Step |
Hyp |
Ref |
Expression |
1 |
|
supmul1.1 |
|- C = { z | E. v e. B z = ( A x. v ) } |
2 |
|
supmul1.2 |
|- ( ph <-> ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) ) |
3 |
|
vex |
|- w e. _V |
4 |
|
oveq2 |
|- ( v = b -> ( A x. v ) = ( A x. b ) ) |
5 |
4
|
eqeq2d |
|- ( v = b -> ( z = ( A x. v ) <-> z = ( A x. b ) ) ) |
6 |
5
|
cbvrexvw |
|- ( E. v e. B z = ( A x. v ) <-> E. b e. B z = ( A x. b ) ) |
7 |
|
eqeq1 |
|- ( z = w -> ( z = ( A x. b ) <-> w = ( A x. b ) ) ) |
8 |
7
|
rexbidv |
|- ( z = w -> ( E. b e. B z = ( A x. b ) <-> E. b e. B w = ( A x. b ) ) ) |
9 |
6 8
|
syl5bb |
|- ( z = w -> ( E. v e. B z = ( A x. v ) <-> E. b e. B w = ( A x. b ) ) ) |
10 |
3 9 1
|
elab2 |
|- ( w e. C <-> E. b e. B w = ( A x. b ) ) |
11 |
|
simpr |
|- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) |
12 |
2 11
|
sylbi |
|- ( ph -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) |
13 |
12
|
simp1d |
|- ( ph -> B C_ RR ) |
14 |
13
|
sselda |
|- ( ( ph /\ b e. B ) -> b e. RR ) |
15 |
|
suprcl |
|- ( ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) -> sup ( B , RR , < ) e. RR ) |
16 |
12 15
|
syl |
|- ( ph -> sup ( B , RR , < ) e. RR ) |
17 |
16
|
adantr |
|- ( ( ph /\ b e. B ) -> sup ( B , RR , < ) e. RR ) |
18 |
|
simpl1 |
|- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> A e. RR ) |
19 |
2 18
|
sylbi |
|- ( ph -> A e. RR ) |
20 |
|
simpl2 |
|- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> 0 <_ A ) |
21 |
2 20
|
sylbi |
|- ( ph -> 0 <_ A ) |
22 |
19 21
|
jca |
|- ( ph -> ( A e. RR /\ 0 <_ A ) ) |
23 |
22
|
adantr |
|- ( ( ph /\ b e. B ) -> ( A e. RR /\ 0 <_ A ) ) |
24 |
|
suprub |
|- ( ( ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) /\ b e. B ) -> b <_ sup ( B , RR , < ) ) |
25 |
12 24
|
sylan |
|- ( ( ph /\ b e. B ) -> b <_ sup ( B , RR , < ) ) |
26 |
|
lemul2a |
|- ( ( ( b e. RR /\ sup ( B , RR , < ) e. RR /\ ( A e. RR /\ 0 <_ A ) ) /\ b <_ sup ( B , RR , < ) ) -> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) ) |
27 |
14 17 23 25 26
|
syl31anc |
|- ( ( ph /\ b e. B ) -> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) ) |
28 |
|
breq1 |
|- ( w = ( A x. b ) -> ( w <_ ( A x. sup ( B , RR , < ) ) <-> ( A x. b ) <_ ( A x. sup ( B , RR , < ) ) ) ) |
29 |
27 28
|
syl5ibrcom |
|- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) ) |
30 |
29
|
rexlimdva |
|- ( ph -> ( E. b e. B w = ( A x. b ) -> w <_ ( A x. sup ( B , RR , < ) ) ) ) |
31 |
10 30
|
syl5bi |
|- ( ph -> ( w e. C -> w <_ ( A x. sup ( B , RR , < ) ) ) ) |
32 |
31
|
ralrimiv |
|- ( ph -> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) |
33 |
19
|
adantr |
|- ( ( ph /\ b e. B ) -> A e. RR ) |
34 |
33 14
|
remulcld |
|- ( ( ph /\ b e. B ) -> ( A x. b ) e. RR ) |
35 |
|
eleq1a |
|- ( ( A x. b ) e. RR -> ( w = ( A x. b ) -> w e. RR ) ) |
36 |
34 35
|
syl |
|- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> w e. RR ) ) |
37 |
36
|
rexlimdva |
|- ( ph -> ( E. b e. B w = ( A x. b ) -> w e. RR ) ) |
38 |
10 37
|
syl5bi |
|- ( ph -> ( w e. C -> w e. RR ) ) |
39 |
38
|
ssrdv |
|- ( ph -> C C_ RR ) |
40 |
|
simpr2 |
|- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> B =/= (/) ) |
41 |
2 40
|
sylbi |
|- ( ph -> B =/= (/) ) |
42 |
|
ovex |
|- ( A x. b ) e. _V |
43 |
42
|
isseti |
|- E. w w = ( A x. b ) |
44 |
43
|
rgenw |
|- A. b e. B E. w w = ( A x. b ) |
45 |
|
r19.2z |
|- ( ( B =/= (/) /\ A. b e. B E. w w = ( A x. b ) ) -> E. b e. B E. w w = ( A x. b ) ) |
46 |
41 44 45
|
sylancl |
|- ( ph -> E. b e. B E. w w = ( A x. b ) ) |
47 |
10
|
exbii |
|- ( E. w w e. C <-> E. w E. b e. B w = ( A x. b ) ) |
48 |
|
n0 |
|- ( C =/= (/) <-> E. w w e. C ) |
49 |
|
rexcom4 |
|- ( E. b e. B E. w w = ( A x. b ) <-> E. w E. b e. B w = ( A x. b ) ) |
50 |
47 48 49
|
3bitr4i |
|- ( C =/= (/) <-> E. b e. B E. w w = ( A x. b ) ) |
51 |
46 50
|
sylibr |
|- ( ph -> C =/= (/) ) |
52 |
19 16
|
remulcld |
|- ( ph -> ( A x. sup ( B , RR , < ) ) e. RR ) |
53 |
|
brralrspcev |
|- ( ( ( A x. sup ( B , RR , < ) ) e. RR /\ A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) -> E. x e. RR A. w e. C w <_ x ) |
54 |
52 32 53
|
syl2anc |
|- ( ph -> E. x e. RR A. w e. C w <_ x ) |
55 |
39 51 54
|
3jca |
|- ( ph -> ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) ) |
56 |
|
suprleub |
|- ( ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) /\ ( A x. sup ( B , RR , < ) ) e. RR ) -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) <-> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) ) |
57 |
55 52 56
|
syl2anc |
|- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) <-> A. w e. C w <_ ( A x. sup ( B , RR , < ) ) ) ) |
58 |
32 57
|
mpbird |
|- ( ph -> sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) ) |
59 |
|
simpr |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) |
60 |
|
suprcl |
|- ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) -> sup ( C , RR , < ) e. RR ) |
61 |
55 60
|
syl |
|- ( ph -> sup ( C , RR , < ) e. RR ) |
62 |
61
|
adantr |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( C , RR , < ) e. RR ) |
63 |
16
|
adantr |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> sup ( B , RR , < ) e. RR ) |
64 |
19
|
adantr |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A e. RR ) |
65 |
|
n0 |
|- ( B =/= (/) <-> E. b b e. B ) |
66 |
|
0red |
|- ( ( ph /\ b e. B ) -> 0 e. RR ) |
67 |
|
simpl3 |
|- ( ( ( A e. RR /\ 0 <_ A /\ A. x e. B 0 <_ x ) /\ ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) -> A. x e. B 0 <_ x ) |
68 |
2 67
|
sylbi |
|- ( ph -> A. x e. B 0 <_ x ) |
69 |
|
breq2 |
|- ( x = b -> ( 0 <_ x <-> 0 <_ b ) ) |
70 |
69
|
rspccva |
|- ( ( A. x e. B 0 <_ x /\ b e. B ) -> 0 <_ b ) |
71 |
68 70
|
sylan |
|- ( ( ph /\ b e. B ) -> 0 <_ b ) |
72 |
66 14 17 71 25
|
letrd |
|- ( ( ph /\ b e. B ) -> 0 <_ sup ( B , RR , < ) ) |
73 |
72
|
ex |
|- ( ph -> ( b e. B -> 0 <_ sup ( B , RR , < ) ) ) |
74 |
73
|
exlimdv |
|- ( ph -> ( E. b b e. B -> 0 <_ sup ( B , RR , < ) ) ) |
75 |
65 74
|
syl5bi |
|- ( ph -> ( B =/= (/) -> 0 <_ sup ( B , RR , < ) ) ) |
76 |
41 75
|
mpd |
|- ( ph -> 0 <_ sup ( B , RR , < ) ) |
77 |
76
|
adantr |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 <_ sup ( B , RR , < ) ) |
78 |
|
0red |
|- ( ( ph /\ w e. C ) -> 0 e. RR ) |
79 |
38
|
imp |
