Step |
Hyp |
Ref |
Expression |
1 |
|
tan4thpi |
⊢ ( tan ‘ ( π / 4 ) ) = 1 |
2 |
1
|
fveq2i |
⊢ ( arctan ‘ ( tan ‘ ( π / 4 ) ) ) = ( arctan ‘ 1 ) |
3 |
|
pire |
⊢ π ∈ ℝ |
4 |
|
4nn |
⊢ 4 ∈ ℕ |
5 |
|
nndivre |
⊢ ( ( π ∈ ℝ ∧ 4 ∈ ℕ ) → ( π / 4 ) ∈ ℝ ) |
6 |
3 4 5
|
mp2an |
⊢ ( π / 4 ) ∈ ℝ |
7 |
6
|
recni |
⊢ ( π / 4 ) ∈ ℂ |
8 |
|
rere |
⊢ ( ( π / 4 ) ∈ ℝ → ( ℜ ‘ ( π / 4 ) ) = ( π / 4 ) ) |
9 |
6 8
|
ax-mp |
⊢ ( ℜ ‘ ( π / 4 ) ) = ( π / 4 ) |
10 |
|
pirp |
⊢ π ∈ ℝ+ |
11 |
|
rphalfcl |
⊢ ( π ∈ ℝ+ → ( π / 2 ) ∈ ℝ+ ) |
12 |
10 11
|
ax-mp |
⊢ ( π / 2 ) ∈ ℝ+ |
13 |
|
rpgt0 |
⊢ ( ( π / 2 ) ∈ ℝ+ → 0 < ( π / 2 ) ) |
14 |
12 13
|
ax-mp |
⊢ 0 < ( π / 2 ) |
15 |
|
halfpire |
⊢ ( π / 2 ) ∈ ℝ |
16 |
|
lt0neg2 |
⊢ ( ( π / 2 ) ∈ ℝ → ( 0 < ( π / 2 ) ↔ - ( π / 2 ) < 0 ) ) |
17 |
15 16
|
ax-mp |
⊢ ( 0 < ( π / 2 ) ↔ - ( π / 2 ) < 0 ) |
18 |
14 17
|
mpbi |
⊢ - ( π / 2 ) < 0 |
19 |
|
nnrp |
⊢ ( 4 ∈ ℕ → 4 ∈ ℝ+ ) |
20 |
4 19
|
ax-mp |
⊢ 4 ∈ ℝ+ |
21 |
|
rpdivcl |
⊢ ( ( π ∈ ℝ+ ∧ 4 ∈ ℝ+ ) → ( π / 4 ) ∈ ℝ+ ) |
22 |
10 20 21
|
mp2an |
⊢ ( π / 4 ) ∈ ℝ+ |
23 |
|
rpgt0 |
⊢ ( ( π / 4 ) ∈ ℝ+ → 0 < ( π / 4 ) ) |
24 |
22 23
|
ax-mp |
⊢ 0 < ( π / 4 ) |
25 |
|
neghalfpire |
⊢ - ( π / 2 ) ∈ ℝ |
26 |
|
0re |
⊢ 0 ∈ ℝ |
27 |
25 26 6
|
lttri |
⊢ ( ( - ( π / 2 ) < 0 ∧ 0 < ( π / 4 ) ) → - ( π / 2 ) < ( π / 4 ) ) |
28 |
18 24 27
|
mp2an |
⊢ - ( π / 2 ) < ( π / 4 ) |
29 |
3
|
recni |
⊢ π ∈ ℂ |
30 |
|
2cnne0 |
⊢ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) |
31 |
|
divdiv1 |
⊢ ( ( π ∈ ℂ ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) → ( ( π / 2 ) / 2 ) = ( π / ( 2 · 2 ) ) ) |
32 |
29 30 30 31
|
mp3an |
⊢ ( ( π / 2 ) / 2 ) = ( π / ( 2 · 2 ) ) |
33 |
|
2t2e4 |
⊢ ( 2 · 2 ) = 4 |
34 |
33
|
oveq2i |
⊢ ( π / ( 2 · 2 ) ) = ( π / 4 ) |
35 |
32 34
|
eqtri |
⊢ ( ( π / 2 ) / 2 ) = ( π / 4 ) |
36 |
|
rphalflt |
⊢ ( ( π / 2 ) ∈ ℝ+ → ( ( π / 2 ) / 2 ) < ( π / 2 ) ) |
37 |
12 36
|
ax-mp |
⊢ ( ( π / 2 ) / 2 ) < ( π / 2 ) |
38 |
35 37
|
eqbrtrri |
⊢ ( π / 4 ) < ( π / 2 ) |
39 |
25
|
rexri |
⊢ - ( π / 2 ) ∈ ℝ* |
40 |
15
|
rexri |
⊢ ( π / 2 ) ∈ ℝ* |
41 |
|
elioo2 |
⊢ ( ( - ( π / 2 ) ∈ ℝ* ∧ ( π / 2 ) ∈ ℝ* ) → ( ( π / 4 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ↔ ( ( π / 4 ) ∈ ℝ ∧ - ( π / 2 ) < ( π / 4 ) ∧ ( π / 4 ) < ( π / 2 ) ) ) ) |
42 |
39 40 41
|
mp2an |
⊢ ( ( π / 4 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ↔ ( ( π / 4 ) ∈ ℝ ∧ - ( π / 2 ) < ( π / 4 ) ∧ ( π / 4 ) < ( π / 2 ) ) ) |
43 |
6 28 38 42
|
mpbir3an |
⊢ ( π / 4 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) |
44 |
9 43
|
eqeltri |
⊢ ( ℜ ‘ ( π / 4 ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) |
45 |
|
atantan |
⊢ ( ( ( π / 4 ) ∈ ℂ ∧ ( ℜ ‘ ( π / 4 ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) → ( arctan ‘ ( tan ‘ ( π / 4 ) ) ) = ( π / 4 ) ) |
46 |
7 44 45
|
mp2an |
⊢ ( arctan ‘ ( tan ‘ ( π / 4 ) ) ) = ( π / 4 ) |
47 |
2 46
|
eqtr3i |
⊢ ( arctan ‘ 1 ) = ( π / 4 ) |