Proof of Theorem abstri
Step | Hyp | Ref
| Expression |
1 | | 2re 11904 |
. . . . . 6
⊢ 2 ∈
ℝ |
2 | 1 | a1i 11 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 2 ∈
ℝ) |
3 | | simpl 486 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈
ℂ) |
4 | | simpr 488 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈
ℂ) |
5 | 4 | cjcld 14759 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(∗‘𝐵) ∈
ℂ) |
6 | 3, 5 | mulcld 10853 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · (∗‘𝐵)) ∈
ℂ) |
7 | 6 | recld 14757 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(ℜ‘(𝐴 ·
(∗‘𝐵))) ∈
ℝ) |
8 | 2, 7 | remulcld 10863 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2
· (ℜ‘(𝐴
· (∗‘𝐵)))) ∈ ℝ) |
9 | | abscl 14842 |
. . . . . . 7
⊢ (𝐴 ∈ ℂ →
(abs‘𝐴) ∈
ℝ) |
10 | 3, 9 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(abs‘𝐴) ∈
ℝ) |
11 | | abscl 14842 |
. . . . . . 7
⊢ (𝐵 ∈ ℂ →
(abs‘𝐵) ∈
ℝ) |
12 | 4, 11 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(abs‘𝐵) ∈
ℝ) |
13 | 10, 12 | remulcld 10863 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘𝐴) ·
(abs‘𝐵)) ∈
ℝ) |
14 | 2, 13 | remulcld 10863 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2
· ((abs‘𝐴)
· (abs‘𝐵)))
∈ ℝ) |
15 | 10 | resqcld 13817 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘𝐴)↑2)
∈ ℝ) |
16 | 12 | resqcld 13817 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘𝐵)↑2)
∈ ℝ) |
17 | 15, 16 | readdcld 10862 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(((abs‘𝐴)↑2) +
((abs‘𝐵)↑2))
∈ ℝ) |
18 | | releabs 14885 |
. . . . . . 7
⊢ ((𝐴 · (∗‘𝐵)) ∈ ℂ →
(ℜ‘(𝐴 ·
(∗‘𝐵))) ≤
(abs‘(𝐴 ·
(∗‘𝐵)))) |
19 | 6, 18 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(ℜ‘(𝐴 ·
(∗‘𝐵))) ≤
(abs‘(𝐴 ·
(∗‘𝐵)))) |
20 | | absmul 14858 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(∗‘𝐵) ∈
ℂ) → (abs‘(𝐴 · (∗‘𝐵))) = ((abs‘𝐴) · (abs‘(∗‘𝐵)))) |
21 | 3, 5, 20 | syl2anc 587 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(abs‘(𝐴 ·
(∗‘𝐵))) =
((abs‘𝐴) ·
(abs‘(∗‘𝐵)))) |
22 | | abscj 14843 |
. . . . . . . . 9
⊢ (𝐵 ∈ ℂ →
(abs‘(∗‘𝐵)) = (abs‘𝐵)) |
23 | 4, 22 | syl 17 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(abs‘(∗‘𝐵)) = (abs‘𝐵)) |
24 | 23 | oveq2d 7229 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘𝐴) ·
(abs‘(∗‘𝐵))) = ((abs‘𝐴) · (abs‘𝐵))) |
25 | 21, 24 | eqtrd 2777 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(abs‘(𝐴 ·
(∗‘𝐵))) =
((abs‘𝐴) ·
(abs‘𝐵))) |
26 | 19, 25 | breqtrd 5079 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(ℜ‘(𝐴 ·
(∗‘𝐵))) ≤
((abs‘𝐴) ·
(abs‘𝐵))) |
27 | | 2rp 12591 |
. . . . . . 7
⊢ 2 ∈
ℝ+ |
28 | 27 | a1i 11 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 2 ∈
ℝ+) |
29 | 7, 13, 28 | lemul2d 12672 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((ℜ‘(𝐴 ·
(∗‘𝐵))) ≤
((abs‘𝐴) ·
(abs‘𝐵)) ↔ (2
· (ℜ‘(𝐴
· (∗‘𝐵)))) ≤ (2 · ((abs‘𝐴) · (abs‘𝐵))))) |
30 | 26, 29 | mpbid 235 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2
