| Step |
Hyp |
Ref |
Expression |
| 1 |
|
iccpnfhmeo.f |
⊢ 𝐹 = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) |
| 2 |
|
0xr |
⊢ 0 ∈ ℝ* |
| 3 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
| 4 |
|
0lepnf |
⊢ 0 ≤ +∞ |
| 5 |
|
ubicc2 |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞ ) → +∞ ∈ ( 0 [,] +∞ ) ) |
| 6 |
2 3 4 5
|
mp3an |
⊢ +∞ ∈ ( 0 [,] +∞ ) |
| 7 |
6
|
a1i |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 = 1 ) → +∞ ∈ ( 0 [,] +∞ ) ) |
| 8 |
|
icossicc |
⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) |
| 9 |
|
1xr |
⊢ 1 ∈ ℝ* |
| 10 |
|
0le1 |
⊢ 0 ≤ 1 |
| 11 |
|
snunico |
⊢ ( ( 0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1 ) → ( ( 0 [,) 1 ) ∪ { 1 } ) = ( 0 [,] 1 ) ) |
| 12 |
2 9 10 11
|
mp3an |
⊢ ( ( 0 [,) 1 ) ∪ { 1 } ) = ( 0 [,] 1 ) |
| 13 |
12
|
eleq2i |
⊢ ( 𝑥 ∈ ( ( 0 [,) 1 ) ∪ { 1 } ) ↔ 𝑥 ∈ ( 0 [,] 1 ) ) |
| 14 |
|
elun |
⊢ ( 𝑥 ∈ ( ( 0 [,) 1 ) ∪ { 1 } ) ↔ ( 𝑥 ∈ ( 0 [,) 1 ) ∨ 𝑥 ∈ { 1 } ) ) |
| 15 |
13 14
|
bitr3i |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) ↔ ( 𝑥 ∈ ( 0 [,) 1 ) ∨ 𝑥 ∈ { 1 } ) ) |
| 16 |
|
pm2.53 |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∨ 𝑥 ∈ { 1 } ) → ( ¬ 𝑥 ∈ ( 0 [,) 1 ) → 𝑥 ∈ { 1 } ) ) |
| 17 |
15 16
|
sylbi |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → ( ¬ 𝑥 ∈ ( 0 [,) 1 ) → 𝑥 ∈ { 1 } ) ) |
| 18 |
|
elsni |
⊢ ( 𝑥 ∈ { 1 } → 𝑥 = 1 ) |
| 19 |
17 18
|
syl6 |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → ( ¬ 𝑥 ∈ ( 0 [,) 1 ) → 𝑥 = 1 ) ) |
| 20 |
19
|
con1d |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → ( ¬ 𝑥 = 1 → 𝑥 ∈ ( 0 [,) 1 ) ) ) |
| 21 |
20
|
imp |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 = 1 ) → 𝑥 ∈ ( 0 [,) 1 ) ) |
| 22 |
|
eqid |
⊢ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) = ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) |
| 23 |
22
|
icopnfcnv |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) 1 ) –1-1-onto→ ( 0 [,) +∞ ) ∧ ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) = ( 𝑦 ∈ ( 0 [,) +∞ ) ↦ ( 𝑦 / ( 1 + 𝑦 ) ) ) ) |
| 24 |
23
|
simpli |
⊢ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) 1 ) –1-1-onto→ ( 0 [,) +∞ ) |
| 25 |
|
f1of |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) 1 ) –1-1-onto→ ( 0 [,) +∞ ) → ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) 1 ) ⟶ ( 0 [,) +∞ ) ) |
| 26 |
24 25
|
ax-mp |
⊢ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) 1 ) ⟶ ( 0 [,) +∞ ) |
| 27 |
22
|
fmpt |
⊢ ( ∀ 𝑥 ∈ ( 0 [,) 1 ) ( 𝑥 / ( 1 − 𝑥 ) ) ∈ ( 0 [,) +∞ ) ↔ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) 1 ) ⟶ ( 0 [,) +∞ ) ) |
| 28 |
26 27
|
mpbir |
⊢ ∀ 𝑥 ∈ ( 0 [,) 1 ) ( 𝑥 / ( 1 − 𝑥 ) ) ∈ ( 0 [,) +∞ ) |
| 29 |
28
|
rspec |
⊢ ( 𝑥 ∈ ( 0 [,) 1 ) → ( 𝑥 / ( 1 − 𝑥 ) ) ∈ ( 0 [,) +∞ ) ) |
| 30 |
21 29
|
syl |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 = 1 ) → ( 𝑥 / ( 1 − 𝑥 ) ) ∈ ( 0 [,) +∞ ) ) |
| 31 |
8 30
|
sseldi |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 = 1 ) → ( 𝑥 / ( 1 − 𝑥 ) ) ∈ ( 0 [,] +∞ ) ) |
| 32 |
7 31
|
ifclda |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 33 |
32
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∈ ( 0 [,] +∞ ) ) |
