Theorem quad2 24566

Description: The quadratic equation, without specifying the particular branch D to the square root. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
quad.a	|- ( ph -> A e. CC )
quad.z	|- ( ph -> A =/= 0 )
quad.b	|- ( ph -> B e. CC )
quad.c	|- ( ph -> C e. CC )
quad.x	|- ( ph -> X e. CC )
quad2.d	|- ( ph -> D e. CC )
quad2.2	|- ( ph -> ( D ^ 2 ) = ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) )

Assertion

Ref	Expression
quad2	|- ( ph -> ( ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) = 0 <-> ( X = ( ( -u B + D ) / ( 2 x. A ) ) \/ X = ( ( -u B - D ) / ( 2 x. A ) ) ) ) )

Proof of Theorem quad2

Step	Hyp	Ref	Expression
1		2cn 11091	. . . . . . . 8 |- 2 e. CC
2		quad.a	. . . . . . . 8 |- ( ph -> A e. CC )
3		mulcl 10020	. . . . . . . 8 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
4	1, 2, 3	sylancr 695	. . . . . . 7 |- ( ph -> ( 2 x. A ) e. CC )
5		quad.x	. . . . . . 7 |- ( ph -> X e. CC )
6	4, 5	mulcld 10060	. . . . . 6 |- ( ph -> ( ( 2 x. A ) x. X ) e. CC )
7		quad.b	. . . . . 6 |- ( ph -> B e. CC )
8	6, 7	addcld 10059	. . . . 5 |- ( ph -> ( ( ( 2 x. A ) x. X ) + B ) e. CC )
9	8	sqcld 13006	. . . 4 |- ( ph -> ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) e. CC )
10		quad2.d	. . . . 5 |- ( ph -> D e. CC )
11	10	sqcld 13006	. . . 4 |- ( ph -> ( D ^ 2 ) e. CC )
12	9, 11	subeq0ad 10402	. . 3 |- ( ph -> ( ( ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) - ( D ^ 2 ) ) = 0 <-> ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) = ( D ^ 2 ) ) )
13	5	sqcld 13006	. . . . . . 7 |- ( ph -> ( X ^ 2 ) e. CC )
14	2, 13	mulcld 10060	. . . . . 6 |- ( ph -> ( A x. ( X ^ 2 ) ) e. CC )
15	7, 5	mulcld 10060	. . . . . . 7 |- ( ph -> ( B x. X ) e. CC )
16		quad.c	. . . . . . 7 |- ( ph -> C e. CC )
17	15, 16	addcld 10059	. . . . . 6 |- ( ph -> ( ( B x. X ) + C ) e. CC )
18	14, 17	addcld 10059	. . . . 5 |- ( ph -> ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) e. CC )
19		0cnd 10033	. . . . 5 |- ( ph -> 0 e. CC )
20		4cn 11098	. . . . . 6 |- 4 e. CC
21		mulcl 10020	. . . . . 6 |- ( ( 4 e. CC /\ A e. CC ) -> ( 4 x. A ) e. CC )
22	20, 2, 21	sylancr 695	. . . . 5 |- ( ph -> ( 4 x. A ) e. CC )
23	20	a1i 11	. . . . . 6 |- ( ph -> 4 e. CC )
24		4ne0 11117	. . . . . . 7 |- 4 =/= 0
25	24	a1i 11	. . . . . 6 |- ( ph -> 4 =/= 0 )
26		quad.z	. . . . . 6 |- ( ph -> A =/= 0 )
27	23, 2, 25, 26	mulne0d 10679	. . . . 5 |- ( ph -> ( 4 x. A ) =/= 0 )
28	18, 19, 22, 27	mulcand 10660	. . . 4 |- ( ph -> ( ( ( 4 x. A ) x. ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) ) = ( ( 4 x. A ) x. 0 ) <-> ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) = 0 ) )
29	6	sqcld 13006	. . . . . . . 8 |- ( ph -> ( ( ( 2 x. A ) x. X ) ^ 2 ) e. CC )
30	6, 7	mulcld 10060	. . . . . . . . 9 |- ( ph -> ( ( ( 2 x. A ) x. X ) x. B ) e. CC )
