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Metamath Proof Explorer

Theorem discr

Description: If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is nonpositive. (Contributed by NM, 10-Aug-1999) (Revised by Mario Carneiro, 4-Jun-2014)

Ref Expression
Hypotheses discr.1 
|- ( ph -> A e. RR )
discr.2 
|- ( ph -> B e. RR )
discr.3 
|- ( ph -> C e. RR )
discr.4 
|- ( ( ph /\ x e. RR ) -> 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) )
Assertion discr 
|- ( ph -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0 )

Proof

Step Hyp Ref Expression
1 discr.1 
 |-  ( ph -> A e. RR )
2 discr.2 
 |-  ( ph -> B e. RR )
3 discr.3 
 |-  ( ph -> C e. RR )
4 discr.4 
 |-  ( ( ph /\ x e. RR ) -> 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) )
5 2 adantr 
 |-  ( ( ph /\ 0 < A ) -> B e. RR )
6 resqcl 
 |-  ( B e. RR -> ( B ^ 2 ) e. RR )
7 5 6 syl 
 |-  ( ( ph /\ 0 < A ) -> ( B ^ 2 ) e. RR )
8 7 recnd 
 |-  ( ( ph /\ 0 < A ) -> ( B ^ 2 ) e. CC )
9 4re 
 |-  4 e. RR
10 1 adantr 
 |-  ( ( ph /\ 0 < A ) -> A e. RR )
11 3 adantr 
 |-  ( ( ph /\ 0 < A ) -> C e. RR )
12 10 11 remulcld 
 |-  ( ( ph /\ 0 < A ) -> ( A x. C ) e. RR )
13 remulcl 
 |-  ( ( 4 e. RR /\ ( A x. C ) e. RR ) -> ( 4 x. ( A x. C ) ) e. RR )
14 9 12 13 sylancr 
 |-  ( ( ph /\ 0 < A ) -> ( 4 x. ( A x. C ) ) e. RR )
15 14 recnd 
 |-  ( ( ph /\ 0 < A ) -> ( 4 x. ( A x. C ) ) e. CC )
16 4pos 
 |-  0 < 4
17 9 16 elrpii 
 |-  4 e. RR+
18 simpr 
 |-  ( ( ph /\ 0 < A ) -> 0 < A )
19 10 18 elrpd 
 |-  ( ( ph /\ 0 < A ) -> A e. RR+ )
20 rpmulcl 
 |-  ( ( 4 e. RR+ /\ A e. RR+ ) -> ( 4 x. A ) e. RR+ )
21 17 19 20 sylancr 
 |-  ( ( ph /\ 0 < A ) -> ( 4 x. A ) e. RR+ )
22 21 rpcnd 
 |-  ( ( ph /\ 0 < A ) -> ( 4 x. A ) e. CC )
23 21 rpne0d 
 |-  ( ( ph /\ 0 < A ) -> ( 4 x. A ) =/= 0 )
24 8 15 22 23 divsubdird 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) / ( 4 x. A ) ) = ( ( ( B ^ 2 ) / ( 4 x. A ) ) - ( ( 4 x. ( A x. C ) ) / ( 4 x. A ) ) ) )
25 12 recnd 
 |-  ( ( ph /\ 0 < A ) -> ( A x. C ) e. CC )
26 10 recnd 
 |-  ( ( ph /\ 0 < A ) -> A e. CC )
27 4cn 
 |-  4 e. CC
28 27 a1i 
 |-  ( ( ph /\ 0 < A ) -> 4 e. CC )
29 19 rpne0d 
 |-  ( ( ph /\ 0 < A ) -> A =/= 0 )
30 4ne0 
 |-  4 =/= 0
31 30 a1i 
 |-  ( ( ph /\ 0 < A ) -> 4 =/= 0 )
32 25 26 28 29 31 divcan5d 
 |-  ( ( ph /\ 0 < A ) -> ( ( 4 x. ( A x. C ) ) / ( 4 x. A ) ) = ( ( A x. C ) / A ) )
33 11 recnd 
 |-  ( ( ph /\ 0 < A ) -> C e. CC )
34 33 26 29 divcan3d 
 |-  ( ( ph /\ 0 < A ) -> ( ( A x. C ) / A ) = C )
35 32 34 eqtrd 
 |-  ( ( ph /\ 0 < A ) -> ( ( 4 x. ( A x. C ) ) / ( 4 x. A ) ) = C )
36 35 oveq2d 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) / ( 4 x. A ) ) - ( ( 4 x. ( A x. C ) ) / ( 4 x. A ) ) ) = ( ( ( B ^ 2 ) / ( 4 x. A ) ) - C ) )
