Step |
Hyp |
Ref |
Expression |
1 |
|
xralrple3.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
2 |
|
xralrple3.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
xralrple3.c |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
4 |
|
xralrple3.g |
⊢ ( 𝜑 → 0 ≤ 𝐶 ) |
5 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → 𝐴 ∈ ℝ* ) |
6 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
7 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → 𝐵 ∈ ℝ* ) |
8 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → 𝐵 ∈ ℝ ) |
9 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → 𝐶 ∈ ℝ ) |
10 |
|
rpre |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ ) |
11 |
10
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ ) |
12 |
9 11
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → ( 𝐶 · 𝑥 ) ∈ ℝ ) |
13 |
8 12
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → ( 𝐵 + ( 𝐶 · 𝑥 ) ) ∈ ℝ ) |
14 |
13
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → ( 𝐵 + ( 𝐶 · 𝑥 ) ) ∈ ℝ* ) |
15 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → 𝐴 ≤ 𝐵 ) |
16 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐶 ∈ ℝ ) |
17 |
10
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ ) |
18 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 0 ≤ 𝐶 ) |
19 |
|
rpge0 |
⊢ ( 𝑥 ∈ ℝ+ → 0 ≤ 𝑥 ) |
20 |
19
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 0 ≤ 𝑥 ) |
21 |
16 17 18 20
|
mulge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 0 ≤ ( 𝐶 · 𝑥 ) ) |
22 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐵 ∈ ℝ ) |
23 |
16 17
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝐶 · 𝑥 ) ∈ ℝ ) |
24 |
22 23
|
addge01d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 0 ≤ ( 𝐶 · 𝑥 ) ↔ 𝐵 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ) |
25 |
21 24
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐵 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) |
26 |
25
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → 𝐵 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) |
27 |
5 7 14 15 26
|
xrletrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) → 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) |
28 |
27
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) |
29 |
28
|
ex |
⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 → ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ) |
30 |
|
1rp |
⊢ 1 ∈ ℝ+ |
31 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 𝐶 · 𝑥 ) = ( 𝐶 · 1 ) ) |
32 |
31
|
oveq2d |
⊢ ( 𝑥 = 1 → ( 𝐵 + ( 𝐶 · 𝑥 ) ) = ( 𝐵 + ( 𝐶 · 1 ) ) ) |
33 |
32
|
breq2d |
⊢ ( 𝑥 = 1 → ( 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ↔ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 1 ) ) ) ) |
34 |
33
|
rspcva |
⊢ ( ( 1 ∈ ℝ+ ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) → 𝐴 ≤ ( 𝐵 + ( 𝐶 · 1 ) ) ) |
35 |
30 34
|
mpan |
⊢ ( ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) → 𝐴 ≤ ( 𝐵 + ( 𝐶 · 1 ) ) ) |
36 |
35
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 = 0 ) → 𝐴 ≤ ( 𝐵 + ( 𝐶 · 1 ) ) ) |
37 |
|
oveq1 |
⊢ ( 𝐶 = 0 → ( 𝐶 · 1 ) = ( 0 · 1 ) ) |
38 |
|
0cn |
⊢ 0 ∈ ℂ |
39 |
38
|
mulid1i |
⊢ ( 0 · 1 ) = 0 |
40 |
39
|
a1i |
⊢ ( 𝐶 = 0 → ( 0 · 1 ) = 0 ) |
41 |
37 40
|
eqtrd |
⊢ ( 𝐶 = 0 → ( 𝐶 · 1 ) = 0 ) |
42 |
41
|
oveq2d |
⊢ ( 𝐶 = 0 → ( 𝐵 + ( 𝐶 · 1 ) ) = ( 𝐵 + 0 ) ) |
43 |
42
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐶 = 0 ) → ( 𝐵 + ( 𝐶 · 1 ) ) = ( 𝐵 + 0 ) ) |
44 |
2
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 = 0 ) → 𝐵 ∈ ℂ ) |
46 |
45
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝐶 = 0 ) → ( 𝐵 + 0 ) = 𝐵 ) |
47 |
43 46
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐶 = 0 ) → ( 𝐵 + ( 𝐶 · 1 ) ) = 𝐵 ) |
48 |
47
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 = 0 ) → ( 𝐵 + ( 𝐶 · 1 ) ) = 𝐵 ) |
