Proof of Theorem zringlpirlem3
Step | Hyp | Ref
| Expression |
1 | | zringlpirlem.i |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈
(LIdeal‘ℤring)) |
2 | | zringbas 19643 |
. . . . . . . . . 10
⊢ ℤ =
(Base‘ℤring) |
3 | | eqid 2610 |
. . . . . . . . . 10
⊢
(LIdeal‘ℤring) =
(LIdeal‘ℤring) |
4 | 2, 3 | lidlss 19031 |
. . . . . . . . 9
⊢ (𝐼 ∈
(LIdeal‘ℤring) → 𝐼 ⊆ ℤ) |
5 | 1, 4 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐼 ⊆ ℤ) |
6 | | zringlpirlem.x |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ 𝐼) |
7 | 5, 6 | sseldd 3569 |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ ℤ) |
8 | 7 | zred 11358 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ ℝ) |
9 | | inss2 3796 |
. . . . . . . 8
⊢ (𝐼 ∩ ℕ) ⊆
ℕ |
10 | | zringlpirlem.g |
. . . . . . . . 9
⊢ 𝐺 = inf((𝐼 ∩ ℕ), ℝ, <
) |
11 | | nnuz 11599 |
. . . . . . . . . . 11
⊢ ℕ =
(ℤ≥‘1) |
12 | 9, 11 | sseqtri 3600 |
. . . . . . . . . 10
⊢ (𝐼 ∩ ℕ) ⊆
(ℤ≥‘1) |
13 | | zringlpirlem.n0 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ≠ {0}) |
14 | 1, 13 | zringlpirlem1 19651 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐼 ∩ ℕ) ≠
∅) |
15 | | infssuzcl 11648 |
. . . . . . . . . 10
⊢ (((𝐼 ∩ ℕ) ⊆
(ℤ≥‘1) ∧ (𝐼 ∩ ℕ) ≠ ∅) →
inf((𝐼 ∩ ℕ),
ℝ, < ) ∈ (𝐼
∩ ℕ)) |
16 | 12, 14, 15 | sylancr 694 |
. . . . . . . . 9
⊢ (𝜑 → inf((𝐼 ∩ ℕ), ℝ, < ) ∈
(𝐼 ∩
ℕ)) |
17 | 10, 16 | syl5eqel 2692 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ (𝐼 ∩ ℕ)) |
18 | 9, 17 | sseldi 3566 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ ℕ) |
19 | 18 | nnrpd 11746 |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈
ℝ+) |
20 | | modlt 12541 |
. . . . . 6
⊢ ((𝑋 ∈ ℝ ∧ 𝐺 ∈ ℝ+)
→ (𝑋 mod 𝐺) < 𝐺) |
21 | 8, 19, 20 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → (𝑋 mod 𝐺) < 𝐺) |
22 | 7, 18 | zmodcld 12553 |
. . . . . . 7
⊢ (𝜑 → (𝑋 mod 𝐺) ∈
ℕ0) |
23 | 22 | nn0red 11229 |
. . . . . 6
⊢ (𝜑 → (𝑋 mod 𝐺) ∈ ℝ) |
24 | 18 | nnred 10912 |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ ℝ) |
25 | 23, 24 | ltnled 10063 |
. . . . 5
⊢ (𝜑 → ((𝑋 mod 𝐺) < 𝐺 ↔ ¬ 𝐺 ≤ (𝑋 mod 𝐺))) |
26 | 21, 25 | mpbid 221 |
. . . 4
⊢ (𝜑 → ¬ 𝐺 ≤ (𝑋 mod 𝐺)) |
27 | 7 | zcnd 11359 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈ ℂ) |
28 | 18 | nncnd 10913 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 ∈ ℂ) |
29 | 8, 18 | nndivred 10946 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑋 / 𝐺) ∈ ℝ) |
30 | 29 | flcld 12461 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (⌊‘(𝑋 / 𝐺)) ∈ ℤ) |
31 | 30 | zcnd 11359 |
. . . . . . . . . . . 12
⊢ (𝜑 → (⌊‘(𝑋 / 𝐺)) ∈ ℂ) |
32 | 28, 31 | mulcld 9939 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺 · (⌊‘(𝑋 / 𝐺))) ∈ ℂ) |
33 | 27, 32 | negsubd 10277 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑋 + -(𝐺 · (⌊‘(𝑋 / 𝐺)))) = (𝑋 − (𝐺 · (⌊‘(𝑋 / 𝐺))))) |
34 | 30 | znegcld 11360 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → -(⌊‘(𝑋 / 𝐺)) ∈ ℤ) |
35 | 34 | zcnd 11359 |
. . . . . . . . . . . . 13
⊢ (𝜑 → -(⌊‘(𝑋 / 𝐺)) ∈ ℂ) |
