Step |
Hyp |
Ref |
Expression |
1 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
2 |
|
1re |
⊢ 1 ∈ ℝ |
3 |
|
0re |
⊢ 0 ∈ ℝ |
4 |
2 3
|
lttri2i |
⊢ ( 1 ≠ 0 ↔ ( 1 < 0 ∨ 0 < 1 ) ) |
5 |
1 4
|
mpbi |
⊢ ( 1 < 0 ∨ 0 < 1 ) |
6 |
|
breq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 < 0 ↔ 1 < 0 ) ) |
7 |
6
|
imbi2d |
⊢ ( 𝑥 = 1 → ( ( 1 < 0 → 𝑥 < 0 ) ↔ ( 1 < 0 → 1 < 0 ) ) ) |
8 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 < 0 ↔ 𝑦 < 0 ) ) |
9 |
8
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 1 < 0 → 𝑥 < 0 ) ↔ ( 1 < 0 → 𝑦 < 0 ) ) ) |
10 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 < 0 ↔ ( 𝑦 + 1 ) < 0 ) ) |
11 |
10
|
imbi2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 1 < 0 → 𝑥 < 0 ) ↔ ( 1 < 0 → ( 𝑦 + 1 ) < 0 ) ) ) |
12 |
|
breq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 < 0 ↔ 𝐴 < 0 ) ) |
13 |
12
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( 1 < 0 → 𝑥 < 0 ) ↔ ( 1 < 0 → 𝐴 < 0 ) ) ) |
14 |
|
id |
⊢ ( 1 < 0 → 1 < 0 ) |
15 |
|
simp1 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → 𝑦 ∈ ℕ ) |
16 |
15
|
nnred |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → 𝑦 ∈ ℝ ) |
17 |
|
1red |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → 1 ∈ ℝ ) |
18 |
16 17
|
readdcld |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → ( 𝑦 + 1 ) ∈ ℝ ) |
19 |
3 2
|
readdcli |
⊢ ( 0 + 1 ) ∈ ℝ |
20 |
19
|
a1i |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → ( 0 + 1 ) ∈ ℝ ) |
21 |
|
0red |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → 0 ∈ ℝ ) |
22 |
|
simp3 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → 𝑦 < 0 ) |
23 |
16 21 17 22
|
ltadd1dd |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → ( 𝑦 + 1 ) < ( 0 + 1 ) ) |
24 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
25 |
24
|
addlidi |
⊢ ( 0 + 1 ) = 1 |
26 |
|
simp2 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → 1 < 0 ) |
27 |
25 26
|
eqbrtrid |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → ( 0 + 1 ) < 0 ) |
28 |
18 20 21 23 27
|
lttrd |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 < 0 ∧ 𝑦 < 0 ) → ( 𝑦 + 1 ) < 0 ) |
29 |
28
|
3exp |
⊢ ( 𝑦 ∈ ℕ → ( 1 < 0 → ( 𝑦 < 0 → ( 𝑦 + 1 ) < 0 ) ) ) |
30 |
29
|
a2d |
⊢ ( 𝑦 ∈ ℕ → ( ( 1 < 0 → 𝑦 < 0 ) → ( 1 < 0 → ( 𝑦 + 1 ) < 0 ) ) ) |
31 |
7 9 11 13 14 30
|
nnind |
⊢ ( 𝐴 ∈ ℕ → ( 1 < 0 → 𝐴 < 0 ) ) |
32 |
31
|
imp |
⊢ ( ( 𝐴 ∈ ℕ ∧ 1 < 0 ) → 𝐴 < 0 ) |
33 |
32
|
lt0ne0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 1 < 0 ) → 𝐴 ≠ 0 ) |
34 |
|
breq2 |
⊢ ( 𝑥 = 1 → ( 0 < 𝑥 ↔ 0 < 1 ) ) |
35 |
34
|
imbi2d |
⊢ ( 𝑥 = 1 → ( ( 0 < 1 → 0 < 𝑥 ) ↔ ( 0 < 1 → 0 < 1 ) ) ) |
36 |
|
breq2 |
⊢ ( 𝑥 = 𝑦 → ( 0 < 𝑥 ↔ 0 < 𝑦 ) ) |
37 |
36
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 0 < 1 → 0 < 𝑥 ) ↔ ( 0 < 1 → 0 < 𝑦 ) ) ) |
38 |
|
breq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 0 < 𝑥 ↔ 0 < ( 𝑦 + 1 ) ) ) |
39 |
38
|
imbi2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 0 < 1 → 0 < 𝑥 ) ↔ ( 0 < 1 → 0 < ( 𝑦 + 1 ) ) ) ) |
40 |
|
breq2 |
⊢ ( 𝑥 = 𝐴 → ( 0 < 𝑥 ↔ 0 < 𝐴 ) ) |
41 |
40
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( 0 < 1 → 0 < 𝑥 ) ↔ ( 0 < 1 → 0 < 𝐴 ) ) ) |
42 |
|
id |
⊢ ( 0 < 1 → 0 < 1 ) |
43 |
|
simp1 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 0 < 1 ∧ 0 < 𝑦 ) → 𝑦 ∈ ℕ ) |
44 |
43
|
nnred |
⊢ ( ( 𝑦 ∈ ℕ ∧ 0 < 1 ∧ 0 < 𝑦 ) → 𝑦 ∈ ℝ ) |
45 |
|
1red |
⊢ ( ( 𝑦 ∈ ℕ ∧ 0 < 1 ∧ 0 < 𝑦 ) → 1 ∈ ℝ ) |
46 |
|
simp3 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 0 < 1 ∧ 0 < 𝑦 ) → 0 < 𝑦 ) |
47 |
|
simp2 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 0 < 1 ∧ 0 < 𝑦 ) → 0 < 1 ) |
48 |
44 45 46 47
|
addgt0d |
⊢ ( ( 𝑦 ∈ ℕ ∧ 0 < 1 ∧ 0 < 𝑦 ) → 0 < ( 𝑦 + 1 ) ) |
49 |
48
|
3exp |
⊢ ( 𝑦 ∈ ℕ → ( 0 < 1 → ( 0 < 𝑦 → 0 < ( 𝑦 + 1 ) ) ) ) |
50 |
49
|
a2d |
⊢ ( 𝑦 ∈ ℕ → ( ( 0 < 1 → 0 < 𝑦 ) → ( 0 < 1 → 0 < ( 𝑦 + 1 ) ) ) ) |
51 |
35 37 39 41 42 50
|
nnind |
⊢ ( 𝐴 ∈ ℕ → ( 0 < 1 → 0 < 𝐴 ) ) |
52 |
51
|
imp |
⊢ ( ( 𝐴 ∈ ℕ ∧ 0 < 1 ) → 0 < 𝐴 ) |
53 |
52
|
gt0ne0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 0 < 1 ) → 𝐴 ≠ 0 ) |
54 |
33 53
|
jaodan |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 1 < 0 ∨ 0 < 1 ) ) → 𝐴 ≠ 0 ) |
55 |
5 54
|
mpan2 |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ≠ 0 ) |