37 24 36 eqtrd 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) / ( 4 x. A ) ) = ( ( ( B ^ 2 ) / ( 4 x. A ) ) - C ) )
38 7 21 rerpdivcld 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) / ( 4 x. A ) ) e. RR )
39 38 recnd 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) / ( 4 x. A ) ) e. CC )
40 39 2timesd 
 |-  ( ( ph /\ 0 < A ) -> ( 2 x. ( ( B ^ 2 ) / ( 4 x. A ) ) ) = ( ( ( B ^ 2 ) / ( 4 x. A ) ) + ( ( B ^ 2 ) / ( 4 x. A ) ) ) )
41 2t2e4 
 |-  ( 2 x. 2 ) = 4
42 41 oveq1i 
 |-  ( ( 2 x. 2 ) x. A ) = ( 4 x. A )
43 2cnd 
 |-  ( ( ph /\ 0 < A ) -> 2 e. CC )
44 43 43 26 mulassd 
 |-  ( ( ph /\ 0 < A ) -> ( ( 2 x. 2 ) x. A ) = ( 2 x. ( 2 x. A ) ) )
45 42 44 eqtr3id 
 |-  ( ( ph /\ 0 < A ) -> ( 4 x. A ) = ( 2 x. ( 2 x. A ) ) )
46 45 oveq2d 
 |-  ( ( ph /\ 0 < A ) -> ( ( 2 x. ( B ^ 2 ) ) / ( 4 x. A ) ) = ( ( 2 x. ( B ^ 2 ) ) / ( 2 x. ( 2 x. A ) ) ) )
47 43 8 22 23 divassd 
 |-  ( ( ph /\ 0 < A ) -> ( ( 2 x. ( B ^ 2 ) ) / ( 4 x. A ) ) = ( 2 x. ( ( B ^ 2 ) / ( 4 x. A ) ) ) )
48 2rp 
 |-  2 e. RR+
49 rpmulcl 
 |-  ( ( 2 e. RR+ /\ A e. RR+ ) -> ( 2 x. A ) e. RR+ )
50 48 19 49 sylancr 
 |-  ( ( ph /\ 0 < A ) -> ( 2 x. A ) e. RR+ )
51 50 rpcnd 
 |-  ( ( ph /\ 0 < A ) -> ( 2 x. A ) e. CC )
52 50 rpne0d 
 |-  ( ( ph /\ 0 < A ) -> ( 2 x. A ) =/= 0 )
53 2ne0 
 |-  2 =/= 0
54 53 a1i 
 |-  ( ( ph /\ 0 < A ) -> 2 =/= 0 )
55 8 51 43 52 54 divcan5d 
 |-  ( ( ph /\ 0 < A ) -> ( ( 2 x. ( B ^ 2 ) ) / ( 2 x. ( 2 x. A ) ) ) = ( ( B ^ 2 ) / ( 2 x. A ) ) )
56 46 47 55 3eqtr3d 
 |-  ( ( ph /\ 0 < A ) -> ( 2 x. ( ( B ^ 2 ) / ( 4 x. A ) ) ) = ( ( B ^ 2 ) / ( 2 x. A ) ) )
57 40 56 eqtr3d 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) / ( 4 x. A ) ) + ( ( B ^ 2 ) / ( 4 x. A ) ) ) = ( ( B ^ 2 ) / ( 2 x. A ) ) )
58 oveq1 
 |-  ( x = -u ( B / ( 2 x. A ) ) -> ( x ^ 2 ) = ( -u ( B / ( 2 x. A ) ) ^ 2 ) )
59 58 oveq2d 
 |-  ( x = -u ( B / ( 2 x. A ) ) -> ( A x. ( x ^ 2 ) ) = ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) )
60 oveq2 
 |-  ( x = -u ( B / ( 2 x. A ) ) -> ( B x. x ) = ( B x. -u ( B / ( 2 x. A ) ) ) )
61 59 60 oveq12d 
 |-  ( x = -u ( B / ( 2 x. A ) ) -> ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) = ( ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) + ( B x. -u ( B / ( 2 x. A ) ) ) ) )
62 61 oveq1d 
 |-  ( x = -u ( B / ( 2 x. A ) ) -> ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) = ( ( ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) + ( B x. -u ( B / ( 2 x. A ) ) ) ) + C ) )
63 62 breq2d 
 |-  ( x = -u ( B / ( 2 x. A ) ) -> ( 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) <-> 0 <_ ( ( ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) + ( B x. -u ( B / ( 2 x. A ) ) ) ) + C ) ) )
