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
|
ad2antrr |
|- ( ( ( ph /\ 0 <_ D ) /\ x e. RR ) -> A e. CC ) |
8 |
2
|
ad2antrr |
|- ( ( ( ph /\ 0 <_ D ) /\ x e. RR ) -> A =/= 0 ) |
9 |
3
|
recnd |
|- ( ph -> B e. CC ) |
10 |
9
|
ad2antrr |
|- ( ( ( ph /\ 0 <_ D ) /\ x e. RR ) -> B e. CC ) |
11 |
4
|
recnd |
|- ( ph -> C e. CC ) |
12 |
11
|
ad2antrr |
|- ( ( ( ph /\ 0 <_ D ) /\ x e. RR ) -> C e. CC ) |
13 |
|
recn |
|- ( x e. RR -> x e. CC ) |
14 |
13
|
adantl |
|- ( ( ( ph /\ 0 <_ D ) /\ x e. RR ) -> x e. CC ) |
15 |
5
|
ad2antrr |
|- ( ( ( ph /\ 0 <_ D ) /\ x e. RR ) -> D = ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) ) |
16 |
7 8 10 12 14 15
|
quad |
|- ( ( ( ph /\ 0 <_ D ) /\ x e. RR ) -> ( ( ( 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 ) ) ) ) ) |
17 |
16
|
reubidva |
|- ( ( ph /\ 0 <_ D ) -> ( E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> E! x e. RR ( x = ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) \/ x = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) ) ) ) |
18 |
3
|
renegcld |
|- ( ph -> -u B e. RR ) |
19 |
18
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> -u B e. RR ) |
20 |
3
|
resqcld |
|- ( ph -> ( B ^ 2 ) e. RR ) |
21 |
|
4re |
|- 4 e. RR |
22 |
21
|
a1i |
|- ( ph -> 4 e. RR ) |
23 |
1 4
|
remulcld |
|- ( ph -> ( A x. C ) e. RR ) |
24 |
22 23
|
remulcld |
|- ( ph -> ( 4 x. ( A x. C ) ) e. RR ) |
25 |
20 24
|
resubcld |
|- ( ph -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) e. RR ) |
26 |
5 25
|
eqeltrd |
|- ( ph -> D e. RR ) |
27 |
|
resqrtcl |
|- ( ( D e. RR /\ 0 <_ D ) -> ( sqrt ` D ) e. RR ) |
28 |
26 27
|
sylan |
|- ( ( ph /\ 0 <_ D ) -> ( sqrt ` D ) e. RR ) |
29 |
19 28
|
readdcld |
|- ( ( ph /\ 0 <_ D ) -> ( -u B + ( sqrt ` D ) ) e. RR ) |
30 |
|
2re |
|- 2 e. RR |
31 |
30
|
a1i |
|- ( ph -> 2 e. RR ) |
32 |
31 1
|
remulcld |
|- ( ph -> ( 2 x. A ) e. RR ) |
33 |
32
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( 2 x. A ) e. RR ) |
34 |
|
2cnd |
|- ( ph -> 2 e. CC ) |
35 |
|
2ne0 |
|- 2 =/= 0 |
36 |
35
|
a1i |
|- ( ph -> 2 =/= 0 ) |
37 |
34 6 36 2
|
mulne0d |
|- ( ph -> ( 2 x. A ) =/= 0 ) |
38 |
37
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( 2 x. A ) =/= 0 ) |
39 |
29 33 38
|
redivcld |
|- ( ( ph /\ 0 <_ D ) -> ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) e. RR ) |
40 |
19 28
|
resubcld |
|- ( ( ph /\ 0 <_ D ) -> ( -u B - ( sqrt ` D ) ) e. RR ) |
41 |
40 33 38
|
redivcld |
|- ( ( ph /\ 0 <_ D ) -> ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) e. RR ) |
42 |
|
euoreqb |
|- ( ( ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) e. RR /\ ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) e. RR ) -> ( E! x e. RR ( 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 ) ) ) ) |
43 |
39 41 42
|
syl2anc |
