Step |
Hyp |
Ref |
Expression |
1 |
|
requad2.a |
|- ( ph -> A e. RR ) |
2 |
|
requad2.z |
|- ( ph -> A =/= 0 ) |
3 |
|
requad2.b |
|- ( ph -> B e. RR ) |
4 |
|
requad2.c |
|- ( ph -> C e. RR ) |
5 |
|
requad2.d |
|- ( ph -> D = ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) ) |
6 |
1
|
recnd |
|- ( ph -> A e. CC ) |
7 |
6
|
ad3antrrr |
|- ( ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) /\ x e. p ) -> A e. CC ) |
8 |
2
|
ad3antrrr |
|- ( ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) /\ x e. p ) -> A =/= 0 ) |
9 |
3
|
recnd |
|- ( ph -> B e. CC ) |
10 |
9
|
ad3antrrr |
|- ( ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) /\ x e. p ) -> B e. CC ) |
11 |
4
|
recnd |
|- ( ph -> C e. CC ) |
12 |
11
|
ad3antrrr |
|- ( ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) /\ x e. p ) -> C e. CC ) |
13 |
|
elelpwi |
|- ( ( x e. p /\ p e. ~P RR ) -> x e. RR ) |
14 |
13
|
expcom |
|- ( p e. ~P RR -> ( x e. p -> x e. RR ) ) |
15 |
14
|
adantl |
|- ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) -> ( x e. p -> x e. RR ) ) |
16 |
15
|
imp |
|- ( ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) /\ x e. p ) -> x e. RR ) |
17 |
16
|
recnd |
|- ( ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) /\ x e. p ) -> x e. CC ) |
18 |
5
|
ad3antrrr |
|- ( ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) /\ x e. p ) -> D = ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) ) |
19 |
7 8 10 12 17 18
|
quad |
|- ( ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) /\ x e. p ) -> ( ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) ) |
20 |
19
|
ralbidva |
|- ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) -> ( A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) ) |
21 |
20
|
anbi2d |
|- ( ( ( ph /\ 0 <_ D ) /\ p e. ~P RR ) -> ( ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) <-> ( ( # ` p ) = 2 /\ A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) ) ) |
22 |
21
|
reubidva |
|- ( ( ph /\ 0 <_ D ) -> ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) <-> E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) ) ) |
23 |
|
eqid |
|- { q e. ~P RR | ( # ` q ) = 2 } = { q e. ~P RR | ( # ` q ) = 2 } |
24 |
23
|
pairreueq |
|- ( E! p e. { q e. ~P RR | ( # ` q ) = 2 } A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) <-> E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) ) |
25 |
24
|
bicomi |
|- ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) <-> E! p e. { q e. ~P RR | ( # ` q ) = 2 } A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) |
26 |
25
|
a1i |
|- ( ( ph /\ 0 <_ D ) -> ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) <-> E! p e. { q e. ~P RR | ( # ` q ) = 2 } A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) ) |
27 |
3
|
renegcld |
|- ( ph -> -u B e. RR ) |
28 |
27
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> -u B e. RR ) |
29 |
3
|
resqcld |
|- ( ph -> ( B ^ 2 ) e. RR ) |
30 |
|
4re |
|- 4 e. RR |
31 |
30
|
a1i |
|- ( ph -> 4 e. RR ) |
32 |
1 4
|
remulcld |
|- ( ph -> ( A x. C ) e. RR ) |
33 |
31 32
|
remulcld |
|- ( ph -> ( 4 x. ( A x. C ) ) e. RR ) |
34 |
29 33
|
resubcld |
