| Step | Hyp | Ref
| Expression |
| 1 | | zlmodzxz.z |
. . 3
⊢ 𝑍 = (ℤring
freeLMod {0, 1}) |
| 2 | | eqid 2737 |
. . 3
⊢
(Base‘𝑍) =
(Base‘𝑍) |
| 3 | | zringring 21460 |
. . . 4
⊢
ℤring ∈ Ring |
| 4 | 3 | a1i 11 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
ℤring ∈ Ring) |
| 5 | | prex 5437 |
. . . 4
⊢ {0, 1}
∈ V |
| 6 | 5 | a1i 11 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {0, 1}
∈ V) |
| 7 | | simpl 482 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈
ℤ) |
| 8 | | simpl 482 |
. . . 4
⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → 𝐶 ∈
ℤ) |
| 9 | 1 | zlmodzxzel 48271 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) →
{〈0, 𝐴〉, 〈1,
𝐶〉} ∈
(Base‘𝑍)) |
| 10 | 7, 8, 9 | syl2an 596 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{〈0, 𝐴〉, 〈1,
𝐶〉} ∈
(Base‘𝑍)) |
| 11 | | simpr 484 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈
ℤ) |
| 12 | | simpr 484 |
. . . 4
⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → 𝐷 ∈
ℤ) |
| 13 | 1 | zlmodzxzel 48271 |
. . . 4
⊢ ((𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) →
{〈0, 𝐵〉, 〈1,
𝐷〉} ∈
(Base‘𝑍)) |
| 14 | 11, 12, 13 | syl2an 596 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{〈0, 𝐵〉, 〈1,
𝐷〉} ∈
(Base‘𝑍)) |
| 15 | | eqid 2737 |
. . 3
⊢
(+g‘ℤring) =
(+g‘ℤring) |
| 16 | | zlmodzxzadd.p |
. . 3
⊢ + =
(+g‘𝑍) |
| 17 | 1, 2, 4, 6, 10, 14, 15, 16 | frlmplusgval 21784 |
. 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 11255 |
. . . . . 6
⊢ 0 ∈
V |
| 19 | | 1ex 11257 |
. . . . . 6
⊢ 1 ∈
V |
| 20 | 18, 19 | pm3.2i 470 |
. . . . 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 613 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐴 ∈ ℤ ∧ 𝐶 ∈
ℤ)) |
| 23 | | 0ne1 12337 |
. . . . 5
⊢ 0 ≠
1 |
| 24 | 23 | a1i 11 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 0 ≠
1) |
| 25 | | fnprg 6625 |
. . . 4
⊢ (((0
∈ V ∧ 1 ∈ V) ∧ (𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 0 ≠ 1) →
{〈0, 𝐴〉, 〈1,
𝐶〉} Fn {0,
1}) |
| 26 | 21, 22, 24, 25 | syl3anc 1373 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{〈0, 𝐴〉, 〈1,
𝐶〉} Fn {0,
1}) |
| 27 | 11, 12 | anim12i 613 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐵 ∈ ℤ ∧ 𝐷 ∈
ℤ)) |
| 28 | | fnprg 6625 |
. . . 4
⊢ (((0
∈ V ∧ 1 ∈ V) ∧ (𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ 0 ≠ 1) →
{〈0, 𝐵〉, 〈1,
𝐷〉} Fn {0,
1}) |
| 29 | 21, 27, 24, 28 | syl3anc 1373 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
{〈0, 𝐵〉, 〈1,
𝐷〉} Fn {0,
1}) |
| 30 | 6, 26, 29 | offvalfv 7719 |
. 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 7466 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐴(+g‘ℤring)𝐵) ∈ V) |
| 34 | | ovexd 7466 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐶(+g‘ℤring)𝐷) ∈ V) |