|- ( ( ph /\ 0 <_ D ) -> ( E! x e. RR ( 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 ) ) ) ) |
44 |
9
|
negcld |
|- ( ph -> -u B e. CC ) |
45 |
26
|
recnd |
|- ( ph -> D e. CC ) |
46 |
45
|
sqrtcld |
|- ( ph -> ( sqrt ` D ) e. CC ) |
47 |
32
|
recnd |
|- ( ph -> ( 2 x. A ) e. CC ) |
48 |
44 46 47 37
|
divdird |
|- ( ph -> ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B / ( 2 x. A ) ) + ( ( sqrt ` D ) / ( 2 x. A ) ) ) ) |
49 |
48
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B / ( 2 x. A ) ) + ( ( sqrt ` D ) / ( 2 x. A ) ) ) ) |
50 |
44 46 47 37
|
divsubdird |
|- ( ph -> ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B / ( 2 x. A ) ) - ( ( sqrt ` D ) / ( 2 x. A ) ) ) ) |
51 |
50
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B / ( 2 x. A ) ) - ( ( sqrt ` D ) / ( 2 x. A ) ) ) ) |
52 |
44 47 37
|
divcld |
|- ( ph -> ( -u B / ( 2 x. A ) ) e. CC ) |
53 |
52
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( -u B / ( 2 x. A ) ) e. CC ) |
54 |
46 47 37
|
divcld |
|- ( ph -> ( ( sqrt ` D ) / ( 2 x. A ) ) e. CC ) |
55 |
54
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( ( sqrt ` D ) / ( 2 x. A ) ) e. CC ) |
56 |
53 55
|
negsubd |
|- ( ( ph /\ 0 <_ D ) -> ( ( -u B / ( 2 x. A ) ) + -u ( ( sqrt ` D ) / ( 2 x. A ) ) ) = ( ( -u B / ( 2 x. A ) ) - ( ( sqrt ` D ) / ( 2 x. A ) ) ) ) |
57 |
46
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( sqrt ` D ) e. CC ) |
58 |
47
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( 2 x. A ) e. CC ) |
59 |
57 58 38
|
divnegd |
|- ( ( ph /\ 0 <_ D ) -> -u ( ( sqrt ` D ) / ( 2 x. A ) ) = ( -u ( sqrt ` D ) / ( 2 x. A ) ) ) |
60 |
59
|
oveq2d |
|- ( ( ph /\ 0 <_ D ) -> ( ( -u B / ( 2 x. A ) ) + -u ( ( sqrt ` D ) / ( 2 x. A ) ) ) = ( ( -u B / ( 2 x. A ) ) + ( -u ( sqrt ` D ) / ( 2 x. A ) ) ) ) |
61 |
51 56 60
|
3eqtr2d |
|- ( ( ph /\ 0 <_ D ) -> ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B / ( 2 x. A ) ) + ( -u ( sqrt ` D ) / ( 2 x. A ) ) ) ) |
62 |
49 61
|
eqeq12d |
|- ( ( ph /\ 0 <_ D ) -> ( ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) <-> ( ( -u B / ( 2 x. A ) ) + ( ( sqrt ` D ) / ( 2 x. A ) ) ) = ( ( -u B / ( 2 x. A ) ) + ( -u ( sqrt ` D ) / ( 2 x. A ) ) ) ) ) |
63 |
46
|
negcld |
|- ( ph -> -u ( sqrt ` D ) e. CC ) |
64 |
63 47 37
|
divcld |
|- ( ph -> ( -u ( sqrt ` D ) / ( 2 x. A ) ) e. CC ) |
65 |
64
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( -u ( sqrt ` D ) / ( 2 x. A ) ) e. CC ) |
66 |
53 55 65
|
addcand |
|- ( ( ph /\ 0 <_ D ) -> ( ( ( -u B / ( 2 x. A ) ) + ( ( sqrt ` D ) / ( 2 x. A ) ) ) = ( ( -u B / ( 2 x. A ) ) + ( -u ( sqrt ` D ) / ( 2 x. A ) ) ) <-> ( ( sqrt ` D ) / ( 2 x. A ) ) = ( -u ( sqrt ` D ) / ( 2 x. A ) ) ) ) |
67 |
|
div11 |
|- ( ( ( sqrt ` D ) e. CC /\ -u ( sqrt ` D ) e. CC /\ ( ( 2 x. A ) e. CC /\ ( 2 x. A ) =/= 0 ) ) -> ( ( ( sqrt ` D ) / ( 2 x. A ) ) = ( -u ( sqrt ` D ) / ( 2 x. A ) ) <-> ( sqrt ` D ) = -u ( sqrt ` D ) ) ) |