|- ( ( ph /\ w e. C ) -> w e. RR ) |
80 |
61
|
adantr |
|- ( ( ph /\ w e. C ) -> sup ( C , RR , < ) e. RR ) |
81 |
21
|
adantr |
|- ( ( ph /\ b e. B ) -> 0 <_ A ) |
82 |
33 14 81 71
|
mulge0d |
|- ( ( ph /\ b e. B ) -> 0 <_ ( A x. b ) ) |
83 |
|
breq2 |
|- ( w = ( A x. b ) -> ( 0 <_ w <-> 0 <_ ( A x. b ) ) ) |
84 |
82 83
|
syl5ibrcom |
|- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> 0 <_ w ) ) |
85 |
84
|
rexlimdva |
|- ( ph -> ( E. b e. B w = ( A x. b ) -> 0 <_ w ) ) |
86 |
10 85
|
syl5bi |
|- ( ph -> ( w e. C -> 0 <_ w ) ) |
87 |
86
|
imp |
|- ( ( ph /\ w e. C ) -> 0 <_ w ) |
88 |
|
suprub |
|- ( ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) /\ w e. C ) -> w <_ sup ( C , RR , < ) ) |
89 |
55 88
|
sylan |
|- ( ( ph /\ w e. C ) -> w <_ sup ( C , RR , < ) ) |
90 |
78 79 80 87 89
|
letrd |
|- ( ( ph /\ w e. C ) -> 0 <_ sup ( C , RR , < ) ) |
91 |
90
|
ex |
|- ( ph -> ( w e. C -> 0 <_ sup ( C , RR , < ) ) ) |
92 |
91
|
exlimdv |
|- ( ph -> ( E. w w e. C -> 0 <_ sup ( C , RR , < ) ) ) |
93 |
48 92
|
syl5bi |
|- ( ph -> ( C =/= (/) -> 0 <_ sup ( C , RR , < ) ) ) |
94 |
51 93
|
mpd |
|- ( ph -> 0 <_ sup ( C , RR , < ) ) |
95 |
94
|
anim1i |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) ) |
96 |
|
0red |
|- ( ph -> 0 e. RR ) |
97 |
|
lelttr |
|- ( ( 0 e. RR /\ sup ( C , RR , < ) e. RR /\ ( A x. sup ( B , RR , < ) ) e. RR ) -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) ) |
98 |
96 61 52 97
|
syl3anc |
|- ( ph -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) ) |
99 |
98
|
adantr |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( ( 0 <_ sup ( C , RR , < ) /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) ) |
100 |
95 99
|
mpd |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < ( A x. sup ( B , RR , < ) ) ) |
101 |
|
prodgt02 |
|- ( ( ( A e. RR /\ sup ( B , RR , < ) e. RR ) /\ ( 0 <_ sup ( B , RR , < ) /\ 0 < ( A x. sup ( B , RR , < ) ) ) ) -> 0 < A ) |
102 |
64 63 77 100 101
|
syl22anc |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> 0 < A ) |
103 |
|
ltdivmul |
|- ( ( sup ( C , RR , < ) e. RR /\ sup ( B , RR , < ) e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) <-> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) ) |
104 |
62 63 64 102 103
|
syl112anc |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) <-> sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) ) |
105 |
59 104
|
mpbird |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) ) |
106 |
12
|
adantr |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) ) |
107 |
102
|
gt0ne0d |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> A =/= 0 ) |
108 |
62 64 107
|
redivcld |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( sup ( C , RR , < ) / A ) e. RR ) |
109 |
|
suprlub |
|- ( ( ( B C_ RR /\ B =/= (/) /\ E. x e. RR A. y e. B y <_ x ) /\ ( sup ( C , RR , < ) / A ) e. RR ) -> ( ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) <-> E. b e. B ( sup ( C , RR , < ) / A ) < b ) ) |
110 |
106 108 109
|
syl2anc |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> ( ( sup ( C , RR , < ) / A ) < sup ( B , RR , < ) <-> E. b e. B ( sup ( C , RR , < ) / A ) < b ) ) |