|- ( ph -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) e. RR ) |
35 |
5 34
|
eqeltrd |
|- ( ph -> D e. RR ) |
36 |
35
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> D e. RR ) |
37 |
|
simpr |
|- ( ( ph /\ 0 <_ D ) -> 0 <_ D ) |
38 |
36 37
|
resqrtcld |
|- ( ( ph /\ 0 <_ D ) -> ( sqrt ` D ) e. RR ) |
39 |
28 38
|
readdcld |
|- ( ( ph /\ 0 <_ D ) -> ( -u B + ( sqrt ` D ) ) e. RR ) |
40 |
|
2re |
|- 2 e. RR |
41 |
40
|
a1i |
|- ( ph -> 2 e. RR ) |
42 |
41 1
|
remulcld |
|- ( ph -> ( 2 x. A ) e. RR ) |
43 |
42
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( 2 x. A ) e. RR ) |
44 |
|
2cnne0 |
|- ( 2 e. CC /\ 2 =/= 0 ) |
45 |
44
|
a1i |
|- ( ph -> ( 2 e. CC /\ 2 =/= 0 ) ) |
46 |
|
mulne0 |
|- ( ( ( 2 e. CC /\ 2 =/= 0 ) /\ ( A e. CC /\ A =/= 0 ) ) -> ( 2 x. A ) =/= 0 ) |
47 |
45 6 2 46
|
syl12anc |
|- ( ph -> ( 2 x. A ) =/= 0 ) |
48 |
47
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( 2 x. A ) =/= 0 ) |
49 |
39 43 48
|
redivcld |
|- ( ( ph /\ 0 <_ D ) -> ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) e. RR ) |
50 |
3
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> B e. RR ) |
51 |
50
|
renegcld |
|- ( ( ph /\ 0 <_ D ) -> -u B e. RR ) |
52 |
51 38
|
resubcld |
|- ( ( ph /\ 0 <_ D ) -> ( -u B - ( sqrt ` D ) ) e. RR ) |
53 |
40
|
a1i |
|- ( ( ph /\ 0 <_ D ) -> 2 e. RR ) |
54 |
1
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> A e. RR ) |
55 |
53 54
|
remulcld |
|- ( ( ph /\ 0 <_ D ) -> ( 2 x. A ) e. RR ) |
56 |
52 55 48
|
redivcld |
|- ( ( ph /\ 0 <_ D ) -> ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) e. RR ) |
57 |
|
fveqeq2 |
|- ( q = x -> ( ( # ` q ) = 2 <-> ( # ` x ) = 2 ) ) |
58 |
57
|
cbvrabv |
|- { q e. ~P RR | ( # ` q ) = 2 } = { x e. ~P RR | ( # ` x ) = 2 } |
59 |
49 56 58
|
paireqne |
|- ( ( ph /\ 0 <_ D ) -> ( E! p e. { q e. ~P RR | ( # ` q ) = 2 } A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) <-> ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) =/= ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) |
60 |
9
|
negcld |
|- ( ph -> -u B e. CC ) |
61 |
35
|
recnd |
|- ( ph -> D e. CC ) |
62 |
61
|
sqrtcld |
|- ( ph -> ( sqrt ` D ) e. CC ) |
63 |
60 62
|
addcld |
|- ( ph -> ( -u B + ( sqrt ` D ) ) e. CC ) |
64 |
60 62
|
subcld |
|- ( ph -> ( -u B - ( sqrt ` D ) ) e. CC ) |
65 |
|
2cnd |
|- ( ph -> 2 e. CC ) |
66 |
65 6
|
mulcld |
|- ( ph -> ( 2 x. A ) e. CC ) |
67 |
|
div11 |
|- ( ( ( -u B + ( sqrt ` D ) ) e. CC /\ ( -u B - ( sqrt ` D ) ) e. CC /\ ( ( 2 x. A ) e. CC /\ ( 2 x. A ) =/= 0 ) ) -> ( ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) <-> ( -u B + ( sqrt ` D ) ) = ( -u B - ( sqrt ` D ) ) ) ) |
68 |
63 64 66 47 67
|
syl112anc |
|- ( ph -> ( ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) <-> ( -u B + ( sqrt ` D ) ) = ( -u B - ( sqrt ` D ) ) ) ) |
69 |
60 62
|
negsubd |
|- ( ph -> ( -u B + -u ( sqrt ` D ) ) = ( -u B - ( sqrt ` D ) ) ) |
70 |
69
|
eqcomd |
|- ( ph -> ( -u B - ( sqrt ` D ) ) = ( -u B + -u ( sqrt ` D ) ) ) |
71 |
70
|
eqeq2d |
|- ( ph -> ( ( -u B + ( sqrt ` D ) ) = ( -u B - ( sqrt ` D ) ) <-> ( -u B + ( sqrt ` D ) ) = ( -u B + -u ( sqrt ` D ) ) ) ) |
72 |
62
|
negcld |
|- ( ph -> -u ( sqrt ` D ) e. CC ) |