64 4 ralrimiva 
 |-  ( ph -> A. x e. RR 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) )
65 64 adantr 
 |-  ( ( ph /\ 0 < A ) -> A. x e. RR 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) )
66 5 50 rerpdivcld 
 |-  ( ( ph /\ 0 < A ) -> ( B / ( 2 x. A ) ) e. RR )
67 66 renegcld 
 |-  ( ( ph /\ 0 < A ) -> -u ( B / ( 2 x. A ) ) e. RR )
68 63 65 67 rspcdva 
 |-  ( ( ph /\ 0 < A ) -> 0 <_ ( ( ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) + ( B x. -u ( B / ( 2 x. A ) ) ) ) + C ) )
69 66 recnd 
 |-  ( ( ph /\ 0 < A ) -> ( B / ( 2 x. A ) ) e. CC )
70 sqneg 
 |-  ( ( B / ( 2 x. A ) ) e. CC -> ( -u ( B / ( 2 x. A ) ) ^ 2 ) = ( ( B / ( 2 x. A ) ) ^ 2 ) )
71 69 70 syl 
 |-  ( ( ph /\ 0 < A ) -> ( -u ( B / ( 2 x. A ) ) ^ 2 ) = ( ( B / ( 2 x. A ) ) ^ 2 ) )
72 5 recnd 
 |-  ( ( ph /\ 0 < A ) -> B e. CC )
73 sqdiv 
 |-  ( ( B e. CC /\ ( 2 x. A ) e. CC /\ ( 2 x. A ) =/= 0 ) -> ( ( B / ( 2 x. A ) ) ^ 2 ) = ( ( B ^ 2 ) / ( ( 2 x. A ) ^ 2 ) ) )
74 72 51 52 73 syl3anc 
 |-  ( ( ph /\ 0 < A ) -> ( ( B / ( 2 x. A ) ) ^ 2 ) = ( ( B ^ 2 ) / ( ( 2 x. A ) ^ 2 ) ) )
75 sqval 
 |-  ( ( 2 x. A ) e. CC -> ( ( 2 x. A ) ^ 2 ) = ( ( 2 x. A ) x. ( 2 x. A ) ) )
76 51 75 syl 
 |-  ( ( ph /\ 0 < A ) -> ( ( 2 x. A ) ^ 2 ) = ( ( 2 x. A ) x. ( 2 x. A ) ) )
77 51 43 26 mulassd 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( 2 x. A ) x. 2 ) x. A ) = ( ( 2 x. A ) x. ( 2 x. A ) ) )
78 43 26 43 mul32d 
 |-  ( ( ph /\ 0 < A ) -> ( ( 2 x. A ) x. 2 ) = ( ( 2 x. 2 ) x. A ) )
79 78 42 eqtrdi 
 |-  ( ( ph /\ 0 < A ) -> ( ( 2 x. A ) x. 2 ) = ( 4 x. A ) )
80 79 oveq1d 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( 2 x. A ) x. 2 ) x. A ) = ( ( 4 x. A ) x. A ) )
81 76 77 80 3eqtr2d 
 |-  ( ( ph /\ 0 < A ) -> ( ( 2 x. A ) ^ 2 ) = ( ( 4 x. A ) x. A ) )
82 81 oveq2d 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) / ( ( 2 x. A ) ^ 2 ) ) = ( ( B ^ 2 ) / ( ( 4 x. A ) x. A ) ) )
83 71 74 82 3eqtrd 
 |-  ( ( ph /\ 0 < A ) -> ( -u ( B / ( 2 x. A ) ) ^ 2 ) = ( ( B ^ 2 ) / ( ( 4 x. A ) x. A ) ) )
84 8 22 26 23 29 divdiv1d 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) / ( 4 x. A ) ) / A ) = ( ( B ^ 2 ) / ( ( 4 x. A ) x. A ) ) )
85 83 84 eqtr4d 
 |-  ( ( ph /\ 0 < A ) -> ( -u ( B / ( 2 x. A ) ) ^ 2 ) = ( ( ( B ^ 2 ) / ( 4 x. A ) ) / A ) )
86 85 oveq2d 
 |-  ( ( ph /\ 0 < A ) -> ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) = ( A x. ( ( ( B ^ 2 ) / ( 4 x. A ) ) / A ) ) )
87 39 26 29 divcan2d 
 |-  ( ( ph /\ 0 < A ) -> ( A x. ( ( ( B ^ 2 ) / ( 4 x. A ) ) / A ) ) = ( ( B ^ 2 ) / ( 4 x. A ) ) )
88 86 87 eqtrd 
 |-  ( ( ph /\ 0 < A ) -> ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) = ( ( B ^ 2 ) / ( 4 x. A ) ) )
