Step |
Hyp |
Ref |
Expression |
1 |
|
supxrgere.xph |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
supxrgere.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ* ) |
3 |
|
supxrgere.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
4 |
|
supxrgere.y |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 𝑥 ) < 𝑦 ) |
5 |
|
rexr |
⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* ) |
6 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
7 |
6
|
a1i |
⊢ ( 𝐵 ∈ ℝ → +∞ ∈ ℝ* ) |
8 |
|
ltpnf |
⊢ ( 𝐵 ∈ ℝ → 𝐵 < +∞ ) |
9 |
5 7 8
|
xrltled |
⊢ ( 𝐵 ∈ ℝ → 𝐵 ≤ +∞ ) |
10 |
3 9
|
syl |
⊢ ( 𝜑 → 𝐵 ≤ +∞ ) |
11 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) = +∞ ) → 𝐵 ≤ +∞ ) |
12 |
|
id |
⊢ ( sup ( 𝐴 , ℝ* , < ) = +∞ → sup ( 𝐴 , ℝ* , < ) = +∞ ) |
13 |
12
|
eqcomd |
⊢ ( sup ( 𝐴 , ℝ* , < ) = +∞ → +∞ = sup ( 𝐴 , ℝ* , < ) ) |
14 |
13
|
adantl |
⊢ ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) = +∞ ) → +∞ = sup ( 𝐴 , ℝ* , < ) ) |
15 |
11 14
|
breqtrd |
⊢ ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) = +∞ ) → 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) |
16 |
|
simpl |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → 𝜑 ) |
17 |
|
1rp |
⊢ 1 ∈ ℝ+ |
18 |
|
nfcv |
⊢ Ⅎ 𝑥 1 |
19 |
|
nfv |
⊢ Ⅎ 𝑥 1 ∈ ℝ+ |
20 |
1 19
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 1 ∈ ℝ+ ) |
21 |
|
nfv |
⊢ Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 1 ) < 𝑦 |
22 |
20 21
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 1 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 1 ) < 𝑦 ) |
23 |
|
eleq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 ∈ ℝ+ ↔ 1 ∈ ℝ+ ) ) |
24 |
23
|
anbi2d |
⊢ ( 𝑥 = 1 → ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ↔ ( 𝜑 ∧ 1 ∈ ℝ+ ) ) ) |
25 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 𝐵 − 𝑥 ) = ( 𝐵 − 1 ) ) |
26 |
25
|
breq1d |
⊢ ( 𝑥 = 1 → ( ( 𝐵 − 𝑥 ) < 𝑦 ↔ ( 𝐵 − 1 ) < 𝑦 ) ) |
27 |
26
|
rexbidv |
⊢ ( 𝑥 = 1 → ( ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 𝑥 ) < 𝑦 ↔ ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 1 ) < 𝑦 ) ) |
28 |
24 27
|
imbi12d |
⊢ ( 𝑥 = 1 → ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 𝑥 ) < 𝑦 ) ↔ ( ( 𝜑 ∧ 1 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 1 ) < 𝑦 ) ) ) |
29 |
18 22 28 4
|
vtoclgf |
⊢ ( 1 ∈ ℝ+ → ( ( 𝜑 ∧ 1 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 1 ) < 𝑦 ) ) |
30 |
17 29
|
ax-mp |
⊢ ( ( 𝜑 ∧ 1 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 1 ) < 𝑦 ) |
31 |
17 30
|
mpan2 |
⊢ ( 𝜑 → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 1 ) < 𝑦 ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 1 ) < 𝑦 ) |
33 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
34 |
33
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) → -∞ ∈ ℝ* ) |
35 |
2
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ ℝ* ) |
36 |
35
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) → 𝑦 ∈ ℝ* ) |
37 |
|
supxrcl |
⊢ ( 𝐴 ⊆ ℝ* → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* ) |
38 |
2 37
|
syl |
⊢ ( 𝜑 → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* ) |
39 |
38
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* ) |
40 |
|
peano2rem |
⊢ ( 𝐵 ∈ ℝ → ( 𝐵 − 1 ) ∈ ℝ ) |
41 |
3 40
|
syl |
⊢ ( 𝜑 → ( 𝐵 − 1 ) ∈ ℝ ) |
42 |
41
|
rexrd |
⊢ ( 𝜑 → ( 𝐵 − 1 ) ∈ ℝ* ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ -∞ < 𝑦 ) → ( 𝐵 − 1 ) ∈ ℝ* ) |
44 |
43
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) ∧ ¬ -∞ < 𝑦 ) → ( 𝐵 − 1 ) ∈ ℝ* ) |
