Step |
Hyp |
Ref |
Expression |
1 |
|
sgn3da.0 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
2 |
|
sgn3da.1 |
⊢ ( ( sgn ‘ 𝐴 ) = 0 → ( 𝜓 ↔ 𝜒 ) ) |
3 |
|
sgn3da.2 |
⊢ ( ( sgn ‘ 𝐴 ) = 1 → ( 𝜓 ↔ 𝜃 ) ) |
4 |
|
sgn3da.3 |
⊢ ( ( sgn ‘ 𝐴 ) = - 1 → ( 𝜓 ↔ 𝜏 ) ) |
5 |
|
sgn3da.4 |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → 𝜒 ) |
6 |
|
sgn3da.5 |
⊢ ( ( 𝜑 ∧ 0 < 𝐴 ) → 𝜃 ) |
7 |
|
sgn3da.6 |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ) → 𝜏 ) |
8 |
|
sgnval |
⊢ ( 𝐴 ∈ ℝ* → ( sgn ‘ 𝐴 ) = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) ) |
9 |
1 8
|
syl |
⊢ ( 𝜑 → ( sgn ‘ 𝐴 ) = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) ) |
10 |
9
|
eqeq2d |
⊢ ( 𝜑 → ( 0 = ( sgn ‘ 𝐴 ) ↔ 0 = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) ) ) |
11 |
10
|
pm5.32i |
⊢ ( ( 𝜑 ∧ 0 = ( sgn ‘ 𝐴 ) ) ↔ ( 𝜑 ∧ 0 = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) ) ) |
12 |
2
|
eqcoms |
⊢ ( 0 = ( sgn ‘ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
13 |
12
|
bicomd |
⊢ ( 0 = ( sgn ‘ 𝐴 ) → ( 𝜒 ↔ 𝜓 ) ) |
14 |
13
|
adantl |
⊢ ( ( 𝜑 ∧ 0 = ( sgn ‘ 𝐴 ) ) → ( 𝜒 ↔ 𝜓 ) ) |
15 |
11 14
|
sylbir |
⊢ ( ( 𝜑 ∧ 0 = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) ) → ( 𝜒 ↔ 𝜓 ) ) |
16 |
15
|
expcom |
⊢ ( 0 = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) → ( 𝜑 → ( 𝜒 ↔ 𝜓 ) ) ) |
17 |
16
|
pm5.74d |
⊢ ( 0 = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜑 → 𝜓 ) ) ) |
18 |
9
|
eqeq2d |
⊢ ( 𝜑 → ( if ( 𝐴 < 0 , - 1 , 1 ) = ( sgn ‘ 𝐴 ) ↔ if ( 𝐴 < 0 , - 1 , 1 ) = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) ) ) |
19 |
18
|
pm5.32i |
⊢ ( ( 𝜑 ∧ if ( 𝐴 < 0 , - 1 , 1 ) = ( sgn ‘ 𝐴 ) ) ↔ ( 𝜑 ∧ if ( 𝐴 < 0 , - 1 , 1 ) = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) ) ) |
20 |
|
eqeq1 |
⊢ ( - 1 = if ( 𝐴 < 0 , - 1 , 1 ) → ( - 1 = ( sgn ‘ 𝐴 ) ↔ if ( 𝐴 < 0 , - 1 , 1 ) = ( sgn ‘ 𝐴 ) ) ) |
21 |
20
|
imbi1d |
⊢ ( - 1 = if ( 𝐴 < 0 , - 1 , 1 ) → ( ( - 1 = ( sgn ‘ 𝐴 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) ↔ ( if ( 𝐴 < 0 , - 1 , 1 ) = ( sgn ‘ 𝐴 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) ) ) |
22 |
|
eqeq1 |
⊢ ( 1 = if ( 𝐴 < 0 , - 1 , 1 ) → ( 1 = ( sgn ‘ 𝐴 ) ↔ if ( 𝐴 < 0 , - 1 , 1 ) = ( sgn ‘ 𝐴 ) ) ) |
23 |
22
|
imbi1d |
⊢ ( 1 = if ( 𝐴 < 0 , - 1 , 1 ) → ( ( 1 = ( sgn ‘ 𝐴 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) ↔ ( if ( 𝐴 < 0 , - 1 , 1 ) = ( sgn ‘ 𝐴 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) ) ) |
24 |
7
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 < 0 ) ∧ ( 𝐴 < 0 → 𝜏 ) ) → 𝜏 ) |
25 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ 𝐴 < 0 ) ∧ 𝜏 ∧ 𝐴 < 0 ) → 𝜏 ) |
26 |
25
|
3expia |
⊢ ( ( ( 𝜑 ∧ 𝐴 < 0 ) ∧ 𝜏 ) → ( 𝐴 < 0 → 𝜏 ) ) |
27 |
24 26
|