31		mulcl 10020	. . . . . . . . 9 |- ( ( 2 e. CC /\ ( ( ( 2 x. A ) x. X ) x. B ) e. CC ) -> ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) e. CC )
32	1, 30, 31	sylancr 695	. . . . . . . 8 |- ( ph -> ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) e. CC )
33	2, 16	mulcld 10060	. . . . . . . . 9 |- ( ph -> ( A x. C ) e. CC )
34		mulcl 10020	. . . . . . . . 9 |- ( ( 4 e. CC /\ ( A x. C ) e. CC ) -> ( 4 x. ( A x. C ) ) e. CC )
35	20, 33, 34	sylancr 695	. . . . . . . 8 |- ( ph -> ( 4 x. ( A x. C ) ) e. CC )
36	29, 32, 35	addassd 10062	. . . . . . 7 |- ( ph -> ( ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) + ( 4 x. ( A x. C ) ) ) = ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) + ( 4 x. ( A x. C ) ) ) ) )
37	7	sqcld 13006	. . . . . . . 8 |- ( ph -> ( B ^ 2 ) e. CC )
38	29, 32	addcld 10059	. . . . . . . 8 |- ( ph -> ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) e. CC )
39	37, 38, 35	pnncand 10431	. . . . . . 7 |- ( ph -> ( ( ( B ^ 2 ) + ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) ) - ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) ) = ( ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) + ( 4 x. ( A x. C ) ) ) )
40	4, 5	sqmuld 13020	. . . . . . . . 9 |- ( ph -> ( ( ( 2 x. A ) x. X ) ^ 2 ) = ( ( ( 2 x. A ) ^ 2 ) x. ( X ^ 2 ) ) )
41		sq2 12960	. . . . . . . . . . . . 13 |- ( 2 ^ 2 ) = 4
42	41	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( 2 ^ 2 ) = 4 )
43	2	sqvald 13005	. . . . . . . . . . . 12 |- ( ph -> ( A ^ 2 ) = ( A x. A ) )
44	42, 43	oveq12d 6668	. . . . . . . . . . 11 |- ( ph -> ( ( 2 ^ 2 ) x. ( A ^ 2 ) ) = ( 4 x. ( A x. A ) ) )
45		sqmul 12926	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ A e. CC ) -> ( ( 2 x. A ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( A ^ 2 ) ) )
46	1, 2, 45	sylancr 695	. . . . . . . . . . 11 |- ( ph -> ( ( 2 x. A ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( A ^ 2 ) ) )
47	23, 2, 2	mulassd 10063	. . . . . . . . . . 11 |- ( ph -> ( ( 4 x. A ) x. A ) = ( 4 x. ( A x. A ) ) )
48	44, 46, 47	3eqtr4d 2666	. . . . . . . . . 10 |- ( ph -> ( ( 2 x. A ) ^ 2 ) = ( ( 4 x. A ) x. A ) )
49	48	oveq1d 6665	. . . . . . . . 9 |- ( ph -> ( ( ( 2 x. A ) ^ 2 ) x. ( X ^ 2 ) ) = ( ( ( 4 x. A ) x. A ) x. ( X ^ 2 ) ) )
50	22, 2, 13	mulassd 10063	. . . . . . . . 9 |- ( ph -> ( ( ( 4 x. A ) x. A ) x. ( X ^ 2 ) ) = ( ( 4 x. A ) x. ( A x. ( X ^ 2 ) ) ) )
51	40, 49, 50	3eqtrrd 2661	. . . . . . . 8 |- ( ph -> ( ( 4 x. A ) x. ( A x. ( X ^ 2 ) ) ) = ( ( ( 2 x. A ) x. X ) ^ 2 ) )
52	22, 15, 16	adddid 10064	. . . . . . . . 9 |- ( ph -> ( ( 4 x. A ) x. ( ( B x. X ) + C ) ) = ( ( ( 4 x. A ) x. ( B x. X ) ) + ( ( 4 x. A ) x. C ) ) )
53		2t2e4 11177	. . . . . . . . . . . . . . . . 17 |- ( 2 x. 2 ) = 4