45 |
36
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) ∧ ¬ -∞ < 𝑦 ) → 𝑦 ∈ ℝ* ) |
46 |
33
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) ∧ ¬ -∞ < 𝑦 ) → -∞ ∈ ℝ* ) |
47 |
|
simpl3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) ∧ ¬ -∞ < 𝑦 ) → ( 𝐵 − 1 ) < 𝑦 ) |
48 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ -∞ < 𝑦 ) → ¬ -∞ < 𝑦 ) |
49 |
35
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ -∞ < 𝑦 ) → 𝑦 ∈ ℝ* ) |
50 |
33
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ -∞ < 𝑦 ) → -∞ ∈ ℝ* ) |
51 |
|
xrlenlt |
⊢ ( ( 𝑦 ∈ ℝ* ∧ -∞ ∈ ℝ* ) → ( 𝑦 ≤ -∞ ↔ ¬ -∞ < 𝑦 ) ) |
52 |
49 50 51
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ -∞ < 𝑦 ) → ( 𝑦 ≤ -∞ ↔ ¬ -∞ < 𝑦 ) ) |
53 |
48 52
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ -∞ < 𝑦 ) → 𝑦 ≤ -∞ ) |
54 |
53
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) ∧ ¬ -∞ < 𝑦 ) → 𝑦 ≤ -∞ ) |
55 |
44 45 46 47 54
|
xrltletrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) ∧ ¬ -∞ < 𝑦 ) → ( 𝐵 − 1 ) < -∞ ) |
56 |
|
nltmnf |
⊢ ( ( 𝐵 − 1 ) ∈ ℝ* → ¬ ( 𝐵 − 1 ) < -∞ ) |
57 |
42 56
|
syl |
⊢ ( 𝜑 → ¬ ( 𝐵 − 1 ) < -∞ ) |
58 |
57
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ -∞ < 𝑦 ) → ¬ ( 𝐵 − 1 ) < -∞ ) |
59 |
58
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) ∧ ¬ -∞ < 𝑦 ) → ¬ ( 𝐵 − 1 ) < -∞ ) |
60 |
55 59
|
condan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) → -∞ < 𝑦 ) |
61 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝐴 ⊆ ℝ* ) |
62 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) |
63 |
|
supxrub |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ≤ sup ( 𝐴 , ℝ* , < ) ) |
64 |
61 62 63
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ≤ sup ( 𝐴 , ℝ* , < ) ) |
65 |
64
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) → 𝑦 ≤ sup ( 𝐴 , ℝ* , < ) ) |
66 |
34 36 39 60 65
|
xrltletrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ( 𝐵 − 1 ) < 𝑦 ) → -∞ < sup ( 𝐴 , ℝ* , < ) ) |
67 |
66
|
3exp |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐴 → ( ( 𝐵 − 1 ) < 𝑦 → -∞ < sup ( 𝐴 , ℝ* , < ) ) ) ) |
68 |
67
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → ( 𝑦 ∈ 𝐴 → ( ( 𝐵 − 1 ) < 𝑦 → -∞ < sup ( 𝐴 , ℝ* , < ) ) ) ) |
69 |
68
|
rexlimdv |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → ( ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 1 ) < 𝑦 → -∞ < sup ( 𝐴 , ℝ* , < ) ) ) |
70 |
32 69
|
mpd |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → -∞ < sup ( 𝐴 , ℝ* , < ) ) |
71 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) |
72 |
|
nltpnft |
⊢ ( sup ( 𝐴 , ℝ* , < ) ∈ ℝ* → ( sup ( 𝐴 , ℝ* , < ) = +∞ ↔ ¬ sup ( 𝐴 , ℝ* , < ) < +∞ ) ) |
73 |
38 72
|
syl |
⊢ ( 𝜑 → ( sup ( 𝐴 , ℝ* , < ) = +∞ ↔ ¬ sup ( 𝐴 , ℝ* , < ) < +∞ ) ) |
74 |
73
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → ( sup ( 𝐴 , ℝ* , < ) = +∞ ↔ ¬ sup ( 𝐴 , ℝ* , < ) < +∞ ) ) |
75 |
71 74
|
mtbid |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → ¬ ¬ sup ( 𝐴 , ℝ* , < ) < +∞ ) |
76 |
75
|
notnotrd |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → sup ( 𝐴 , ℝ* , < ) < +∞ ) |
77 |
70 76
|
jca |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → ( -∞ < sup ( 𝐴 , ℝ* , < ) ∧ sup ( 𝐴 , ℝ* , < ) < +∞ ) ) |
78 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* ) |
79 |
|
xrrebnd |
⊢ ( sup ( 𝐴 , ℝ* , < ) ∈ ℝ* → ( sup ( 𝐴 , ℝ* , < ) ∈ ℝ ↔ ( -∞ < sup ( 𝐴 , ℝ* , < ) ∧ sup ( 𝐴 , ℝ* , < ) < +∞ ) ) ) |
80 |
78 79
|
