Theorem discrlem3 4715

Description: Lemma for discriminant theorem.

Hypotheses

Ref	Expression
discrlem.1	|- A e. RR
discrlem.2	|- B e. RR
discrlem.3	|- C e. RR
discrlem3.4	|- D = ((C + 1) / -uB)
discrlem3.5	|- (D e. RR -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))

Assertion

Ref	Expression
discrlem3	|- (0 = A -> ((B^2) - (4 x. (A x. C))) <_ 0)

Proof of Theorem discrlem3

Step	Hyp	Ref	Expression
1		discrlem.3	. . . . . . . . . 10 |- C e. RR
2	1	ltplus1 4384	. . . . . . . . 9 |- C < (C + 1)
3		df-ne 1192	. . . . . . . . . . 11 |- (B =/= 0 <-> -. B = 0)
4		discrlem.2	. . . . . . . . . . . . 13 |- B e. RR
5	4	recn 4098	. . . . . . . . . . . 12 |- B e. CC
6	5	negne0 4379	. . . . . . . . . . 11 |- (B =/= 0 <-> -uB =/= 0)
7	3, 6	bitr3 153	. . . . . . . . . 10 |- (-. B = 0 <-> -uB =/= 0)
8		ax1re 4064	. . . . . . . . . . . . . . . . . . . 20 |- 1 e. RR
9	1, 8	readdcl 4118	. . . . . . . . . . . . . . . . . . 19 |- (C + 1) e. RR
10	4	renegcl 4171	. . . . . . . . . . . . . . . . . . 19 |- -uB e. RR
11	9, 10	redivclz 4275	. . . . . . . . . . . . . . . . . 18 |- (-uB =/= 0 -> ((C + 1) / -uB) e. RR)
12		discrlem3.4	. . . . . . . . . . . . . . . . . . 19 |- D = ((C + 1) / -uB)
13	12	eleq1i 1152	. . . . . . . . . . . . . . . . . 18 |- (D e. RR <-> ((C + 1) / -uB) e. RR)
14	11, 13	sylibr 175	. . . . . . . . . . . . . . . . 17 |- (-uB =/= 0 -> D e. RR)
15		discrlem3.5	. . . . . . . . . . . . . . . . 17 |- (D e. RR -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
16	14, 15	syl 12	. . . . . . . . . . . . . . . 16 |- (-uB =/= 0 -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
17	16	adantr 306	. . . . . . . . . . . . . . 15 |- ((-uB =/= 0 /\ 0 = A) -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
18		opreq1 3006	. . . . . . . . . . . . . . . . . . . 20 |- (0 = A -> (0 x. (D^2)) = (A x. (D^2)))
19	18	cleqcomd 1106	. . . . . . . . . . . . . . . . . . 19 |- (0 = A -> (A x. (D^2)) = (0 x. (D^2)))
20	14	recnd 4099	. . . . . . . . . . . . . . . . . . . 20 |- (-uB =/= 0 -> D e. CC)
21		sqclt 4684	. . . . . . . . . . . . . . . . . . . 20 |- (D e. CC -> (D^2) e. CC)
22		mulzer2t 4189	. . . . . . . . . . . . . . . . . . . 20 |- ((D^2) e. CC -> (0 x. (D^2)) = 0)
23	20, 21, 22	3syl 21	. . . . . . . . . . . . . . . . . . 19 |- (-uB =/= 0 -> (0 x. (D^2)) = 0)
24	19, 23	sylan9eqr 1145	. . . . . . . . . . . . . . . . . 18 |- ((-uB =/= 0 /\ 0 = A) -> (A x. (D^2)) = 0)
25	24	opreq1d 3012	. . . . . . . . . . . . . . . . 17 |- ((-uB =/= 0 /\ 0 = A) -> ((A x. (D^2)) + (B x. D)) = (0 + (B x. D)))