| 34 |
|
1elunit |
⊢ 1 ∈ ( 0 [,] 1 ) |
| 35 |
34
|
a1i |
⊢ ( ( 𝑦 ∈ ( 0 [,] +∞ ) ∧ 𝑦 = +∞ ) → 1 ∈ ( 0 [,] 1 ) ) |
| 36 |
|
icossicc |
⊢ ( 0 [,) 1 ) ⊆ ( 0 [,] 1 ) |
| 37 |
|
snunico |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞ ) → ( ( 0 [,) +∞ ) ∪ { +∞ } ) = ( 0 [,] +∞ ) ) |
| 38 |
2 3 4 37
|
mp3an |
⊢ ( ( 0 [,) +∞ ) ∪ { +∞ } ) = ( 0 [,] +∞ ) |
| 39 |
38
|
eleq2i |
⊢ ( 𝑦 ∈ ( ( 0 [,) +∞ ) ∪ { +∞ } ) ↔ 𝑦 ∈ ( 0 [,] +∞ ) ) |
| 40 |
|
elun |
⊢ ( 𝑦 ∈ ( ( 0 [,) +∞ ) ∪ { +∞ } ) ↔ ( 𝑦 ∈ ( 0 [,) +∞ ) ∨ 𝑦 ∈ { +∞ } ) ) |
| 41 |
39 40
|
bitr3i |
⊢ ( 𝑦 ∈ ( 0 [,] +∞ ) ↔ ( 𝑦 ∈ ( 0 [,) +∞ ) ∨ 𝑦 ∈ { +∞ } ) ) |
| 42 |
|
pm2.53 |
⊢ ( ( 𝑦 ∈ ( 0 [,) +∞ ) ∨ 𝑦 ∈ { +∞ } ) → ( ¬ 𝑦 ∈ ( 0 [,) +∞ ) → 𝑦 ∈ { +∞ } ) ) |
| 43 |
41 42
|
sylbi |
⊢ ( 𝑦 ∈ ( 0 [,] +∞ ) → ( ¬ 𝑦 ∈ ( 0 [,) +∞ ) → 𝑦 ∈ { +∞ } ) ) |
| 44 |
|
elsni |
⊢ ( 𝑦 ∈ { +∞ } → 𝑦 = +∞ ) |
| 45 |
43 44
|
syl6 |
⊢ ( 𝑦 ∈ ( 0 [,] +∞ ) → ( ¬ 𝑦 ∈ ( 0 [,) +∞ ) → 𝑦 = +∞ ) ) |
| 46 |
45
|
con1d |
⊢ ( 𝑦 ∈ ( 0 [,] +∞ ) → ( ¬ 𝑦 = +∞ → 𝑦 ∈ ( 0 [,) +∞ ) ) ) |
| 47 |
46
|
imp |
⊢ ( ( 𝑦 ∈ ( 0 [,] +∞ ) ∧ ¬ 𝑦 = +∞ ) → 𝑦 ∈ ( 0 [,) +∞ ) ) |
| 48 |
|
f1ocnv |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) 1 ) –1-1-onto→ ( 0 [,) +∞ ) → ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) +∞ ) –1-1-onto→ ( 0 [,) 1 ) ) |
| 49 |
|
f1of |
⊢ ( ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) +∞ ) –1-1-onto→ ( 0 [,) 1 ) → ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) +∞ ) ⟶ ( 0 [,) 1 ) ) |
| 50 |
24 48 49
|
mp2b |
⊢ ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) +∞ ) ⟶ ( 0 [,) 1 ) |
| 51 |
23
|
simpri |
⊢ ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) = ( 𝑦 ∈ ( 0 [,) +∞ ) ↦ ( 𝑦 / ( 1 + 𝑦 ) ) ) |
| 52 |
51
|
fmpt |
⊢ ( ∀ 𝑦 ∈ ( 0 [,) +∞ ) ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ( 0 [,) 1 ) ↔ ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) +∞ ) ⟶ ( 0 [,) 1 ) ) |
| 53 |
50 52
|
mpbir |
⊢ ∀ 𝑦 ∈ ( 0 [,) +∞ ) ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ( 0 [,) 1 ) |
| 54 |
53
|
rspec |
⊢ ( 𝑦 ∈ ( 0 [,) +∞ ) → ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ( 0 [,) 1 ) ) |
| 55 |
47 54
|
syl |
⊢ ( ( 𝑦 ∈ ( 0 [,] +∞ ) ∧ ¬ 𝑦 = +∞ ) → ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ( 0 [,) 1 ) ) |
| 56 |
36 55
|
sseldi |
⊢ ( ( 𝑦 ∈ ( 0 [,] +∞ ) ∧ ¬ 𝑦 = +∞ ) → ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ( 0 [,] 1 ) ) |
| 57 |
35 56
|
ifclda |
⊢ ( 𝑦 ∈ ( 0 [,] +∞ ) → if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ∈ ( 0 [,] 1 ) ) |
| 58 |
57
|
adantl |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) → if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ∈ ( 0 [,] 1 ) ) |
| 59 |
|
eqeq2 |
⊢ ( 1 = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) → ( 𝑥 = 1 ↔ 𝑥 = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ) ) |
| 60 |
59
|
bibi1d |
⊢ ( 1 = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) → ( ( 𝑥 = 1 ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ↔ ( 𝑥 = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) ) |
| 61 |
|
eqeq2 |
⊢ ( ( 𝑦 / ( 1 + 𝑦 ) ) = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) → ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑥 = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ) ) |