73 |
60 62 72
|
addcand |
|- ( ph -> ( ( -u B + ( sqrt ` D ) ) = ( -u B + -u ( sqrt ` D ) ) <-> ( sqrt ` D ) = -u ( sqrt ` D ) ) ) |
74 |
68 71 73
|
3bitrd |
|- ( ph -> ( ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) <-> ( sqrt ` D ) = -u ( sqrt ` D ) ) ) |
75 |
74
|
necon3bid |
|- ( ph -> ( ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) =/= ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) <-> ( sqrt ` D ) =/= -u ( sqrt ` D ) ) ) |
76 |
75
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) =/= ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) <-> ( sqrt ` D ) =/= -u ( sqrt ` D ) ) ) |
77 |
|
cnsqrt00 |
|- ( D e. CC -> ( ( sqrt ` D ) = 0 <-> D = 0 ) ) |
78 |
61 77
|
syl |
|- ( ph -> ( ( sqrt ` D ) = 0 <-> D = 0 ) ) |
79 |
78
|
necon3bid |
|- ( ph -> ( ( sqrt ` D ) =/= 0 <-> D =/= 0 ) ) |
80 |
79
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( ( sqrt ` D ) =/= 0 <-> D =/= 0 ) ) |
81 |
62
|
eqnegd |
|- ( ph -> ( ( sqrt ` D ) = -u ( sqrt ` D ) <-> ( sqrt ` D ) = 0 ) ) |
82 |
81
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( ( sqrt ` D ) = -u ( sqrt ` D ) <-> ( sqrt ` D ) = 0 ) ) |
83 |
82
|
necon3bid |
|- ( ( ph /\ 0 <_ D ) -> ( ( sqrt ` D ) =/= -u ( sqrt ` D ) <-> ( sqrt ` D ) =/= 0 ) ) |
84 |
|
0red |
|- ( ( ph /\ 0 <_ D ) -> 0 e. RR ) |
85 |
84 36 37
|
leltned |
|- ( ( ph /\ 0 <_ D ) -> ( 0 < D <-> D =/= 0 ) ) |
86 |
80 83 85
|
3bitr4d |
|- ( ( ph /\ 0 <_ D ) -> ( ( sqrt ` D ) =/= -u ( sqrt ` D ) <-> 0 < D ) ) |
87 |
76 86
|
bitrd |
|- ( ( ph /\ 0 <_ D ) -> ( ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) =/= ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) <-> 0 < D ) ) |
88 |
26 59 87
|
3bitrd |
|- ( ( ph /\ 0 <_ D ) -> ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) <-> 0 < D ) ) |
89 |
22 88
|
bitrd |
|- ( ( ph /\ 0 <_ D ) -> ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) <-> 0 < D ) ) |
90 |
89
|
expcom |
|- ( 0 <_ D -> ( ph -> ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) <-> 0 < D ) ) ) |
91 |
|
hash2prb |
|- ( p e. ~P RR -> ( ( # ` p ) = 2 <-> E. a e. p E. b e. p ( a =/= b /\ p = { a , b } ) ) ) |
92 |
91
|
adantl |
|- ( ( ph /\ p e. ~P RR ) -> ( ( # ` p ) = 2 <-> E. a e. p E. b e. p ( a =/= b /\ p = { a , b } ) ) ) |
93 |
|
raleq |
|- ( p = { a , b } -> ( A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> A. x e. { a , b } ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) |
94 |
|
vex |
|- a e. _V |
95 |
|
vex |
|- b e. _V |
96 |
|
oveq1 |
|- ( x = a -> ( x ^ 2 ) = ( a ^ 2 ) ) |
97 |
96
|
oveq2d |
|- ( x = a -> ( A x. ( x ^ 2 ) ) = ( A x. ( a ^ 2 ) ) ) |
98 |
|
oveq2 |
|- ( x = a -> ( B x. x ) = ( B x. a ) ) |
99 |
98
|
oveq1d |
|- ( x = a -> ( ( B x. x ) + C ) = ( ( B x. a ) + C ) ) |
100 |
97 99
|
oveq12d |
|- ( x = a -> ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = ( ( A x. ( a ^ 2 ) ) + ( ( B x. a ) + C ) ) ) |
101 |
100
|
eqeq1d |
|- ( x = a -> ( ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> ( ( A x. ( a ^ 2 ) ) + ( ( B x. a ) + C ) ) = 0 ) ) |
102 |
|
oveq1 |
|- ( x = b -> ( x ^ 2 ) = ( b ^ 2 ) ) |
103 |
102
|
oveq2d |
|- ( x = b -> ( A x. ( x ^ 2 ) ) = ( A x. ( b ^ 2 ) ) ) |