26	14, 4	jctil 240	. . . . . . . . . . . . . . . . . . . . 21 |- (-uB =/= 0 -> (B e. RR /\ D e. RR))
27		axmulrcl 4069	. . . . . . . . . . . . . . . . . . . . 21 |- ((B e. RR /\ D e. RR) -> (B x. D) e. RR)
28	26, 27	syl 12	. . . . . . . . . . . . . . . . . . . 20 |- (-uB =/= 0 -> (B x. D) e. RR)
29	28	recnd 4099	. . . . . . . . . . . . . . . . . . 19 |- (-uB =/= 0 -> (B x. D) e. CC)
30		addid2t 4132	. . . . . . . . . . . . . . . . . . 19 |- ((B x. D) e. CC -> (0 + (B x. D)) = (B x. D))
31	29, 30	syl 12	. . . . . . . . . . . . . . . . . 18 |- (-uB =/= 0 -> (0 + (B x. D)) = (B x. D))
32	31	adantr 306	. . . . . . . . . . . . . . . . 17 |- ((-uB =/= 0 /\ 0 = A) -> (0 + (B x. D)) = (B x. D))
33	25, 32	eqtrd 1128	. . . . . . . . . . . . . . . 16 |- ((-uB =/= 0 /\ 0 = A) -> ((A x. (D^2)) + (B x. D)) = (B x. D))
34	33	opreq1d 3012	. . . . . . . . . . . . . . 15 |- ((-uB =/= 0 /\ 0 = A) -> (((A x. (D^2)) + (B x. D)) + C) = ((B x. D) + C))
35	17, 34	breqtrd 2081	. . . . . . . . . . . . . 14 |- ((-uB =/= 0 /\ 0 = A) -> 0 <_ ((B x. D) + C))
36		ax0re 4063	. . . . . . . . . . . . . . . . 17 |- 0 e. RR
37		lesubadd2t 4356	. . . . . . . . . . . . . . . . . 18 |- ((0 e. RR /\ (B x. D) e. RR /\ C e. RR) -> ((0 - (B x. D)) <_ C <-> 0 <_ ((B x. D) + C)))
38	1, 37	mp3an3 641	. . . . . . . . . . . . . . . . 17 |- ((0 e. RR /\ (B x. D) e. RR) -> ((0 - (B x. D)) <_ C <-> 0 <_ ((B x. D) + C)))
39	36, 38	mpan 518	. . . . . . . . . . . . . . . 16 |- ((B x. D) e. RR -> ((0 - (B x. D)) <_ C <-> 0 <_ ((B x. D) + C)))
40	26, 27, 39	3syl 21	. . . . . . . . . . . . . . 15 |- (-uB =/= 0 -> ((0 - (B x. D)) <_ C <-> 0 <_ ((B x. D) + C)))
41	40	adantr 306	. . . . . . . . . . . . . 14 |- ((-uB =/= 0 /\ 0 = A) -> ((0 - (B x. D)) <_ C <-> 0 <_ ((B x. D) + C)))
42	35, 41	mpbird 171	. . . . . . . . . . . . 13 |- ((-uB =/= 0 /\ 0 = A) -> (0 - (B x. D)) <_ C)
43		recnt 4097	. . . . . . . . . . . . . . . . . . . 20 |- (B e. RR -> B e. CC)
44		recnt 4097	. . . . . . . . . . . . . . . . . . . 20 |- (D e. RR -> D e. CC)
45	43, 44	anim12i 268	. . . . . . . . . . . . . . . . . . 19 |- ((B e. RR /\ D e. RR) -> (B e. CC /\ D e. CC))
46		mulneg1t 4196	. . . . . . . . . . . . . . . . . . 19 |- ((B e. CC /\ D e. CC) -> (-uB x. D) = -u(B x. D))
47	26, 45, 46	3syl 21	. . . . . . . . . . . . . . . . . 18 |- (-uB =/= 0 -> (-uB x. D) = -u(B x. D))
48	47	cleqcomd 1106	. . . . . . . . . . . . . . . . 17 |- (-uB =/= 0 -> -u(B x. D) = (-uB x. D))
49		df-neg 4135	. . . . . . . . . . . . . . . . 17 |- -u(B x. D) = (0 - (B x. D))