impbida |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ) → ( ( 𝐴 < 0 → 𝜏 ) ↔ 𝜏 ) ) |
28 |
|
pm3.24 |
⊢ ¬ ( 𝐴 < 0 ∧ ¬ 𝐴 < 0 ) |
29 |
28
|
pm2.21i |
⊢ ( ( 𝐴 < 0 ∧ ¬ 𝐴 < 0 ) → 𝜃 ) |
30 |
29
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝐴 < 0 ∧ ¬ 𝐴 < 0 ) ) → 𝜃 ) |
31 |
30
|
expr |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ) → ( ¬ 𝐴 < 0 → 𝜃 ) ) |
32 |
|
tbtru |
⊢ ( ( ¬ 𝐴 < 0 → 𝜃 ) ↔ ( ( ¬ 𝐴 < 0 → 𝜃 ) ↔ ⊤ ) ) |
33 |
31 32
|
sylib |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ) → ( ( ¬ 𝐴 < 0 → 𝜃 ) ↔ ⊤ ) ) |
34 |
27 33
|
anbi12d |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ ( 𝜏 ∧ ⊤ ) ) ) |
35 |
|
ancom |
⊢ ( ( 𝜏 ∧ ⊤ ) ↔ ( ⊤ ∧ 𝜏 ) ) |
36 |
|
truan |
⊢ ( ( ⊤ ∧ 𝜏 ) ↔ 𝜏 ) |
37 |
35 36
|
bitri |
⊢ ( ( 𝜏 ∧ ⊤ ) ↔ 𝜏 ) |
38 |
34 37
|
bitrdi |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜏 ) ) |
39 |
38
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ∧ - 1 = ( sgn ‘ 𝐴 ) ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜏 ) ) |
40 |
4
|
eqcoms |
⊢ ( - 1 = ( sgn ‘ 𝐴 ) → ( 𝜓 ↔ 𝜏 ) ) |
41 |
40
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ∧ - 1 = ( sgn ‘ 𝐴 ) ) → ( 𝜓 ↔ 𝜏 ) ) |
42 |
39 41
|
bitr4d |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ∧ - 1 = ( sgn ‘ 𝐴 ) ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) |
43 |
42
|
3expia |
⊢ ( ( 𝜑 ∧ 𝐴 < 0 ) → ( - 1 = ( sgn ‘ 𝐴 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) ) |
44 |
7
|
3adant2 |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ∧ 𝐴 < 0 ) → 𝜏 ) |
45 |
44
|
3expia |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ) → ( 𝐴 < 0 → 𝜏 ) ) |
46 |
|
tbtru |
⊢ ( ( 𝐴 < 0 → 𝜏 ) ↔ ( ( 𝐴 < 0 → 𝜏 ) ↔ ⊤ ) ) |
47 |
45 46
|
sylib |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ) → ( ( 𝐴 < 0 → 𝜏 ) ↔ ⊤ ) ) |
48 |
|
pm3.35 |
⊢ ( ( ¬ 𝐴 < 0 ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) → 𝜃 ) |
49 |
48
|
adantll |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 0 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) → 𝜃 ) |
50 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 0 ) ∧ 𝜃 ∧ ¬ 𝐴 < 0 ) → 𝜃 ) |
51 |
50
|
3expia |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 0 ) ∧ 𝜃 ) → ( ¬ 𝐴 < 0 → 𝜃 ) ) |
52 |
49 51
|
impbida |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ) → ( ( ¬ 𝐴 < 0 → 𝜃 ) ↔ 𝜃 ) ) |
53 |
47 52
|
anbi12d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ ( ⊤ ∧ 𝜃 ) ) ) |
54 |
|
truan |
⊢ ( ( ⊤ ∧ 𝜃 ) ↔ 𝜃 ) |
55 |
53 54
|
bitrdi |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜃 ) ) |
56 |
55
|
3adant3 |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = ( sgn ‘ 𝐴 ) ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜃 ) ) |
57 |
3
|
eqcoms |