syl |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → ( sup ( 𝐴 , ℝ* , < ) ∈ ℝ ↔ ( -∞ < sup ( 𝐴 , ℝ* , < ) ∧ sup ( 𝐴 , ℝ* , < ) < +∞ ) ) ) |
81 |
77 80
|
mpbird |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) |
82 |
|
simpl |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) → ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ) |
83 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) → ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) |
84 |
82
|
simprd |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) → sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) |
85 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) → 𝐵 ∈ ℝ ) |
86 |
84 85
|
ltnled |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) → ( sup ( 𝐴 , ℝ* , < ) < 𝐵 ↔ ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) ) |
87 |
83 86
|
mpbird |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) → sup ( 𝐴 , ℝ* , < ) < 𝐵 ) |
88 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → 𝜑 ) |
89 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) → 𝐵 ∈ ℝ ) |
90 |
|
simpr |
⊢ ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) → sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) |
91 |
89 90
|
resubcld |
⊢ ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) → ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ ) |
92 |
91
|
adantr |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ ) |
93 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → sup ( 𝐴 , ℝ* , < ) < 𝐵 ) |
94 |
90
|
adantr |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) |
95 |
88 3
|
syl |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → 𝐵 ∈ ℝ ) |
96 |
94 95
|
posdifd |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → ( sup ( 𝐴 , ℝ* , < ) < 𝐵 ↔ 0 < ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) ) |
97 |
93 96
|
mpbid |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → 0 < ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) |
98 |
92 97
|
elrpd |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ+ ) |
99 |
|
ovex |
⊢ ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ V |
100 |
|
nfcv |
⊢ Ⅎ 𝑥 ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) |
101 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ+ |
102 |
1 101
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ+ ) |
103 |
|
nfv |
⊢ Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 |
104 |
102 103
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) |
105 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) → ( 𝑥 ∈ ℝ+ ↔ ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ+ ) ) |
106 |
105
|
anbi2d |
⊢ ( 𝑥 = ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) → ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ↔ ( 𝜑 ∧ ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ+ ) ) ) |
107 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) → ( 𝐵 − 𝑥 ) = ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) ) |
108 |
107
|
breq1d |
⊢ ( 𝑥 = ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) → ( ( 𝐵 − 𝑥 ) < 𝑦 ↔ ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) ) |
109 |
108
|
rexbidv |
⊢ ( 𝑥 = ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) → ( ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 𝑥 ) < 𝑦 ↔ ∃ 𝑦 ∈ 𝐴 ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) ) |
110 |
106 109
|
imbi12d |
⊢ ( 𝑥 = ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) → ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − 𝑥 ) < 𝑦 ) ↔ ( ( 𝜑 ∧ ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) ) ) |