| 62 |
61
|
bibi1d |
⊢ ( ( 𝑦 / ( 1 + 𝑦 ) ) = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) → ( ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ↔ ( 𝑥 = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) ) |
| 63 |
|
simpr |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → 𝑦 = +∞ ) |
| 64 |
|
iftrue |
⊢ ( 𝑥 = 1 → if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) = +∞ ) |
| 65 |
64
|
eqeq2d |
⊢ ( 𝑥 = 1 → ( 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ↔ 𝑦 = +∞ ) ) |
| 66 |
63 65
|
syl5ibrcom |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → ( 𝑥 = 1 → 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) |
| 67 |
|
pnfnre |
⊢ +∞ ∉ ℝ |
| 68 |
|
neleq1 |
⊢ ( 𝑦 = +∞ → ( 𝑦 ∉ ℝ ↔ +∞ ∉ ℝ ) ) |
| 69 |
68
|
adantl |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → ( 𝑦 ∉ ℝ ↔ +∞ ∉ ℝ ) ) |
| 70 |
67 69
|
mpbiri |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → 𝑦 ∉ ℝ ) |
| 71 |
|
neleq1 |
⊢ ( 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) → ( 𝑦 ∉ ℝ ↔ if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∉ ℝ ) ) |
| 72 |
70 71
|
syl5ibcom |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → ( 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) → if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∉ ℝ ) ) |
| 73 |
|
df-nel |
⊢ ( if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∉ ℝ ↔ ¬ if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∈ ℝ ) |
| 74 |
|
iffalse |
⊢ ( ¬ 𝑥 = 1 → if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) = ( 𝑥 / ( 1 − 𝑥 ) ) ) |
| 75 |
74
|
adantl |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 = 1 ) → if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) = ( 𝑥 / ( 1 − 𝑥 ) ) ) |
| 76 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
| 77 |
76 30
|
sseldi |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 = 1 ) → ( 𝑥 / ( 1 − 𝑥 ) ) ∈ ℝ ) |
| 78 |
75 77
|
eqeltrd |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 = 1 ) → if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∈ ℝ ) |
| 79 |
78
|
ex |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → ( ¬ 𝑥 = 1 → if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∈ ℝ ) ) |
| 80 |
79
|
ad2antrr |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → ( ¬ 𝑥 = 1 → if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∈ ℝ ) ) |
| 81 |
80
|
con1d |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → ( ¬ if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∈ ℝ → 𝑥 = 1 ) ) |
| 82 |
73 81
|
syl5bi |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → ( if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ∉ ℝ → 𝑥 = 1 ) ) |
| 83 |
72 82
|
syld |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → ( 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) → 𝑥 = 1 ) ) |
| 84 |
66 83
|
impbid |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 𝑦 = +∞ ) → ( 𝑥 = 1 ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) |
| 85 |
|
eqeq2 |