| 35 | | fveq2 6906 |
. . . . . 6
⊢ (𝑥 = 0 → ({〈0, 𝐴〉, 〈1, 𝐶〉}‘𝑥) = ({〈0, 𝐴〉, 〈1, 𝐶〉}‘0)) |
| 36 | | fveq2 6906 |
. . . . . 6
⊢ (𝑥 = 0 → ({〈0, 𝐵〉, 〈1, 𝐷〉}‘𝑥) = ({〈0, 𝐵〉, 〈1, 𝐷〉}‘0)) |
| 37 | 35, 36 | oveq12d 7449 |
. . . . 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 480 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐴 ∈
ℤ) |
| 39 | | fvpr1g 7210 |
. . . . . . 7
⊢ ((0
∈ V ∧ 𝐴 ∈
ℤ ∧ 0 ≠ 1) → ({〈0, 𝐴〉, 〈1, 𝐶〉}‘0) = 𝐴) |
| 40 | 31, 38, 24, 39 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐴〉,
〈1, 𝐶〉}‘0)
= 𝐴) |
| 41 | 11 | adantr 480 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐵 ∈
ℤ) |
| 42 | | fvpr1g 7210 |
. . . . . . 7
⊢ ((0
∈ V ∧ 𝐵 ∈
ℤ ∧ 0 ≠ 1) → ({〈0, 𝐵〉, 〈1, 𝐷〉}‘0) = 𝐵) |
| 43 | 31, 41, 24, 42 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐵〉,
〈1, 𝐷〉}‘0)
= 𝐵) |
| 44 | 40, 43 | oveq12d 7449 |
. . . . 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 2799 |
. . . 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 6906 |
. . . . . 6
⊢ (𝑥 = 1 → ({〈0, 𝐴〉, 〈1, 𝐶〉}‘𝑥) = ({〈0, 𝐴〉, 〈1, 𝐶〉}‘1)) |
| 47 | | fveq2 6906 |
. . . . . 6
⊢ (𝑥 = 1 → ({〈0, 𝐵〉, 〈1, 𝐷〉}‘𝑥) = ({〈0, 𝐵〉, 〈1, 𝐷〉}‘1)) |
| 48 | 46, 47 | oveq12d 7449 |
. . . . 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 481 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐶 ∈
ℤ) |
| 50 | | fvpr2g 7211 |
. . . . . . 7
⊢ ((1
∈ V ∧ 𝐶 ∈
ℤ ∧ 0 ≠ 1) → ({〈0, 𝐴〉, 〈1, 𝐶〉}‘1) = 𝐶) |
| 51 | 32, 49, 24, 50 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐴〉,
〈1, 𝐶〉}‘1)
= 𝐶) |
| 52 | 12 | adantl 481 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝐷 ∈
ℤ) |
| 53 | | fvpr2g 7211 |
. . . . . . 7
⊢ ((1
∈ V ∧ 𝐷 ∈
ℤ ∧ 0 ≠ 1) → ({〈0, 𝐵〉, 〈1, 𝐷〉}‘1) = 𝐷) |
| 54 | 32, 52, 24, 53 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐵〉,
〈1, 𝐷〉}‘1)
= 𝐷) |
| 55 | 51, 54 | oveq12d 7449 |
. . . . 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 2799 |
. . . 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 7192 |
. . 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 21465 |
. . . . . . 7
⊢ + =
(+g‘ℤring) |
| 59 | 58 | eqcomi 2746 |
. . . . . 6
⊢
(+g‘ℤring) = + |
| 60 | 59 | oveqi 7444 |
. . . . 5
⊢ (𝐴(+g‘ℤring)𝐵) = (𝐴 + 𝐵) |
| 61 | 60 | opeq2i 4877 |
. . . 4
⊢ 〈0,
(𝐴(+g‘ℤring)𝐵)〉 = 〈0, (𝐴 + 𝐵)〉 |
| 62 | 59 | oveqi 7444 |
. . . . 5
⊢ (𝐶(+g‘ℤring)𝐷) = (𝐶 + 𝐷) |
| 63 | 62 | opeq2i 4877 |
. . . 4
⊢ 〈1,
(𝐶(+g‘ℤring)𝐷)〉 = 〈1, (𝐶 + 𝐷)〉 |
| 64 | 61, 63 | preq12i 4738 |
. . 3
⊢ {〈0,
(𝐴(+g‘ℤring)𝐵)〉, 〈1, (𝐶(+g‘ℤring)𝐷)〉} = {〈0, (𝐴 + 𝐵)〉, 〈1, (𝐶 + 𝐷)〉} |
| 65 | 57, 64 | eqtr3di 2792 |
. 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 2781 |
1
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) →
({〈0, 𝐴〉,
〈1, 𝐶〉} + {〈0,
𝐵〉, 〈1, 𝐷〉}) = {〈0, (𝐴 + 𝐵)〉, 〈1, (𝐶 + 𝐷)〉}) |