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