Step |
Hyp |
Ref |
Expression |
1 |
|
eluzelcn |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 𝐵 ∈ ℂ ) |
2 |
1
|
exp0d |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐵 ↑ 0 ) = 1 ) |
3 |
2
|
eqcomd |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 1 = ( 𝐵 ↑ 0 ) ) |
4 |
3
|
ad2antrl |
⊢ ( ( 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → 1 = ( 𝐵 ↑ 0 ) ) |
5 |
4
|
oveq2d |
⊢ ( ( 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → ( 𝐾 ( digit ‘ 𝐵 ) 1 ) = ( 𝐾 ( digit ‘ 𝐵 ) ( 𝐵 ↑ 0 ) ) ) |
6 |
|
simprl |
⊢ ( ( 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → 𝐵 ∈ ( ℤ≥ ‘ 2 ) ) |
7 |
|
simpr |
⊢ ( ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) → 𝐾 ∈ ℤ ) |
8 |
7
|
anim2i |
⊢ ( ( 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → ( 0 ≤ 𝐾 ∧ 𝐾 ∈ ℤ ) ) |
9 |
8
|
ancomd |
⊢ ( ( 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → ( 𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ) ) |
10 |
|
elnn0z |
⊢ ( 𝐾 ∈ ℕ0 ↔ ( 𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ) ) |
11 |
9 10
|
sylibr |
⊢ ( ( 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → 𝐾 ∈ ℕ0 ) |
12 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
13 |
12
|
a1i |
⊢ ( ( 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → 0 ∈ ℕ0 ) |
14 |
|
digexp |
⊢ ( ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℕ0 ∧ 0 ∈ ℕ0 ) → ( 𝐾 ( digit ‘ 𝐵 ) ( 𝐵 ↑ 0 ) ) = if ( 𝐾 = 0 , 1 , 0 ) ) |
15 |
6 11 13 14
|
syl3anc |
⊢ ( ( 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → ( 𝐾 ( digit ‘ 𝐵 ) ( 𝐵 ↑ 0 ) ) = if ( 𝐾 = 0 , 1 , 0 ) ) |
16 |
5 15
|
eqtrd |
⊢ ( ( 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → ( 𝐾 ( digit ‘ 𝐵 ) 1 ) = if ( 𝐾 = 0 , 1 , 0 ) ) |
17 |
|
eluz2nn |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) → 𝐵 ∈ ℕ ) |
18 |
17
|
ad2antrl |
⊢ ( ( ¬ 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → 𝐵 ∈ ℕ ) |
19 |
|
simprr |
⊢ ( ( ¬ 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → 𝐾 ∈ ℤ ) |
20 |
|
nn0ge0 |
⊢ ( 𝐾 ∈ ℕ0 → 0 ≤ 𝐾 ) |
21 |
20
|
a1i |
⊢ ( ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) → ( 𝐾 ∈ ℕ0 → 0 ≤ 𝐾 ) ) |
22 |
21
|
con3d |
⊢ ( ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) → ( ¬ 0 ≤ 𝐾 → ¬ 𝐾 ∈ ℕ0 ) ) |
23 |
22
|
impcom |
⊢ ( ( ¬ 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → ¬ 𝐾 ∈ ℕ0 ) |
24 |
19 23
|
eldifd |
⊢ ( ( ¬ 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → 𝐾 ∈ ( ℤ ∖ ℕ0 ) ) |
25 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
26 |
25
|
a1i |
⊢ ( ( ¬ 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → 1 ∈ ℕ0 ) |
27 |
|
dignn0fr |
⊢ ( ( 𝐵 ∈ ℕ ∧ 𝐾 ∈ ( ℤ ∖ ℕ0 ) ∧ 1 ∈ ℕ0 ) → ( 𝐾 ( digit ‘ 𝐵 ) 1 ) = 0 ) |
28 |
18 24 26 27
|
syl3anc |
⊢ ( ( ¬ 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → ( 𝐾 ( digit ‘ 𝐵 ) 1 ) = 0 ) |
29 |
|
0le0 |
⊢ 0 ≤ 0 |
30 |
|
breq2 |
⊢ ( 𝐾 = 0 → ( 0 ≤ 𝐾 ↔ 0 ≤ 0 ) ) |
31 |
29 30
|
mpbiri |
⊢ ( 𝐾 = 0 → 0 ≤ 𝐾 ) |
32 |
31
|
a1i |
⊢ ( ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) → ( 𝐾 = 0 → 0 ≤ 𝐾 ) ) |
33 |
32
|
con3d |
⊢ ( ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) → ( ¬ 0 ≤ 𝐾 → ¬ 𝐾 = 0 ) ) |
34 |
33
|
impcom |
⊢ ( ( ¬ 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → ¬ 𝐾 = 0 ) |
35 |
34
|
iffalsed |
⊢ ( ( ¬ 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → if ( 𝐾 = 0 , 1 , 0 ) = 0 ) |
36 |
28 35
|
eqtr4d |
⊢ ( ( ¬ 0 ≤ 𝐾 ∧ ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) ) → ( 𝐾 ( digit ‘ 𝐵 ) 1 ) = if ( 𝐾 = 0 , 1 , 0 ) ) |
37 |
16 36
|
pm2.61ian |
⊢ ( ( 𝐵 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐾 ∈ ℤ ) → ( 𝐾 ( digit ‘ 𝐵 ) 1 ) = if ( 𝐾 = 0 , 1 , 0 ) ) |