89 72 69 mulneg2d 
 |-  ( ( ph /\ 0 < A ) -> ( B x. -u ( B / ( 2 x. A ) ) ) = -u ( B x. ( B / ( 2 x. A ) ) ) )
90 sqval 
 |-  ( B e. CC -> ( B ^ 2 ) = ( B x. B ) )
91 72 90 syl 
 |-  ( ( ph /\ 0 < A ) -> ( B ^ 2 ) = ( B x. B ) )
92 91 oveq1d 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) / ( 2 x. A ) ) = ( ( B x. B ) / ( 2 x. A ) ) )
93 72 72 51 52 divassd 
 |-  ( ( ph /\ 0 < A ) -> ( ( B x. B ) / ( 2 x. A ) ) = ( B x. ( B / ( 2 x. A ) ) ) )
94 92 93 eqtrd 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) / ( 2 x. A ) ) = ( B x. ( B / ( 2 x. A ) ) ) )
95 94 negeqd 
 |-  ( ( ph /\ 0 < A ) -> -u ( ( B ^ 2 ) / ( 2 x. A ) ) = -u ( B x. ( B / ( 2 x. A ) ) ) )
96 89 95 eqtr4d 
 |-  ( ( ph /\ 0 < A ) -> ( B x. -u ( B / ( 2 x. A ) ) ) = -u ( ( B ^ 2 ) / ( 2 x. A ) ) )
97 88 96 oveq12d 
 |-  ( ( ph /\ 0 < A ) -> ( ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) + ( B x. -u ( B / ( 2 x. A ) ) ) ) = ( ( ( B ^ 2 ) / ( 4 x. A ) ) + -u ( ( B ^ 2 ) / ( 2 x. A ) ) ) )
98 7 50 rerpdivcld 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) / ( 2 x. A ) ) e. RR )
99 98 recnd 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) / ( 2 x. A ) ) e. CC )
100 39 99 negsubd 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) / ( 4 x. A ) ) + -u ( ( B ^ 2 ) / ( 2 x. A ) ) ) = ( ( ( B ^ 2 ) / ( 4 x. A ) ) - ( ( B ^ 2 ) / ( 2 x. A ) ) ) )
101 97 100 eqtrd 
 |-  ( ( ph /\ 0 < A ) -> ( ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) + ( B x. -u ( B / ( 2 x. A ) ) ) ) = ( ( ( B ^ 2 ) / ( 4 x. A ) ) - ( ( B ^ 2 ) / ( 2 x. A ) ) ) )
102 101 oveq1d 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) + ( B x. -u ( B / ( 2 x. A ) ) ) ) + C ) = ( ( ( ( B ^ 2 ) / ( 4 x. A ) ) - ( ( B ^ 2 ) / ( 2 x. A ) ) ) + C ) )
103 39 33 99 addsubd 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( ( B ^ 2 ) / ( 4 x. A ) ) + C ) - ( ( B ^ 2 ) / ( 2 x. A ) ) ) = ( ( ( ( B ^ 2 ) / ( 4 x. A ) ) - ( ( B ^ 2 ) / ( 2 x. A ) ) ) + C ) )
104 102 103 eqtr4d 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( A x. ( -u ( B / ( 2 x. A ) ) ^ 2 ) ) + ( B x. -u ( B / ( 2 x. A ) ) ) ) + C ) = ( ( ( ( B ^ 2 ) / ( 4 x. A ) ) + C ) - ( ( B ^ 2 ) / ( 2 x. A ) ) ) )
105 68 104 breqtrd 
 |-  ( ( ph /\ 0 < A ) -> 0 <_ ( ( ( ( B ^ 2 ) / ( 4 x. A ) ) + C ) - ( ( B ^ 2 ) / ( 2 x. A ) ) ) )
106 38 11 readdcld 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) / ( 4 x. A ) ) + C ) e. RR )
107 106 98 subge0d 
 |-  ( ( ph /\ 0 < A ) -> ( 0 <_ ( ( ( ( B ^ 2 ) / ( 4 x. A ) ) + C ) - ( ( B ^ 2 ) / ( 2 x. A ) ) ) <-> ( ( B ^ 2 ) / ( 2 x. A ) ) <_ ( ( ( B ^ 2 ) / ( 4 x. A ) ) + C ) ) )