54	53	oveq1i 6660	. . . . . . . . . . . . . . . 16 |- ( ( 2 x. 2 ) x. A ) = ( 4 x. A )
55	1	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> 2 e. CC )
56	55, 55, 2	mulassd 10063	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( 2 x. 2 ) x. A ) = ( 2 x. ( 2 x. A ) ) )
57	54, 56	syl5eqr 2670	. . . . . . . . . . . . . . 15 |- ( ph -> ( 4 x. A ) = ( 2 x. ( 2 x. A ) ) )
58	57	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ph -> ( ( 4 x. A ) x. B ) = ( ( 2 x. ( 2 x. A ) ) x. B ) )
59	55, 4, 7	mulassd 10063	. . . . . . . . . . . . . 14 |- ( ph -> ( ( 2 x. ( 2 x. A ) ) x. B ) = ( 2 x. ( ( 2 x. A ) x. B ) ) )
60	58, 59	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ph -> ( ( 4 x. A ) x. B ) = ( 2 x. ( ( 2 x. A ) x. B ) ) )
61	60	oveq1d 6665	. . . . . . . . . . . 12 |- ( ph -> ( ( ( 4 x. A ) x. B ) x. X ) = ( ( 2 x. ( ( 2 x. A ) x. B ) ) x. X ) )
62	4, 7	mulcld 10060	. . . . . . . . . . . . 13 |- ( ph -> ( ( 2 x. A ) x. B ) e. CC )
63	55, 62, 5	mulassd 10063	. . . . . . . . . . . 12 |- ( ph -> ( ( 2 x. ( ( 2 x. A ) x. B ) ) x. X ) = ( 2 x. ( ( ( 2 x. A ) x. B ) x. X ) ) )
64	61, 63	eqtrd 2656	. . . . . . . . . . 11 |- ( ph -> ( ( ( 4 x. A ) x. B ) x. X ) = ( 2 x. ( ( ( 2 x. A ) x. B ) x. X ) ) )
65	22, 7, 5	mulassd 10063	. . . . . . . . . . 11 |- ( ph -> ( ( ( 4 x. A ) x. B ) x. X ) = ( ( 4 x. A ) x. ( B x. X ) ) )
66	4, 7, 5	mul32d 10246	. . . . . . . . . . . 12 |- ( ph -> ( ( ( 2 x. A ) x. B ) x. X ) = ( ( ( 2 x. A ) x. X ) x. B ) )
67	66	oveq2d 6666	. . . . . . . . . . 11 |- ( ph -> ( 2 x. ( ( ( 2 x. A ) x. B ) x. X ) ) = ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) )
68	64, 65, 67	3eqtr3d 2664	. . . . . . . . . 10 |- ( ph -> ( ( 4 x. A ) x. ( B x. X ) ) = ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) )
69	23, 2, 16	mulassd 10063	. . . . . . . . . 10 |- ( ph -> ( ( 4 x. A ) x. C ) = ( 4 x. ( A x. C ) ) )
70	68, 69	oveq12d 6668	. . . . . . . . 9 |- ( ph -> ( ( ( 4 x. A ) x. ( B x. X ) ) + ( ( 4 x. A ) x. C ) ) = ( ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) + ( 4 x. ( A x. C ) ) ) )
71	52, 70	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( ( 4 x. A ) x. ( ( B x. X ) + C ) ) = ( ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) + ( 4 x. ( A x. C ) ) ) )
72	51, 71	oveq12d 6668	. . . . . . 7 |- ( ph -> ( ( ( 4 x. A ) x. ( A x. ( X ^ 2 ) ) ) + ( ( 4 x. A ) x. ( ( B x. X ) + C ) ) ) = ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) + ( 4 x. ( A x. C ) ) ) ) )
73	36, 39, 72	3eqtr4rd 2667	. . . . . 6 |- ( ph -> ( ( ( 4 x. A ) x. ( A x. ( X ^ 2 ) ) ) + ( ( 4 x. A ) x. ( ( B x. X ) + C ) ) ) = ( ( ( B ^ 2 ) + ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) ) - ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) ) )