⊢ ( +∞ = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) → ( 𝑦 = +∞ ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) |
| 86 |
85
|
bibi2d |
⊢ ( +∞ = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) → ( ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = +∞ ) ↔ ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) ) |
| 87 |
|
eqeq2 |
⊢ ( ( 𝑥 / ( 1 − 𝑥 ) ) = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) → ( 𝑦 = ( 𝑥 / ( 1 − 𝑥 ) ) ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) |
| 88 |
87
|
bibi2d |
⊢ ( ( 𝑥 / ( 1 − 𝑥 ) ) = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) → ( ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = ( 𝑥 / ( 1 − 𝑥 ) ) ) ↔ ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) ) |
| 89 |
|
0re |
⊢ 0 ∈ ℝ |
| 90 |
|
elico2 |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ* ) → ( ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ( 0 [,) 1 ) ↔ ( ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ℝ ∧ 0 ≤ ( 𝑦 / ( 1 + 𝑦 ) ) ∧ ( 𝑦 / ( 1 + 𝑦 ) ) < 1 ) ) ) |
| 91 |
89 9 90
|
mp2an |
⊢ ( ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ( 0 [,) 1 ) ↔ ( ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ℝ ∧ 0 ≤ ( 𝑦 / ( 1 + 𝑦 ) ) ∧ ( 𝑦 / ( 1 + 𝑦 ) ) < 1 ) ) |
| 92 |
55 91
|
sylib |
⊢ ( ( 𝑦 ∈ ( 0 [,] +∞ ) ∧ ¬ 𝑦 = +∞ ) → ( ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ℝ ∧ 0 ≤ ( 𝑦 / ( 1 + 𝑦 ) ) ∧ ( 𝑦 / ( 1 + 𝑦 ) ) < 1 ) ) |
| 93 |
92
|
simp1d |
⊢ ( ( 𝑦 ∈ ( 0 [,] +∞ ) ∧ ¬ 𝑦 = +∞ ) → ( 𝑦 / ( 1 + 𝑦 ) ) ∈ ℝ ) |
| 94 |
92
|
simp3d |
⊢ ( ( 𝑦 ∈ ( 0 [,] +∞ ) ∧ ¬ 𝑦 = +∞ ) → ( 𝑦 / ( 1 + 𝑦 ) ) < 1 ) |
| 95 |
93 94
|
gtned |
⊢ ( ( 𝑦 ∈ ( 0 [,] +∞ ) ∧ ¬ 𝑦 = +∞ ) → 1 ≠ ( 𝑦 / ( 1 + 𝑦 ) ) ) |
| 96 |
95
|
adantll |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑦 = +∞ ) → 1 ≠ ( 𝑦 / ( 1 + 𝑦 ) ) ) |
| 97 |
96
|
neneqd |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑦 = +∞ ) → ¬ 1 = ( 𝑦 / ( 1 + 𝑦 ) ) ) |
| 98 |
|
eqeq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 1 = ( 𝑦 / ( 1 + 𝑦 ) ) ) ) |
| 99 |
98
|
notbid |
⊢ ( 𝑥 = 1 → ( ¬ 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ ¬ 1 = ( 𝑦 / ( 1 + 𝑦 ) ) ) ) |
| 100 |
97 99
|
syl5ibrcom |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑦 = +∞ ) → ( 𝑥 = 1 → ¬ 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ) ) |
| 101 |
100
|
imp |
⊢ ( ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑦 = +∞ ) ∧ 𝑥 = 1 ) → ¬ 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ) |
| 102 |
|
simplr |
⊢ ( ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑦 = +∞ ) ∧ 𝑥 = 1 ) → ¬ 𝑦 = +∞ ) |
| 103 |
101 102
|
2falsed |
⊢ ( ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑦 = +∞ ) ∧ 𝑥 = 1 ) → ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = +∞ ) ) |
| 104 |
|
f1ocnvfvb |
⊢ ( ( ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) : ( 0 [,) 1 ) –1-1-onto→ ( 0 [,) +∞ ) ∧ 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( ( ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑥 ) = 𝑦 ↔ ( ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑦 ) = 𝑥 ) ) |
| 105 |
24 104
|
mp3an1 |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( ( ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑥 ) = 𝑦 ↔ ( ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑦 ) = 𝑥 ) ) |