108 105 107 mpbid 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) / ( 2 x. A ) ) <_ ( ( ( B ^ 2 ) / ( 4 x. A ) ) + C ) )
109 57 108 eqbrtrd 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) / ( 4 x. A ) ) + ( ( B ^ 2 ) / ( 4 x. A ) ) ) <_ ( ( ( B ^ 2 ) / ( 4 x. A ) ) + C ) )
110 38 11 38 leadd2d 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) / ( 4 x. A ) ) <_ C <-> ( ( ( B ^ 2 ) / ( 4 x. A ) ) + ( ( B ^ 2 ) / ( 4 x. A ) ) ) <_ ( ( ( B ^ 2 ) / ( 4 x. A ) ) + C ) ) )
111 109 110 mpbird 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) / ( 4 x. A ) ) <_ C )
112 38 11 suble0d 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( ( B ^ 2 ) / ( 4 x. A ) ) - C ) <_ 0 <-> ( ( B ^ 2 ) / ( 4 x. A ) ) <_ C ) )
113 111 112 mpbird 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) / ( 4 x. A ) ) - C ) <_ 0 )
114 37 113 eqbrtrd 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) / ( 4 x. A ) ) <_ 0 )
115 7 14 resubcld 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) e. RR )
116 0red 
 |-  ( ( ph /\ 0 < A ) -> 0 e. RR )
117 115 116 21 ledivmuld 
 |-  ( ( ph /\ 0 < A ) -> ( ( ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) / ( 4 x. A ) ) <_ 0 <-> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ ( ( 4 x. A ) x. 0 ) ) )
118 114 117 mpbid 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ ( ( 4 x. A ) x. 0 ) )
119 22 mul01d 
 |-  ( ( ph /\ 0 < A ) -> ( ( 4 x. A ) x. 0 ) = 0 )
120 118 119 breqtrd 
 |-  ( ( ph /\ 0 < A ) -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0 )
121 3 adantr 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> C e. RR )
122 121 ltp1d 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> C < ( C + 1 ) )
123 peano2re 
 |-  ( C e. RR -> ( C + 1 ) e. RR )
124 121 123 syl 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( C + 1 ) e. RR )
125 121 124 ltnegd 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( C < ( C + 1 ) <-> -u ( C + 1 ) < -u C ) )
126 122 125 mpbid 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> -u ( C + 1 ) < -u C )
127 df-neg 
 |-  -u C = ( 0 - C )
128 126 127 breqtrdi 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> -u ( C + 1 ) < ( 0 - C ) )
129 124 renegcld 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> -u ( C + 1 ) e. RR )
130 0red 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> 0 e. RR )
131 129 121 130 ltaddsubd 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( ( -u ( C + 1 ) + C ) < 0 <-> -u ( C + 1 ) < ( 0 - C ) ) )
132 128 131 mpbird 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( -u ( C + 1 ) + C ) < 0 )
133 132 expr 
 |-  ( ( ph /\ 0 = A ) -> ( B =/= 0 -> ( -u ( C + 1 ) + C ) < 0 ) )
134 oveq1 
 |-  ( x = ( -u ( C + 1 ) / B ) -> ( x ^ 2 ) = ( ( -u ( C + 1 ) / B ) ^ 2 ) )