68 |
46 63 47 37 67
|
syl112anc |
|- ( ph -> ( ( ( sqrt ` D ) / ( 2 x. A ) ) = ( -u ( sqrt ` D ) / ( 2 x. A ) ) <-> ( sqrt ` D ) = -u ( sqrt ` D ) ) ) |
69 |
68
|
adantr |
|- ( ( ph /\ 0 <_ D ) -> ( ( ( sqrt ` D ) / ( 2 x. A ) ) = ( -u ( sqrt ` D ) / ( 2 x. A ) ) <-> ( sqrt ` D ) = -u ( sqrt ` D ) ) ) |
70 |
57
|
eqnegd |
|- ( ( ph /\ 0 <_ D ) -> ( ( sqrt ` D ) = -u ( sqrt ` D ) <-> ( sqrt ` D ) = 0 ) ) |
71 |
|
sqrt00 |
|- ( ( D e. RR /\ 0 <_ D ) -> ( ( sqrt ` D ) = 0 <-> D = 0 ) ) |
72 |
26 71
|
sylan |
|- ( ( ph /\ 0 <_ D ) -> ( ( sqrt ` D ) = 0 <-> D = 0 ) ) |
73 |
70 72
|
bitrd |
|- ( ( ph /\ 0 <_ D ) -> ( ( sqrt ` D ) = -u ( sqrt ` D ) <-> D = 0 ) ) |
74 |
66 69 73
|
3bitrd |
|- ( ( ph /\ 0 <_ D ) -> ( ( ( -u B / ( 2 x. A ) ) + ( ( sqrt ` D ) / ( 2 x. A ) ) ) = ( ( -u B / ( 2 x. A ) ) + ( -u ( sqrt ` D ) / ( 2 x. A ) ) ) <-> D = 0 ) ) |
75 |
62 74
|
bitrd |
|- ( ( ph /\ 0 <_ D ) -> ( ( ( -u B + ( sqrt ` D ) ) / ( 2 x. A ) ) = ( ( -u B - ( sqrt ` D ) ) / ( 2 x. A ) ) <-> D = 0 ) ) |
76 |
17 43 75
|
3bitrd |
|- ( ( ph /\ 0 <_ D ) -> ( E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> D = 0 ) ) |
77 |
76
|
expcom |
|- ( 0 <_ D -> ( ph -> ( E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> D = 0 ) ) ) |
78 |
1 2 3 4 5
|
requad01 |
|- ( ph -> ( E. x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> 0 <_ D ) ) |
79 |
78
|
notbid |
|- ( ph -> ( -. E. x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> -. 0 <_ D ) ) |
80 |
79
|
biimparc |
|- ( ( -. 0 <_ D /\ ph ) -> -. E. x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) |
81 |
|
reurex |
|- ( E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 -> E. x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) |
82 |
80 81
|
nsyl |
|- ( ( -. 0 <_ D /\ ph ) -> -. E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) |
83 |
82
|
pm2.21d |
|- ( ( -. 0 <_ D /\ ph ) -> ( E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 -> D = 0 ) ) |
84 |
|
0red |
|- ( ph -> 0 e. RR ) |
85 |
26 84
|
ltnled |
|- ( ph -> ( D < 0 <-> -. 0 <_ D ) ) |
86 |
85
|
biimparc |
|- ( ( -. 0 <_ D /\ ph ) -> D < 0 ) |
87 |
86
|
lt0ne0d |
|- ( ( -. 0 <_ D /\ ph ) -> D =/= 0 ) |
88 |
|
eqneqall |
|- ( D = 0 -> ( D =/= 0 -> E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) |
89 |
87 88
|
syl5com |
|- ( ( -. 0 <_ D /\ ph ) -> ( D = 0 -> E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 ) ) |
90 |
83 89
|
impbid |
|- ( ( -. 0 <_ D /\ ph ) -> ( E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> D = 0 ) ) |
91 |
90
|
ex |
|- ( -. 0 <_ D -> ( ph -> ( E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> D = 0 ) ) ) |
92 |
77 91
|
pm2.61i |
|- ( ph -> ( E! x e. RR ( ( A x. ( x ^ 2 ) ) + ( ( B x. x ) + C ) ) = 0 <-> D = 0 ) ) |