104 |
|
oveq2 |
|- ( x = b -> ( B x. x ) = ( B x. b ) ) |
105 |
104
|
oveq1d |
|- ( x = b -> ( ( B x. x ) + C ) = ( ( B x. b ) + C ) ) |
106 |
103 105
|
oveq12d |
|- ( x = b -> ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) ) |
107 |
106
|
eqeq1d |
|- ( x = b -> ( ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 ) ) |
108 |
94 95 101 107
|
ralpr |
|- ( A. x e. { a , b } ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> ( ( ( A x. ( a ^ 2 ) ) + ( ( B x. a ) + C ) ) = 0 /\ ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 ) ) |
109 |
93 108
|
bitrdi |
|- ( p = { a , b } -> ( A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> ( ( ( A x. ( a ^ 2 ) ) + ( ( B x. a ) + C ) ) = 0 /\ ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 ) ) ) |
110 |
109
|
adantl |
|- ( ( a =/= b /\ p = { a , b } ) -> ( A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> ( ( ( A x. ( a ^ 2 ) ) + ( ( B x. a ) + C ) ) = 0 /\ ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 ) ) ) |
111 |
110
|
adantl |
|- ( ( ( ( ph /\ p e. ~P RR ) /\ ( a e. p /\ b e. p ) ) /\ ( a =/= b /\ p = { a , b } ) ) -> ( A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> ( ( ( A x. ( a ^ 2 ) ) + ( ( B x. a ) + C ) ) = 0 /\ ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 ) ) ) |
112 |
|
elelpwi |
|- ( ( b e. p /\ p e. ~P RR ) -> b e. RR ) |
113 |
112
|
ex |
|- ( b e. p -> ( p e. ~P RR -> b e. RR ) ) |
114 |
113
|
adantl |
|- ( ( a e. p /\ b e. p ) -> ( p e. ~P RR -> b e. RR ) ) |
115 |
114
|
com12 |
|- ( p e. ~P RR -> ( ( a e. p /\ b e. p ) -> b e. RR ) ) |
116 |
115
|
adantl |
|- ( ( ph /\ p e. ~P RR ) -> ( ( a e. p /\ b e. p ) -> b e. RR ) ) |
117 |
116
|
imp |
|- ( ( ( ph /\ p e. ~P RR ) /\ ( a e. p /\ b e. p ) ) -> b e. RR ) |
118 |
|
oveq1 |
|- ( y = b -> ( y ^ 2 ) = ( b ^ 2 ) ) |
119 |
118
|
oveq2d |
|- ( y = b -> ( A x. ( y ^ 2 ) ) = ( A x. ( b ^ 2 ) ) ) |
120 |
|
oveq2 |
|- ( y = b -> ( B x. y ) = ( B x. b ) ) |
121 |
120
|
oveq1d |
|- ( y = b -> ( ( B x. y ) + C ) = ( ( B x. b ) + C ) ) |
122 |
119 121
|
oveq12d |
|- ( y = b -> ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) ) |
123 |
122
|
eqeq1d |
|- ( y = b -> ( ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 <-> ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 ) ) |
124 |
123
|
adantl |
|- ( ( ( ( ph /\ p e. ~P RR ) /\ ( a e. p /\ b e. p ) ) /\ y = b ) -> ( ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 <-> ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 ) ) |
125 |
117 124
|
rspcedv |
|- ( ( ( ph /\ p e. ~P RR ) /\ ( a e. p /\ b e. p ) ) -> ( ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 -> E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 ) ) |
126 |
125
|
adantr |
|- ( ( ( ( ph /\ p e. ~P RR ) /\ ( a e. p /\ b e. p ) ) /\ ( a =/= b /\ p = { a , b } ) ) -> ( ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 -> E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 ) ) |
127 |
126
|
adantld |
|- ( ( ( ( ph /\ p e. ~P RR ) /\ ( a e. p /\ b e. p ) ) /\ ( a =/= b /\ p = { a , b } ) ) -> ( ( ( ( A x. ( a ^ 2 ) ) + ( ( B x. a ) + C ) ) = 0 /\ ( ( A x. ( b ^ 2 ) ) + ( ( B x. b ) + C ) ) = 0 ) -> E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 ) ) |
