Proof of Theorem sgnmulsgp
| Step | Hyp | Ref
| Expression |
| 1 | | 0lt1 11785 |
. . . . 5
⊢ 0 <
1 |
| 2 | | breq2 5147 |
. . . . 5
⊢
((sgn‘(𝐴
· 𝐵)) = 1 → (0
< (sgn‘(𝐴 ·
𝐵)) ↔ 0 <
1)) |
| 3 | 1, 2 | mpbiri 258 |
. . . 4
⊢
((sgn‘(𝐴
· 𝐵)) = 1 → 0
< (sgn‘(𝐴 ·
𝐵))) |
| 4 | 3 | adantl 481 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧
(sgn‘(𝐴 ·
𝐵)) = 1) → 0 <
(sgn‘(𝐴 ·
𝐵))) |
| 5 | | simplr 769 |
. . . . . 6
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = -1) → 0 <
(sgn‘(𝐴 ·
𝐵))) |
| 6 | | simpr 484 |
. . . . . 6
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = -1) →
(sgn‘(𝐴 ·
𝐵)) = -1) |
| 7 | 5, 6 | breqtrd 5169 |
. . . . 5
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = -1) → 0 <
-1) |
| 8 | | 1nn0 12542 |
. . . . . . . 8
⊢ 1 ∈
ℕ0 |
| 9 | | nn0nlt0 12552 |
. . . . . . . 8
⊢ (1 ∈
ℕ0 → ¬ 1 < 0) |
| 10 | 8, 9 | ax-mp 5 |
. . . . . . 7
⊢ ¬ 1
< 0 |
| 11 | | 1re 11261 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
| 12 | | lt0neg1 11769 |
. . . . . . . 8
⊢ (1 ∈
ℝ → (1 < 0 ↔ 0 < -1)) |
| 13 | 11, 12 | ax-mp 5 |
. . . . . . 7
⊢ (1 < 0
↔ 0 < -1) |
| 14 | 10, 13 | mtbi 322 |
. . . . . 6
⊢ ¬ 0
< -1 |
| 15 | 14 | a1i 11 |
. . . . 5
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = -1) → ¬ 0
< -1) |
| 16 | 7, 15 | pm2.21dd 195 |
. . . 4
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = -1) →
(sgn‘(𝐴 ·
𝐵)) = 1) |
| 17 | | simpr 484 |
. . . . 5
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = 0) →
(sgn‘(𝐴 ·
𝐵)) = 0) |
| 18 | | simplr 769 |
. . . . . 6
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = 0) → 0 <
(sgn‘(𝐴 ·
𝐵))) |
| 19 | 18 | gt0ne0d 11827 |
. . . . 5
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = 0) →
(sgn‘(𝐴 ·
𝐵)) ≠
0) |
| 20 | 17, 19 | pm2.21ddne 3026 |
. . . 4
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = 0) →
(sgn‘(𝐴 ·
𝐵)) = 1) |
| 21 | | simpr 484 |
. . . 4
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) ∧
(sgn‘(𝐴 ·
𝐵)) = 1) →
(sgn‘(𝐴 ·
𝐵)) = 1) |
| 22 | | remulcl 11240 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
| 23 | 22 | rexrd 11311 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈
ℝ*) |
| 24 | 23 | adantr 480 |
. . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) → (𝐴 · 𝐵) ∈
ℝ*) |
| 25 | | sgncl 34541 |
. . . . 5
⊢ ((𝐴 · 𝐵) ∈ ℝ* →
(sgn‘(𝐴 ·
𝐵)) ∈ {-1, 0,
1}) |
| 26 | | eltpi 4688 |
. . . . 5
⊢
((sgn‘(𝐴
· 𝐵)) ∈ {-1, 0,
1} → ((sgn‘(𝐴
· 𝐵)) = -1 ∨
(sgn‘(𝐴 ·
𝐵)) = 0 ∨
(sgn‘(𝐴 ·
𝐵)) = 1)) |
| 27 | 24, 25, 26 | 3syl 18 |
. . . 4
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) →
((sgn‘(𝐴 ·
𝐵)) = -1 ∨
(sgn‘(𝐴 ·
𝐵)) = 0 ∨
(sgn‘(𝐴 ·
𝐵)) = 1)) |
| 28 | 16, 20, 21, 27 | mpjao3dan 1434 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 0 <
(sgn‘(𝐴 ·
𝐵))) →
(sgn‘(𝐴 ·
𝐵)) = 1) |
| 29 | 4, 28 | impbida 801 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
((sgn‘(𝐴 ·
𝐵)) = 1 ↔ 0 <
(sgn‘(𝐴 ·
𝐵)))) |
| 30 | | sgnpbi 34549 |
. . 3
⊢ ((𝐴 · 𝐵) ∈ ℝ* →
((sgn‘(𝐴 ·
𝐵)) = 1 ↔ 0 <
(𝐴 · 𝐵))) |
| 31 | 23, 30 | syl 17 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
((sgn‘(𝐴 ·
𝐵)) = 1 ↔ 0 <
(𝐴 · 𝐵))) |
| 32 | | sgnmul 34545 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
(sgn‘(𝐴 ·
𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵))) |
| 33 | 32 | breq2d 5155 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 <
(sgn‘(𝐴 ·
𝐵)) ↔ 0 <
((sgn‘𝐴) ·
(sgn‘𝐵)))) |
| 34 | 29, 31, 33 | 3bitr3d 309 |
1
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 <
(𝐴 · 𝐵) ↔ 0 <
((sgn‘𝐴) ·
(sgn‘𝐵)))) |