36 | 35, 28 | mulcomd 9940 |
. . . . . . . . . . . 12
⊢ (𝜑 → (-(⌊‘(𝑋 / 𝐺)) · 𝐺) = (𝐺 · -(⌊‘(𝑋 / 𝐺)))) |
37 | 28, 31 | mulneg2d 10363 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐺 · -(⌊‘(𝑋 / 𝐺))) = -(𝐺 · (⌊‘(𝑋 / 𝐺)))) |
38 | 36, 37 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ (𝜑 → (-(⌊‘(𝑋 / 𝐺)) · 𝐺) = -(𝐺 · (⌊‘(𝑋 / 𝐺)))) |
39 | 38 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑋 + (-(⌊‘(𝑋 / 𝐺)) · 𝐺)) = (𝑋 + -(𝐺 · (⌊‘(𝑋 / 𝐺))))) |
40 | | modval 12532 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ ℝ ∧ 𝐺 ∈ ℝ+)
→ (𝑋 mod 𝐺) = (𝑋 − (𝐺 · (⌊‘(𝑋 / 𝐺))))) |
41 | 8, 19, 40 | syl2anc 691 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑋 mod 𝐺) = (𝑋 − (𝐺 · (⌊‘(𝑋 / 𝐺))))) |
42 | 33, 39, 41 | 3eqtr4rd 2655 |
. . . . . . . . 9
⊢ (𝜑 → (𝑋 mod 𝐺) = (𝑋 + (-(⌊‘(𝑋 / 𝐺)) · 𝐺))) |
43 | | zringring 19640 |
. . . . . . . . . . 11
⊢
ℤring ∈ Ring |
44 | 43 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℤring
∈ Ring) |
45 | 1, 13, 10 | zringlpirlem2 19652 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ∈ 𝐼) |
46 | | zringmulr 19646 |
. . . . . . . . . . . 12
⊢ ·
= (.r‘ℤring) |
47 | 3, 2, 46 | lidlmcl 19038 |
. . . . . . . . . . 11
⊢
(((ℤring ∈ Ring ∧ 𝐼 ∈
(LIdeal‘ℤring)) ∧ (-(⌊‘(𝑋 / 𝐺)) ∈ ℤ ∧ 𝐺 ∈ 𝐼)) → (-(⌊‘(𝑋 / 𝐺)) · 𝐺) ∈ 𝐼) |
48 | 44, 1, 34, 45, 47 | syl22anc 1319 |
. . . . . . . . . 10
⊢ (𝜑 → (-(⌊‘(𝑋 / 𝐺)) · 𝐺) ∈ 𝐼) |
49 | | zringplusg 19644 |
. . . . . . . . . . 11
⊢ + =
(+g‘ℤring) |
50 | 3, 49 | lidlacl 19034 |
. . . . . . . . . 10
⊢
(((ℤring ∈ Ring ∧ 𝐼 ∈
(LIdeal‘ℤring)) ∧ (𝑋 ∈ 𝐼 ∧ (-(⌊‘(𝑋 / 𝐺)) · 𝐺) ∈ 𝐼)) → (𝑋 + (-(⌊‘(𝑋 / 𝐺)) · 𝐺)) ∈ 𝐼) |
51 | 44, 1, 6, 48, 50 | syl22anc 1319 |
. . . . . . . . 9
⊢ (𝜑 → (𝑋 + (-(⌊‘(𝑋 / 𝐺)) · 𝐺)) ∈ 𝐼) |
52 | 42, 51 | eqeltrd 2688 |
. . . . . . . 8
⊢ (𝜑 → (𝑋 mod 𝐺) ∈ 𝐼) |
53 | 52 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑋 mod 𝐺) ∈ ℕ) → (𝑋 mod 𝐺) ∈ 𝐼) |
54 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑋 mod 𝐺) ∈ ℕ) → (𝑋 mod 𝐺) ∈ ℕ) |
55 | 53, 54 | elind 3760 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑋 mod 𝐺) ∈ ℕ) → (𝑋 mod 𝐺) ∈ (𝐼 ∩ ℕ)) |
56 | | infssuzle 11647 |
. . . . . 6
⊢ (((𝐼 ∩ ℕ) ⊆
(ℤ≥‘1) ∧ (𝑋 mod 𝐺) ∈ (𝐼 ∩ ℕ)) → inf((𝐼 ∩ ℕ), ℝ, < )
≤ (𝑋 mod 𝐺)) |
57 | 12, 55, 56 | sylancr 694 |
. . . . 5
⊢ ((𝜑 ∧ (𝑋 mod 𝐺) ∈ ℕ) → inf((𝐼 ∩ ℕ), ℝ, < )
≤ (𝑋 mod 𝐺)) |
58 | 10, 57 | syl5eqbr 4618 |
. . . 4
⊢ ((𝜑 ∧ (𝑋 mod 𝐺) ∈ ℕ) → 𝐺 ≤ (𝑋 mod 𝐺)) |
59 | 26, 58 | mtand 689 |
. . 3
⊢ (𝜑 → ¬ (𝑋 mod 𝐺) ∈ ℕ) |
60 | | elnn0 11171 |
. . . 4
⊢ ((𝑋 mod 𝐺) ∈ ℕ0 ↔ ((𝑋 mod 𝐺) ∈ ℕ ∨ (𝑋 mod 𝐺) = 0)) |
61 | 22, 60 | sylib 207 |
. . 3
⊢ (𝜑 → ((𝑋 mod 𝐺) ∈ ℕ ∨ (𝑋 mod 𝐺) = 0)) |
62 | | orel1 396 |
. . 3
⊢ (¬
(𝑋 mod 𝐺) ∈ ℕ → (((𝑋 mod 𝐺) ∈ ℕ ∨ (𝑋 mod 𝐺) = 0) → (𝑋 mod 𝐺) = 0)) |
63 | 59, 61, 62 | sylc 63 |
. 2
⊢ (𝜑 → (𝑋 mod 𝐺) = 0) |
64 | | dvdsval3 14825 |
. . 3
⊢ ((𝐺 ∈ ℕ ∧ 𝑋 ∈ ℤ) → (𝐺 ∥ 𝑋 ↔ (𝑋 mod 𝐺) = 0)) |
65 | 18, 7, 64 | syl2anc 691 |
. 2
⊢ (𝜑 → (𝐺 ∥ 𝑋 ↔ (𝑋 mod 𝐺) = 0)) |
66 | 63, 65 | mpbird 246 |
1
⊢ (𝜑 → 𝐺 ∥ 𝑋) |