135 134 oveq2d 
 |-  ( x = ( -u ( C + 1 ) / B ) -> ( A x. ( x ^ 2 ) ) = ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) )
136 oveq2 
 |-  ( x = ( -u ( C + 1 ) / B ) -> ( B x. x ) = ( B x. ( -u ( C + 1 ) / B ) ) )
137 135 136 oveq12d 
 |-  ( x = ( -u ( C + 1 ) / B ) -> ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) = ( ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) + ( B x. ( -u ( C + 1 ) / B ) ) ) )
138 137 oveq1d 
 |-  ( x = ( -u ( C + 1 ) / B ) -> ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) = ( ( ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) + ( B x. ( -u ( C + 1 ) / B ) ) ) + C ) )
139 138 breq2d 
 |-  ( x = ( -u ( C + 1 ) / B ) -> ( 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) <-> 0 <_ ( ( ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) + ( B x. ( -u ( C + 1 ) / B ) ) ) + C ) ) )
140 64 adantr 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> A. x e. RR 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) )
141 2 adantr 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> B e. RR )
142 simprr 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> B =/= 0 )
143 129 141 142 redivcld 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( -u ( C + 1 ) / B ) e. RR )
144 139 140 143 rspcdva 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> 0 <_ ( ( ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) + ( B x. ( -u ( C + 1 ) / B ) ) ) + C ) )
145 simprl 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> 0 = A )
146 145 oveq1d 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( 0 x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) = ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) )
147 143 recnd 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( -u ( C + 1 ) / B ) e. CC )
148 sqcl 
 |-  ( ( -u ( C + 1 ) / B ) e. CC -> ( ( -u ( C + 1 ) / B ) ^ 2 ) e. CC )
149 147 148 syl 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( ( -u ( C + 1 ) / B ) ^ 2 ) e. CC )
150 149 mul02d 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( 0 x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) = 0 )
151 146 150 eqtr3d 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) = 0 )
152 129 recnd 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> -u ( C + 1 ) e. CC )
153 141 recnd 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> B e. CC )
154 152 153 142 divcan2d 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( B x. ( -u ( C + 1 ) / B ) ) = -u ( C + 1 ) )
155 151 154 oveq12d 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) + ( B x. ( -u ( C + 1 ) / B ) ) ) = ( 0 + -u ( C + 1 ) ) )