49 |
36 48
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 = 0 ) → 𝐴 ≤ 𝐵 ) |
50 |
|
neqne |
⊢ ( ¬ 𝐶 = 0 → 𝐶 ≠ 0 ) |
51 |
50
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 = 0 ) → 𝐶 ≠ 0 ) |
52 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ≠ 0 ) → 𝐶 ∈ ℝ ) |
53 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝐶 ≠ 0 ) → 0 ∈ ℝ ) |
54 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ≠ 0 ) → 0 ≤ 𝐶 ) |
55 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 ≠ 0 ) → 𝐶 ≠ 0 ) |
56 |
53 52 54 55
|
leneltd |
⊢ ( ( 𝜑 ∧ 𝐶 ≠ 0 ) → 0 < 𝐶 ) |
57 |
52 56
|
elrpd |
⊢ ( ( 𝜑 ∧ 𝐶 ≠ 0 ) → 𝐶 ∈ ℝ+ ) |
58 |
51 57
|
syldan |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 = 0 ) → 𝐶 ∈ ℝ+ ) |
59 |
58
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ ¬ 𝐶 = 0 ) → 𝐶 ∈ ℝ+ ) |
60 |
|
simpr |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → 𝑦 ∈ ℝ+ ) |
61 |
|
simpl |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → 𝐶 ∈ ℝ+ ) |
62 |
60 61
|
rpdivcld |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → ( 𝑦 / 𝐶 ) ∈ ℝ+ ) |
63 |
62
|
adantll |
⊢ ( ( ( ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ∧ 𝐶 ∈ ℝ+ ) ∧ 𝑦 ∈ ℝ+ ) → ( 𝑦 / 𝐶 ) ∈ ℝ+ ) |
64 |
|
simpll |
⊢ ( ( ( ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ∧ 𝐶 ∈ ℝ+ ) ∧ 𝑦 ∈ ℝ+ ) → ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) |
65 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 / 𝐶 ) → ( 𝐶 · 𝑥 ) = ( 𝐶 · ( 𝑦 / 𝐶 ) ) ) |
66 |
65
|
oveq2d |
⊢ ( 𝑥 = ( 𝑦 / 𝐶 ) → ( 𝐵 + ( 𝐶 · 𝑥 ) ) = ( 𝐵 + ( 𝐶 · ( 𝑦 / 𝐶 ) ) ) ) |
67 |
66
|
breq2d |
⊢ ( 𝑥 = ( 𝑦 / 𝐶 ) → ( 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ↔ 𝐴 ≤ ( 𝐵 + ( 𝐶 · ( 𝑦 / 𝐶 ) ) ) ) ) |
68 |
67
|
rspcva |
⊢ ( ( ( 𝑦 / 𝐶 ) ∈ ℝ+ ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) → 𝐴 ≤ ( 𝐵 + ( 𝐶 · ( 𝑦 / 𝐶 ) ) ) ) |
69 |
63 64 68
|
syl2anc |
⊢ ( ( ( ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ∧ 𝐶 ∈ ℝ+ ) ∧ 𝑦 ∈ ℝ+ ) → 𝐴 ≤ ( 𝐵 + ( 𝐶 · ( 𝑦 / 𝐶 ) ) ) ) |
70 |
69
|
adantlll |
⊢ ( ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 ∈ ℝ+ ) ∧ 𝑦 ∈ ℝ+ ) → 𝐴 ≤ ( 𝐵 + ( 𝐶 · ( 𝑦 / 𝐶 ) ) ) ) |
71 |
60
|
rpcnd |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → 𝑦 ∈ ℂ ) |
72 |
61
|
rpcnd |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → 𝐶 ∈ ℂ ) |
73 |
61
|
rpne0d |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → 𝐶 ≠ 0 ) |
74 |
71 72 73
|
divcan2d |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → ( 𝐶 · ( 𝑦 / 𝐶 ) ) = 𝑦 ) |
75 |
74
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 ∈ ℝ+ ) ∧ 𝑦 ∈ ℝ+ ) → ( 𝐶 · ( 𝑦 / 𝐶 ) ) = 𝑦 ) |
76 |
75
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 ∈ ℝ+ ) ∧ 𝑦 ∈ ℝ+ ) → ( 𝐵 + ( 𝐶 · ( 𝑦 / 𝐶 ) ) ) = ( 𝐵 + 𝑦 ) ) |
77 |
70 76
|
breqtrd |
⊢ ( ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 ∈ ℝ+ ) ∧ 𝑦 ∈ ℝ+ ) → 𝐴 ≤ ( 𝐵 + 𝑦 ) ) |
78 |
77
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 ∈ ℝ+ ) → ∀ 𝑦 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + 𝑦 ) ) |
79 |
|
xralrple |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ∀ 𝑦 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + 𝑦 ) ) ) |
80 |
1 2 79
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ∀ 𝑦 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + 𝑦 ) ) ) |
81 |
80
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 ≤ 𝐵 ↔ ∀ 𝑦 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + 𝑦 ) ) ) |
82 |
78 81
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ 𝐶 ∈ ℝ+ ) → 𝐴 ≤ 𝐵 ) |
83 |
59 82
|
syldan |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ∧ ¬ 𝐶 = 0 ) → 𝐴 ≤ 𝐵 ) |
84 |
49 83
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) → 𝐴 ≤ 𝐵 ) |
85 |
84
|
ex |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) → 𝐴 ≤ 𝐵 ) ) |
86 |
29 85
|
impbid |
⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + ( 𝐶 · 𝑥 ) ) ) ) |