74	22, 14, 17	adddid 10064	. . . . . 6 |- ( ph -> ( ( 4 x. A ) x. ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) ) = ( ( ( 4 x. A ) x. ( A x. ( X ^ 2 ) ) ) + ( ( 4 x. A ) x. ( ( B x. X ) + C ) ) ) )
75		binom2 12979	. . . . . . . . 9 |- ( ( ( ( 2 x. A ) x. X ) e. CC /\ B e. CC ) -> ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) = ( ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) + ( B ^ 2 ) ) )
76	6, 7, 75	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) = ( ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) + ( B ^ 2 ) ) )
77	38, 37	addcomd 10238	. . . . . . . 8 |- ( ph -> ( ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) + ( B ^ 2 ) ) = ( ( B ^ 2 ) + ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) ) )
78	76, 77	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) = ( ( B ^ 2 ) + ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) ) )
79		quad2.2	. . . . . . 7 |- ( ph -> ( D ^ 2 ) = ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) )
80	78, 79	oveq12d 6668	. . . . . 6 |- ( ph -> ( ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) - ( D ^ 2 ) ) = ( ( ( B ^ 2 ) + ( ( ( ( 2 x. A ) x. X ) ^ 2 ) + ( 2 x. ( ( ( 2 x. A ) x. X ) x. B ) ) ) ) - ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) ) )
81	73, 74, 80	3eqtr4d 2666	. . . . 5 |- ( ph -> ( ( 4 x. A ) x. ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) ) = ( ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) - ( D ^ 2 ) ) )
82	22	mul01d 10235	. . . . 5 |- ( ph -> ( ( 4 x. A ) x. 0 ) = 0 )
83	81, 82	eqeq12d 2637	. . . 4 |- ( ph -> ( ( ( 4 x. A ) x. ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) ) = ( ( 4 x. A ) x. 0 ) <-> ( ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) - ( D ^ 2 ) ) = 0 ) )
84	28, 83	bitr3d 270	. . 3 |- ( ph -> ( ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) = 0 <-> ( ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) - ( D ^ 2 ) ) = 0 ) )
85	6, 7	subnegd 10399	. . . . 5 |- ( ph -> ( ( ( 2 x. A ) x. X ) - -u B ) = ( ( ( 2 x. A ) x. X ) + B ) )
86	85	oveq1d 6665	. . . 4 |- ( ph -> ( ( ( ( 2 x. A ) x. X ) - -u B ) ^ 2 ) = ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) )
87	86	eqeq1d 2624	. . 3 |- ( ph -> ( ( ( ( ( 2 x. A ) x. X ) - -u B ) ^ 2 ) = ( D ^ 2 ) <-> ( ( ( ( 2 x. A ) x. X ) + B ) ^ 2 ) = ( D ^ 2 ) ) )
88	12, 84, 87	3bitr4d 300	. 2 |- ( ph -> ( ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) = 0 <-> ( ( ( ( 2 x. A ) x. X ) - -u B ) ^ 2 ) = ( D ^ 2 ) ) )
89	7	negcld 10379	. . . 4 |- ( ph -> -u B e. CC )
90	6, 89	subcld 10392	. . 3 |- ( ph -> ( ( ( 2 x. A ) x. X ) - -u B ) e. CC )
91		sqeqor 12978	. . 3 |- ( ( ( ( ( 2 x. A ) x. X ) - -u B ) e. CC /\ D e. CC ) -> ( ( ( ( ( 2 x. A ) x. X ) - -u B ) ^ 2 ) = ( D ^ 2 ) <-> ( ( ( ( 2 x. A ) x. X ) - -u B ) = D \/ ( ( ( 2 x. A ) x. X ) - -u B ) = -u D ) ) )