156 152 addlidd 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( 0 + -u ( C + 1 ) ) = -u ( C + 1 ) )
157 155 156 eqtrd 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) + ( B x. ( -u ( C + 1 ) / B ) ) ) = -u ( C + 1 ) )
158 157 oveq1d 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( ( ( A x. ( ( -u ( C + 1 ) / B ) ^ 2 ) ) + ( B x. ( -u ( C + 1 ) / B ) ) ) + C ) = ( -u ( C + 1 ) + C ) )
159 144 158 breqtrd 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> 0 <_ ( -u ( C + 1 ) + C ) )
160 0re 
 |-  0 e. RR
161 129 121 readdcld 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( -u ( C + 1 ) + C ) e. RR )
162 lenlt 
 |-  ( ( 0 e. RR /\ ( -u ( C + 1 ) + C ) e. RR ) -> ( 0 <_ ( -u ( C + 1 ) + C ) <-> -. ( -u ( C + 1 ) + C ) < 0 ) )
163 160 161 162 sylancr 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> ( 0 <_ ( -u ( C + 1 ) + C ) <-> -. ( -u ( C + 1 ) + C ) < 0 ) )
164 159 163 mpbid 
 |-  ( ( ph /\ ( 0 = A /\ B =/= 0 ) ) -> -. ( -u ( C + 1 ) + C ) < 0 )
165 164 expr 
 |-  ( ( ph /\ 0 = A ) -> ( B =/= 0 -> -. ( -u ( C + 1 ) + C ) < 0 ) )
166 133 165 pm2.65d 
 |-  ( ( ph /\ 0 = A ) -> -. B =/= 0 )
167 nne 
 |-  ( -. B =/= 0 <-> B = 0 )
168 166 167 sylib 
 |-  ( ( ph /\ 0 = A ) -> B = 0 )
169 168 sq0id 
 |-  ( ( ph /\ 0 = A ) -> ( B ^ 2 ) = 0 )
170 simpr 
 |-  ( ( ph /\ 0 = A ) -> 0 = A )
171 170 oveq1d 
 |-  ( ( ph /\ 0 = A ) -> ( 0 x. C ) = ( A x. C ) )
172 3 recnd 
 |-  ( ph -> C e. CC )
173 172 adantr 
 |-  ( ( ph /\ 0 = A ) -> C e. CC )
174 173 mul02d 
 |-  ( ( ph /\ 0 = A ) -> ( 0 x. C ) = 0 )
175 171 174 eqtr3d 
 |-  ( ( ph /\ 0 = A ) -> ( A x. C ) = 0 )
176 175 oveq2d 
 |-  ( ( ph /\ 0 = A ) -> ( 4 x. ( A x. C ) ) = ( 4 x. 0 ) )
177 27 mul01i 
 |-  ( 4 x. 0 ) = 0
178 176 177 eqtrdi 
 |-  ( ( ph /\ 0 = A ) -> ( 4 x. ( A x. C ) ) = 0 )
179 169 178 oveq12d 
 |-  ( ( ph /\ 0 = A ) -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) = ( 0 - 0 ) )
180 0m0e0 
 |-  ( 0 - 0 ) = 0
181 0le0 
 |-  0 <_ 0
182 180 181 eqbrtri 
 |-  ( 0 - 0 ) <_ 0
183 179 182 eqbrtrdi 
 |-  ( ( ph /\ 0 = A ) -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0 )
184 eqid 
 |-  if ( 1 <_ ( ( ( B + if ( 0 <_ C , C , 0 ) ) + 1 ) / -u A ) , ( ( ( B + if ( 0 <_ C , C , 0 ) ) + 1 ) / -u A ) , 1 ) = if ( 1 <_ ( ( ( B + if ( 0 <_ C , C , 0 ) ) + 1 ) / -u A ) , ( ( ( B + if ( 0 <_ C , C , 0 ) ) + 1 ) / -u A ) , 1 )
185 1 2 3 4 184 discr1 
 |-  ( ph -> 0 <_ A )
186 leloe 
 |-  ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
187 160 1 186 sylancr 
 |-  ( ph -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
188 185 187 mpbid 
 |-  ( ph -> ( 0 < A \/ 0 = A ) )
189 120 183 188 mpjaodan 
 |-  ( ph -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0 )