128 |
111 127
|
sylbid |
|- ( ( ( ( ph /\ p e. ~P RR ) /\ ( a e. p /\ b e. p ) ) /\ ( a =/= b /\ p = { a , b } ) ) -> ( A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 -> E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 ) ) |
129 |
128
|
ex |
|- ( ( ( ph /\ p e. ~P RR ) /\ ( a e. p /\ b e. p ) ) -> ( ( a =/= b /\ p = { a , b } ) -> ( A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 -> E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 ) ) ) |
130 |
129
|
rexlimdvva |
|- ( ( ph /\ p e. ~P RR ) -> ( E. a e. p E. b e. p ( a =/= b /\ p = { a , b } ) -> ( A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 -> E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 ) ) ) |
131 |
92 130
|
sylbid |
|- ( ( ph /\ p e. ~P RR ) -> ( ( # ` p ) = 2 -> ( A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 -> E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 ) ) ) |
132 |
131
|
impd |
|- ( ( ph /\ p e. ~P RR ) -> ( ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) -> E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 ) ) |
133 |
132
|
rexlimdva |
|- ( ph -> ( E. p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) -> E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 ) ) |
134 |
1 2 3 4 5
|
requad01 |
|- ( ph -> ( E. y e. RR ( ( A x. ( y ^ 2 ) ) + ( ( B x. y ) + C ) ) = 0 <-> 0 <_ D ) ) |
135 |
133 134
|
sylibd |
|- ( ph -> ( E. p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) -> 0 <_ D ) ) |
136 |
135
|
con3d |
|- ( ph -> ( -. 0 <_ D -> -. E. p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) ) |
137 |
136
|
impcom |
|- ( ( -. 0 <_ D /\ ph ) -> -. E. p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) |
138 |
|
reurex |
|- ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) -> E. p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) |
139 |
137 138
|
nsyl |
|- ( ( -. 0 <_ D /\ ph ) -> -. E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) |
140 |
139
|
pm2.21d |
|- ( ( -. 0 <_ D /\ ph ) -> ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) -> 0 < D ) ) |
141 |
|
0red |
|- ( ph -> 0 e. RR ) |
142 |
|
ltle |
|- ( ( 0 e. RR /\ D e. RR ) -> ( 0 < D -> 0 <_ D ) ) |
143 |
141 35 142
|
syl2anc |
|- ( ph -> ( 0 < D -> 0 <_ D ) ) |
144 |
|
pm2.24 |
|- ( 0 <_ D -> ( -. 0 <_ D -> E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) ) |
145 |
143 144
|
syl6 |
|- ( ph -> ( 0 < D -> ( -. 0 <_ D -> E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) ) ) |
146 |
145
|
com23 |
|- ( ph -> ( -. 0 <_ D -> ( 0 < D -> E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) ) ) |
147 |
146
|
impcom |
|- ( ( -. 0 <_ D /\ ph ) -> ( 0 < D -> E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) ) |
148 |
140 147
|
impbid |
|- ( ( -. 0 <_ D /\ ph ) -> ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) <-> 0 < D ) ) |
149 |
148
|
ex |
|- ( -. 0 <_ D -> ( ph -> ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) <-> 0 < D ) ) ) |
150 |
90 149
|
pm2.61i |
|- ( ph -> ( E! p e. ~P RR ( ( # ` p ) = 2 /\ A. x e. p ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) <-> 0 < D ) ) |