92	90, 10, 91	syl2anc 693	. 2 |- ( ph -> ( ( ( ( ( 2 x. A ) x. X ) - -u B ) ^ 2 ) = ( D ^ 2 ) <-> ( ( ( ( 2 x. A ) x. X ) - -u B ) = D \/ ( ( ( 2 x. A ) x. X ) - -u B ) = -u D ) ) )
93	6, 89, 10	subaddd 10410	. . . 4 |- ( ph -> ( ( ( ( 2 x. A ) x. X ) - -u B ) = D <-> ( -u B + D ) = ( ( 2 x. A ) x. X ) ) )
94	89, 10	addcld 10059	. . . . . 6 |- ( ph -> ( -u B + D ) e. CC )
95		2ne0 11113	. . . . . . . 8 |- 2 =/= 0
96	95	a1i 11	. . . . . . 7 |- ( ph -> 2 =/= 0 )
97	55, 2, 96, 26	mulne0d 10679	. . . . . 6 |- ( ph -> ( 2 x. A ) =/= 0 )
98	94, 4, 5, 97	divmuld 10823	. . . . 5 |- ( ph -> ( ( ( -u B + D ) / ( 2 x. A ) ) = X <-> ( ( 2 x. A ) x. X ) = ( -u B + D ) ) )
99		eqcom 2629	. . . . 5 |- ( X = ( ( -u B + D ) / ( 2 x. A ) ) <-> ( ( -u B + D ) / ( 2 x. A ) ) = X )
100		eqcom 2629	. . . . 5 |- ( ( -u B + D ) = ( ( 2 x. A ) x. X ) <-> ( ( 2 x. A ) x. X ) = ( -u B + D ) )
101	98, 99, 100	3bitr4g 303	. . . 4 |- ( ph -> ( X = ( ( -u B + D ) / ( 2 x. A ) ) <-> ( -u B + D ) = ( ( 2 x. A ) x. X ) ) )
102	93, 101	bitr4d 271	. . 3 |- ( ph -> ( ( ( ( 2 x. A ) x. X ) - -u B ) = D <-> X = ( ( -u B + D ) / ( 2 x. A ) ) ) )
103	89, 10	negsubd 10398	. . . . 5 |- ( ph -> ( -u B + -u D ) = ( -u B - D ) )
104	103	eqeq1d 2624	. . . 4 |- ( ph -> ( ( -u B + -u D ) = ( ( 2 x. A ) x. X ) <-> ( -u B - D ) = ( ( 2 x. A ) x. X ) ) )
105	10	negcld 10379	. . . . 5 |- ( ph -> -u D e. CC )
106	6, 89, 105	subaddd 10410	. . . 4 |- ( ph -> ( ( ( ( 2 x. A ) x. X ) - -u B ) = -u D <-> ( -u B + -u D ) = ( ( 2 x. A ) x. X ) ) )
107	89, 10	subcld 10392	. . . . . 6 |- ( ph -> ( -u B - D ) e. CC )
108	107, 4, 5, 97	divmuld 10823	. . . . 5 |- ( ph -> ( ( ( -u B - D ) / ( 2 x. A ) ) = X <-> ( ( 2 x. A ) x. X ) = ( -u B - D ) ) )
109		eqcom 2629	. . . . 5 |- ( X = ( ( -u B - D ) / ( 2 x. A ) ) <-> ( ( -u B - D ) / ( 2 x. A ) ) = X )
110		eqcom 2629	. . . . 5 |- ( ( -u B - D ) = ( ( 2 x. A ) x. X ) <-> ( ( 2 x. A ) x. X ) = ( -u B - D ) )
111	108, 109, 110	3bitr4g 303	. . . 4 |- ( ph -> ( X = ( ( -u B - D ) / ( 2 x. A ) ) <-> ( -u B - D ) = ( ( 2 x. A ) x. X ) ) )
112	104, 106, 111	3bitr4d 300	. . 3 |- ( ph -> ( ( ( ( 2 x. A ) x. X ) - -u B ) = -u D <-> X = ( ( -u B - D ) / ( 2 x. A ) ) ) )
113	102, 112	orbi12d 746	. 2 |- ( ph -> ( ( ( ( ( 2 x. A ) x. X ) - -u B ) = D \/ ( ( ( 2 x. A ) x. X ) - -u B ) = -u D ) <-> ( X = ( ( -u B + D ) / ( 2 x. A ) ) \/ X = ( ( -u B - D ) / ( 2 x. A ) ) ) ) )
114	88, 92, 113	3bitrd 294	1 |- ( ph -> ( ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) = 0 <-> ( X = ( ( -u B + D ) / ( 2 x. A ) ) \/ X = ( ( -u B - D ) / ( 2 x. A ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   4c4 11072   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by:  quad  24567  dcubic2  24571  dquartlem1  24578