· (ℜ‘(𝐴
· (∗‘𝐵)))) ≤ (2 · ((abs‘𝐴) · (abs‘𝐵)))) |
31 | 8, 14, 17, 30 | leadd2dd 11447 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((((abs‘𝐴)↑2) +
((abs‘𝐵)↑2)) +
(2 · (ℜ‘(𝐴 · (∗‘𝐵))))) ≤ ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) + (2 · ((abs‘𝐴) · (abs‘𝐵))))) |
32 | | sqabsadd 14846 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘(𝐴 + 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) + (2 ·
(ℜ‘(𝐴 ·
(∗‘𝐵)))))) |
33 | 10 | recnd 10861 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(abs‘𝐴) ∈
ℂ) |
34 | 12 | recnd 10861 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(abs‘𝐵) ∈
ℂ) |
35 | | binom2 13785 |
. . . . 5
⊢
(((abs‘𝐴)
∈ ℂ ∧ (abs‘𝐵) ∈ ℂ) → (((abs‘𝐴) + (abs‘𝐵))↑2) = ((((abs‘𝐴)↑2) + (2 · ((abs‘𝐴) · (abs‘𝐵)))) + ((abs‘𝐵)↑2))) |
36 | 33, 34, 35 | syl2anc 587 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(((abs‘𝐴) +
(abs‘𝐵))↑2) =
((((abs‘𝐴)↑2) +
(2 · ((abs‘𝐴)
· (abs‘𝐵)))) +
((abs‘𝐵)↑2))) |
37 | 15 | recnd 10861 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘𝐴)↑2)
∈ ℂ) |
38 | 14 | recnd 10861 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2
· ((abs‘𝐴)
· (abs‘𝐵)))
∈ ℂ) |
39 | 16 | recnd 10861 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘𝐵)↑2)
∈ ℂ) |
40 | 37, 38, 39 | add32d 11059 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((((abs‘𝐴)↑2) +
(2 · ((abs‘𝐴)
· (abs‘𝐵)))) +
((abs‘𝐵)↑2)) =
((((abs‘𝐴)↑2) +
((abs‘𝐵)↑2)) +
(2 · ((abs‘𝐴)
· (abs‘𝐵))))) |
41 | 36, 40 | eqtrd 2777 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(((abs‘𝐴) +
(abs‘𝐵))↑2) =
((((abs‘𝐴)↑2) +
((abs‘𝐵)↑2)) +
(2 · ((abs‘𝐴)
· (abs‘𝐵))))) |
42 | 31, 32, 41 | 3brtr4d 5085 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘(𝐴 + 𝐵))↑2) ≤
(((abs‘𝐴) +
(abs‘𝐵))↑2)) |
43 | | addcl 10811 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
44 | | abscl 14842 |
. . . 4
⊢ ((𝐴 + 𝐵) ∈ ℂ → (abs‘(𝐴 + 𝐵)) ∈ ℝ) |
45 | 43, 44 | syl 17 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(abs‘(𝐴 + 𝐵)) ∈
ℝ) |
46 | 10, 12 | readdcld 10862 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘𝐴) +
(abs‘𝐵)) ∈
ℝ) |
47 | | absge0 14851 |
. . . 4
⊢ ((𝐴 + 𝐵) ∈ ℂ → 0 ≤
(abs‘(𝐴 + 𝐵))) |
48 | 43, 47 | syl 17 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 0 ≤
(abs‘(𝐴 + 𝐵))) |
49 | | absge0 14851 |
. . . . 5
⊢ (𝐴 ∈ ℂ → 0 ≤
(abs‘𝐴)) |
50 | 3, 49 | syl 17 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 0 ≤
(abs‘𝐴)) |
51 | | absge0 14851 |
. . . . 5
⊢ (𝐵 ∈ ℂ → 0 ≤
(abs‘𝐵)) |
52 | 4, 51 | syl 17 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 0 ≤
(abs‘𝐵)) |
53 | 10, 12, 50, 52 | addge0d 11408 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 0 ≤
((abs‘𝐴) +
(abs‘𝐵))) |
54 | 45, 46, 48, 53 | le2sqd 13826 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)) ↔ ((abs‘(𝐴 + 𝐵))↑2) ≤ (((abs‘𝐴) + (abs‘𝐵))↑2))) |
55 | 42, 54 | mpbird 260 |
1
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵))) |