111 |
105 110
|
mpbid |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> E. b e. B ( sup ( C , RR , < ) / A ) < b ) |
112 |
34
|
adantlr |
|- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) e. RR ) |
113 |
61
|
ad2antrr |
|- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> sup ( C , RR , < ) e. RR ) |
114 |
|
rspe |
|- ( ( b e. B /\ w = ( A x. b ) ) -> E. b e. B w = ( A x. b ) ) |
115 |
114 10
|
sylibr |
|- ( ( b e. B /\ w = ( A x. b ) ) -> w e. C ) |
116 |
115
|
adantl |
|- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> w e. C ) |
117 |
|
simplrr |
|- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w = ( A x. b ) ) |
118 |
89
|
adantlr |
|- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> w <_ sup ( C , RR , < ) ) |
119 |
117 118
|
eqbrtrrd |
|- ( ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) /\ w e. C ) -> ( A x. b ) <_ sup ( C , RR , < ) ) |
120 |
116 119
|
mpdan |
|- ( ( ph /\ ( b e. B /\ w = ( A x. b ) ) ) -> ( A x. b ) <_ sup ( C , RR , < ) ) |
121 |
120
|
expr |
|- ( ( ph /\ b e. B ) -> ( w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) ) |
122 |
121
|
exlimdv |
|- ( ( ph /\ b e. B ) -> ( E. w w = ( A x. b ) -> ( A x. b ) <_ sup ( C , RR , < ) ) ) |
123 |
43 122
|
mpi |
|- ( ( ph /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) ) |
124 |
123
|
adantlr |
|- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( A x. b ) <_ sup ( C , RR , < ) ) |
125 |
112 113 124
|
lensymd |
|- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. sup ( C , RR , < ) < ( A x. b ) ) |
126 |
14
|
adantlr |
|- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> b e. RR ) |
127 |
19
|
ad2antrr |
|- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> A e. RR ) |
128 |
102
|
adantr |
|- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> 0 < A ) |
129 |
|
ltdivmul |
|- ( ( sup ( C , RR , < ) e. RR /\ b e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( sup ( C , RR , < ) / A ) < b <-> sup ( C , RR , < ) < ( A x. b ) ) ) |
130 |
113 126 127 128 129
|
syl112anc |
|- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> ( ( sup ( C , RR , < ) / A ) < b <-> sup ( C , RR , < ) < ( A x. b ) ) ) |
131 |
125 130
|
mtbird |
|- ( ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) /\ b e. B ) -> -. ( sup ( C , RR , < ) / A ) < b ) |
132 |
131
|
nrexdv |
|- ( ( ph /\ sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) -> -. E. b e. B ( sup ( C , RR , < ) / A ) < b ) |
133 |
111 132
|
pm2.65da |
|- ( ph -> -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) |
134 |
58 133
|
jca |
|- ( ph -> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) /\ -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) ) |
135 |
61 52
|
eqleltd |
|- ( ph -> ( sup ( C , RR , < ) = ( A x. sup ( B , RR , < ) ) <-> ( sup ( C , RR , < ) <_ ( A x. sup ( B , RR , < ) ) /\ -. sup ( C , RR , < ) < ( A x. sup ( B , RR , < ) ) ) ) ) |
136 |
134 135
|
mpbird |
|- ( ph -> sup ( C , RR , < ) = ( A x. sup ( B , RR , < ) ) ) |
137 |
136
|
eqcomd |
|- ( ph -> ( A x. sup ( B , RR , < ) ) = sup ( C , RR , < ) ) |