . 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))) |
45 | 7 | adantr 473 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐴 ∈
ℤ) |
46 | | fvpr1g 6779 |
. . . . . . 7
⊢ ((0
∈ V ∧ 𝐴 ∈
ℤ ∧ 0 ≠ 1) → ({〈0, 𝐴〉, 〈1, 𝐶〉}‘0) = 𝐴) |
47 | 38, 45, 24, 46 | syl3anc 1352 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐴〉,
〈1, 𝐶〉}‘0)
= 𝐴) |
48 | 11 | adantr 473 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐵 ∈
ℤ) |
49 | | fvpr1g 6779 |
. . . . . . 7
⊢ ((0
∈ V ∧ 𝐵 ∈
ℤ ∧ 0 ≠ 1) → ({〈0, 𝐵〉, 〈1, 𝐷〉}‘0) = 𝐵) |
50 | 38, 48, 24, 49 | syl3anc 1352 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐵〉,
〈1, 𝐷〉}‘0)
= 𝐵) |
51 | 47, 50 | oveq12d 6992 |
. . . . 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)) |
52 | 44, 51 | sylan9eqr 2829 |
. . . 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)) |
53 | | fveq2 6496 |
. . . . . 6
⊢ (𝑥 = 1 → ({〈0, 𝐴〉, 〈1, 𝐶〉}‘𝑥) = ({〈0, 𝐴〉, 〈1, 𝐶〉}‘1)) |
54 | | fveq2 6496 |
. . . . . 6
⊢ (𝑥 = 1 → ({〈0, 𝐵〉, 〈1, 𝐷〉}‘𝑥) = ({〈0, 𝐵〉, 〈1, 𝐷〉}‘1)) |
55 | 53, 54 | oveq12d 6992 |
. . . . 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))) |
56 | 8 | adantl 474 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐶 ∈
ℤ) |
57 | | fvpr2g 6780 |
. . . . . . 7
⊢ ((1
∈ V ∧ 𝐶 ∈
ℤ ∧ 0 ≠ 1) → ({〈0, 𝐴〉, 〈1, 𝐶〉}‘1) = 𝐶) |
58 | 39, 56, 24, 57 | syl3anc 1352 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐴〉,
〈1, 𝐶〉}‘1)
= 𝐶) |
59 | 12 | adantl 474 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐷 ∈
ℤ) |
60 | | fvpr2g 6780 |
. . . . . . 7
⊢ ((1
∈ V ∧ 𝐷 ∈
ℤ ∧ 0 ≠ 1) → ({〈0, 𝐵〉, 〈1, 𝐷〉}‘1) = 𝐷) |
61 | 39, 59, 24, 60 | syl3anc 1352 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐵〉,
〈1, 𝐷〉}‘1)
= 𝐷) |
62 | 58, 61 | oveq12d 6992 |
. . . . 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)) |
63 | 55, 62 | sylan9eqr 2829 |
. . . 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)) |
64 | 38, 39, 40, 41, 52, 63 | fmptpr 6755 |
. . 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)))) |
65 | 37, 64 | syl5reqr 2822 |
. 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 2811 |
1
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐴〉,
〈1, 𝐶〉} + {〈0,
𝐵〉, 〈1, 𝐷〉}) = {〈0, (𝐴 + 𝐵)〉, 〈1, (𝐶 + 𝐷)〉}) |