⊢ ( 1 = ( sgn ‘ 𝐴 ) → ( 𝜓 ↔ 𝜃 ) ) |
58 |
57
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = ( sgn ‘ 𝐴 ) ) → ( 𝜓 ↔ 𝜃 ) ) |
59 |
56 58
|
bitr4d |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = ( sgn ‘ 𝐴 ) ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) |
60 |
59
|
3expia |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 0 ) → ( 1 = ( sgn ‘ 𝐴 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) ) |
61 |
21 23 43 60
|
ifbothda |
⊢ ( 𝜑 → ( if ( 𝐴 < 0 , - 1 , 1 ) = ( sgn ‘ 𝐴 ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) ) |
62 |
61
|
imp |
⊢ ( ( 𝜑 ∧ if ( 𝐴 < 0 , - 1 , 1 ) = ( sgn ‘ 𝐴 ) ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) |
63 |
19 62
|
sylbir |
⊢ ( ( 𝜑 ∧ if ( 𝐴 < 0 , - 1 , 1 ) = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) ) → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) |
64 |
63
|
expcom |
⊢ ( if ( 𝐴 < 0 , - 1 , 1 ) = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) → ( 𝜑 → ( ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ↔ 𝜓 ) ) ) |
65 |
64
|
pm5.74d |
⊢ ( if ( 𝐴 < 0 , - 1 , 1 ) = if ( 𝐴 = 0 , 0 , if ( 𝐴 < 0 , - 1 , 1 ) ) → ( ( 𝜑 → ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ) ↔ ( 𝜑 → 𝜓 ) ) ) |
66 |
5
|
expcom |
⊢ ( 𝐴 = 0 → ( 𝜑 → 𝜒 ) ) |
67 |
66
|
adantl |
⊢ ( ( ⊤ ∧ 𝐴 = 0 ) → ( 𝜑 → 𝜒 ) ) |
68 |
7
|
ex |
⊢ ( 𝜑 → ( 𝐴 < 0 → 𝜏 ) ) |
69 |
68
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 0 ) → ( 𝐴 < 0 → 𝜏 ) ) |
70 |
|
simp1 |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0 ) → 𝜑 ) |
71 |
|
df-ne |
⊢ ( 𝐴 ≠ 0 ↔ ¬ 𝐴 = 0 ) |
72 |
|
0xr |
⊢ 0 ∈ ℝ* |
73 |
|
xrlttri2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝐴 ≠ 0 ↔ ( 𝐴 < 0 ∨ 0 < 𝐴 ) ) ) |
74 |
1 72 73
|
sylancl |
⊢ ( 𝜑 → ( 𝐴 ≠ 0 ↔ ( 𝐴 < 0 ∨ 0 < 𝐴 ) ) ) |
75 |
71 74
|
bitr3id |
⊢ ( 𝜑 → ( ¬ 𝐴 = 0 ↔ ( 𝐴 < 0 ∨ 0 < 𝐴 ) ) ) |
76 |
75
|
biimpa |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 0 ) → ( 𝐴 < 0 ∨ 0 < 𝐴 ) ) |
77 |
76
|
ord |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 0 ) → ( ¬ 𝐴 < 0 → 0 < 𝐴 ) ) |
78 |
77
|
3impia |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0 ) → 0 < 𝐴 ) |
79 |
70 78 6
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0 ) → 𝜃 ) |
80 |
79
|
3expia |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 0 ) → ( ¬ 𝐴 < 0 → 𝜃 ) ) |
81 |
69 80
|
jca |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 0 ) → ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ) |
82 |
81
|
expcom |
⊢ ( ¬ 𝐴 = 0 → ( 𝜑 → ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ) ) |
83 |
82
|
adantl |
⊢ ( ( ⊤ ∧ ¬ 𝐴 = 0 ) → ( 𝜑 → ( ( 𝐴 < 0 → 𝜏 ) ∧ ( ¬ 𝐴 < 0 → 𝜃 ) ) ) ) |
84 |
17 65 67 83
|
ifbothda |
⊢ ( ⊤ → ( 𝜑 → 𝜓 ) ) |
85 |
84
|
mptru |
⊢ ( 𝜑 → 𝜓 ) |