Step |
Hyp |
Ref |
Expression |
1 |
|
breq2 |
⊢ ( 𝑧 = 𝐴 → ( 0 ≤ 𝑧 ↔ 0 ≤ 𝐴 ) ) |
2 |
1
|
elrab |
⊢ ( 𝐴 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) |
3 |
|
ssrab2 |
⊢ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ⊆ ℝ |
4 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
5 |
3 4
|
sstri |
⊢ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ⊆ ℂ |
6 |
|
breq2 |
⊢ ( 𝑧 = 𝑥 → ( 0 ≤ 𝑧 ↔ 0 ≤ 𝑥 ) ) |
7 |
6
|
elrab |
⊢ ( 𝑥 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ) |
8 |
|
breq2 |
⊢ ( 𝑧 = 𝑦 → ( 0 ≤ 𝑧 ↔ 0 ≤ 𝑦 ) ) |
9 |
8
|
elrab |
⊢ ( 𝑦 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) ) |
10 |
|
breq2 |
⊢ ( 𝑧 = ( 𝑥 · 𝑦 ) → ( 0 ≤ 𝑧 ↔ 0 ≤ ( 𝑥 · 𝑦 ) ) ) |
11 |
|
remulcl |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 · 𝑦 ) ∈ ℝ ) |
12 |
11
|
ad2ant2r |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ∧ ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ ) |
13 |
|
mulge0 |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ∧ ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) ) → 0 ≤ ( 𝑥 · 𝑦 ) ) |
14 |
10 12 13
|
elrabd |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ∧ ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) ) → ( 𝑥 · 𝑦 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ) |
15 |
7 9 14
|
syl2anb |
⊢ ( ( 𝑥 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ∧ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ) → ( 𝑥 · 𝑦 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ) |
16 |
|
1re |
⊢ 1 ∈ ℝ |
17 |
|
0le1 |
⊢ 0 ≤ 1 |
18 |
|
breq2 |
⊢ ( 𝑧 = 1 → ( 0 ≤ 𝑧 ↔ 0 ≤ 1 ) ) |
19 |
18
|
elrab |
⊢ ( 1 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( 1 ∈ ℝ ∧ 0 ≤ 1 ) ) |
20 |
16 17 19
|
mpbir2an |
⊢ 1 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } |
21 |
5 15 20
|
expcllem |
⊢ ( ( 𝐴 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ) |
22 |
|
breq2 |
⊢ ( 𝑧 = ( 𝐴 ↑ 𝑁 ) → ( 0 ≤ 𝑧 ↔ 0 ≤ ( 𝐴 ↑ 𝑁 ) ) ) |
23 |
22
|
elrab |
⊢ ( ( 𝐴 ↑ 𝑁 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ↔ ( ( 𝐴 ↑ 𝑁 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 ↑ 𝑁 ) ) ) |
24 |
23
|
simprbi |
⊢ ( ( 𝐴 ↑ 𝑁 ) ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } → 0 ≤ ( 𝐴 ↑ 𝑁 ) ) |
25 |
21 24
|
syl |
⊢ ( ( 𝐴 ∈ { 𝑧 ∈ ℝ ∣ 0 ≤ 𝑧 } ∧ 𝑁 ∈ ℕ0 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) ) |
26 |
2 25
|
sylanbr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝑁 ∈ ℕ0 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) ) |
27 |
26
|
3impa |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝑁 ∈ ℕ0 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) ) |
28 |
27
|
3com23 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) ) |