111 |
100 104 110 4
|
vtoclgf |
⊢ ( ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ V → ( ( 𝜑 ∧ ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) ) |
112 |
99 111
|
ax-mp |
⊢ ( ( 𝜑 ∧ ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) |
113 |
88 98 112
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → ∃ 𝑦 ∈ 𝐴 ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) |
114 |
3
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
115 |
114
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) ∧ ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) → 𝐵 ∈ ℂ ) |
116 |
90
|
recnd |
⊢ ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) → sup ( 𝐴 , ℝ* , < ) ∈ ℂ ) |
117 |
116
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) ∧ ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) → sup ( 𝐴 , ℝ* , < ) ∈ ℂ ) |
118 |
115 117
|
nncand |
⊢ ( ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) ∧ ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) → ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) = sup ( 𝐴 , ℝ* , < ) ) |
119 |
118
|
eqcomd |
⊢ ( ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) ∧ ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) → sup ( 𝐴 , ℝ* , < ) = ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) ) |
120 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) ∧ ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) → ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) |
121 |
119 120
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) ∧ ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 ) → sup ( 𝐴 , ℝ* , < ) < 𝑦 ) |
122 |
121
|
ex |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → ( ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 → sup ( 𝐴 , ℝ* , < ) < 𝑦 ) ) |
123 |
122
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 → sup ( 𝐴 , ℝ* , < ) < 𝑦 ) ) |
124 |
123
|
reximdva |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → ( ∃ 𝑦 ∈ 𝐴 ( 𝐵 − ( 𝐵 − sup ( 𝐴 , ℝ* , < ) ) ) < 𝑦 → ∃ 𝑦 ∈ 𝐴 sup ( 𝐴 , ℝ* , < ) < 𝑦 ) ) |
125 |
113 124
|
mpd |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) → ∃ 𝑦 ∈ 𝐴 sup ( 𝐴 , ℝ* , < ) < 𝑦 ) |
126 |
82 87 125
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) → ∃ 𝑦 ∈ 𝐴 sup ( 𝐴 , ℝ* , < ) < 𝑦 ) |
127 |
61 37
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* ) |
128 |
35 127
|
xrlenltd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑦 ≤ sup ( 𝐴 , ℝ* , < ) ↔ ¬ sup ( 𝐴 , ℝ* , < ) < 𝑦 ) ) |
129 |
64 128
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ¬ sup ( 𝐴 , ℝ* , < ) < 𝑦 ) |
130 |
129
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 ¬ sup ( 𝐴 , ℝ* , < ) < 𝑦 ) |
131 |
|
ralnex |
⊢ ( ∀ 𝑦 ∈ 𝐴 ¬ sup ( 𝐴 , ℝ* , < ) < 𝑦 ↔ ¬ ∃ 𝑦 ∈ 𝐴 sup ( 𝐴 , ℝ* , < ) < 𝑦 ) |
132 |
130 131
|
sylib |
⊢ ( 𝜑 → ¬ ∃ 𝑦 ∈ 𝐴 sup ( 𝐴 , ℝ* , < ) < 𝑦 ) |
133 |
132
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) ∧ ¬ 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) → ¬ ∃ 𝑦 ∈ 𝐴 sup ( 𝐴 , ℝ* , < ) < 𝑦 ) |
134 |
126 133
|
condan |
⊢ ( ( 𝜑 ∧ sup ( 𝐴 , ℝ* , < ) ∈ ℝ ) → 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) |
135 |
16 81 134
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ sup ( 𝐴 , ℝ* , < ) = +∞ ) → 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) |
136 |
15 135
|
pm2.61dan |
⊢ ( 𝜑 → 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) |