| 106 |
|
simpl |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → 𝑥 ∈ ( 0 [,) 1 ) ) |
| 107 |
|
ovex |
⊢ ( 𝑥 / ( 1 − 𝑥 ) ) ∈ V |
| 108 |
22
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ ( 𝑥 / ( 1 − 𝑥 ) ) ∈ V ) → ( ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑥 ) = ( 𝑥 / ( 1 − 𝑥 ) ) ) |
| 109 |
106 107 108
|
sylancl |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑥 ) = ( 𝑥 / ( 1 − 𝑥 ) ) ) |
| 110 |
109
|
eqeq1d |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( ( ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑥 ) = 𝑦 ↔ ( 𝑥 / ( 1 − 𝑥 ) ) = 𝑦 ) ) |
| 111 |
|
simpr |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → 𝑦 ∈ ( 0 [,) +∞ ) ) |
| 112 |
|
ovex |
⊢ ( 𝑦 / ( 1 + 𝑦 ) ) ∈ V |
| 113 |
51
|
fvmpt2 |
⊢ ( ( 𝑦 ∈ ( 0 [,) +∞ ) ∧ ( 𝑦 / ( 1 + 𝑦 ) ) ∈ V ) → ( ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑦 ) = ( 𝑦 / ( 1 + 𝑦 ) ) ) |
| 114 |
111 112 113
|
sylancl |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑦 ) = ( 𝑦 / ( 1 + 𝑦 ) ) ) |
| 115 |
114
|
eqeq1d |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( ( ◡ ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) ‘ 𝑦 ) = 𝑥 ↔ ( 𝑦 / ( 1 + 𝑦 ) ) = 𝑥 ) ) |
| 116 |
105 110 115
|
3bitr3rd |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( ( 𝑦 / ( 1 + 𝑦 ) ) = 𝑥 ↔ ( 𝑥 / ( 1 − 𝑥 ) ) = 𝑦 ) ) |
| 117 |
|
eqcom |
⊢ ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ ( 𝑦 / ( 1 + 𝑦 ) ) = 𝑥 ) |
| 118 |
|
eqcom |
⊢ ( 𝑦 = ( 𝑥 / ( 1 − 𝑥 ) ) ↔ ( 𝑥 / ( 1 − 𝑥 ) ) = 𝑦 ) |
| 119 |
116 117 118
|
3bitr4g |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = ( 𝑥 / ( 1 − 𝑥 ) ) ) ) |
| 120 |
21 47 119
|
syl2an |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 = 1 ) ∧ ( 𝑦 ∈ ( 0 [,] +∞ ) ∧ ¬ 𝑦 = +∞ ) ) → ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = ( 𝑥 / ( 1 − 𝑥 ) ) ) ) |
| 121 |
120
|
an4s |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ ( ¬ 𝑥 = 1 ∧ ¬ 𝑦 = +∞ ) ) → ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = ( 𝑥 / ( 1 − 𝑥 ) ) ) ) |
| 122 |
121
|
anass1rs |
⊢ ( ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑦 = +∞ ) ∧ ¬ 𝑥 = 1 ) → ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = ( 𝑥 / ( 1 − 𝑥 ) ) ) ) |
| 123 |
86 88 103 122
|
ifbothda |
⊢ ( ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑦 = +∞ ) → ( 𝑥 = ( 𝑦 / ( 1 + 𝑦 ) ) ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) |
| 124 |
60 62 84 123
|
ifbothda |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) → ( 𝑥 = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) |
| 125 |
124
|
adantl |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑥 = if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ↔ 𝑦 = if ( 𝑥 = 1 , +∞ , ( 𝑥 / ( 1 − 𝑥 ) ) ) ) ) |
| 126 |
1 33 58 125
|
f1ocnv2d |
⊢ ( ⊤ → ( 𝐹 : ( 0 [,] 1 ) –1-1-onto→ ( 0 [,] +∞ ) ∧ ◡ 𝐹 = ( 𝑦 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ) ) ) |
| 127 |
126
|
mptru |
⊢ ( 𝐹 : ( 0 [,] 1 ) –1-1-onto→ ( 0 [,] +∞ ) ∧ ◡ 𝐹 = ( 𝑦 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑦 = +∞ , 1 , ( 𝑦 / ( 1 + 𝑦 ) ) ) ) ) |