50	12	opreq2i 3010	. . . . . . . . . . . . . . . . 17 |- (-uB x. D) = (-uB x. ((C + 1) / -uB))
51	48, 49, 50	3eqtr3g 1146	. . . . . . . . . . . . . . . 16 |- (-uB =/= 0 -> (0 - (B x. D)) = (-uB x. ((C + 1) / -uB)))
52	5	negcl 4142	. . . . . . . . . . . . . . . . 17 |- -uB e. CC
53	9	recn 4098	. . . . . . . . . . . . . . . . 17 |- (C + 1) e. CC
54	52, 53	divcan2z 4227	. . . . . . . . . . . . . . . 16 |- (-uB =/= 0 -> (-uB x. ((C + 1) / -uB)) = (C + 1))
55	51, 54	eqtrd 1128	. . . . . . . . . . . . . . 15 |- (-uB =/= 0 -> (0 - (B x. D)) = (C + 1))
56	55	breq1d 2071	. . . . . . . . . . . . . 14 |- (-uB =/= 0 -> ((0 - (B x. D)) <_ C <-> (C + 1) <_ C))
57	56	adantr 306	. . . . . . . . . . . . 13 |- ((-uB =/= 0 /\ 0 = A) -> ((0 - (B x. D)) <_ C <-> (C + 1) <_ C))
58	42, 57	mpbid 170	. . . . . . . . . . . 12 |- ((-uB =/= 0 /\ 0 = A) -> (C + 1) <_ C)
59	9, 1	lelt 4301	. . . . . . . . . . . 12 |- ((C + 1) <_ C <-> -. C < (C + 1))
60	58, 59	sylib 173	. . . . . . . . . . 11 |- ((-uB =/= 0 /\ 0 = A) -> -. C < (C + 1))
61	60	exp 291	. . . . . . . . . 10 |- (-uB =/= 0 -> (0 = A -> -. C < (C + 1)))
62	7, 61	sylbi 174	. . . . . . . . 9 |- (-. B = 0 -> (0 = A -> -. C < (C + 1)))
63	2, 62	mt2i 97	. . . . . . . 8 |- (-. B = 0 -> -. 0 = A)
64	63	a3i 69	. . . . . . 7 |- (0 = A -> B = 0)
65	64	opreq1d 3012	. . . . . 6 |- (0 = A -> (B x. B) = (0 x. B))
66	5	mulzer2 4186	. . . . . 6 |- (0 x. B) = 0
67	65, 66	syl6eq 1140	. . . . 5 |- (0 = A -> (B x. B) = 0)
68	5	sqval 4685	. . . . 5 |- (B^2) = (B x. B)
69	67, 68	syl5eq 1136	. . . 4 |- (0 = A -> (B^2) = 0)
70		opreq1 3006	. . . . . . 7 |- (0 = A -> (0 x. C) = (A x. C))
71	1	recn 4098	. . . . . . . 8 |- C e. CC
72	71	mulzer2 4186	. . . . . . 7 |- (0 x. C) = 0
73	70, 72	syl5reqr 1139	. . . . . 6 |- (0 = A -> (A x. C) = 0)
74	73	opreq2d 3013	. . . . 5 |- (0 = A -> (4 x. (A x. C)) = (4 x. 0))
75		4re 4473	. . . . . . 7 |- 4 e. RR
76	75	recn 4098	. . . . . 6 |- 4 e. CC
77	76	mulzer1 4185	. . . . 5 |- (4 x. 0) = 0
78	74, 77	syl6eq 1140	. . . 4 |- (0 = A -> (4 x. (A x. C)) = 0)
79	69, 78	opreq12d 3014	. . 3 |- (0 = A -> ((B^2) - (4 x. (A x. C))) = (0 - 0))
80		0cn 4100	. . . 4 |- 0 e. CC
81	80	subid 4155	. . 3 |- (0 - 0) = 0
82	79, 81	syl6eq 1140	. 2 |- (0 = A -> ((B^2) - (4 x. (A x. C))) = 0)
83	4	sqrecl 4699	. . . 4 |- (B^2) e. RR
84		discrlem.1	. . . . . 6 |- A e. RR
85	84, 1	remulcl 4119	. . . . 5 |- (A x. C) e. RR
86	75, 85	remulcl 4119	. . . 4 |- (4 x. (A x. C)) e. RR
87	83, 86	resubcl 4174	. . 3 |- ((B^2) - (4 x. (A x. C))) e. RR
88	87, 36	eqle 4304	. 2 |- (((B^2) - (4 x. (A x. C))) = 0 -> ((B^2) - (4 x. (A x. C))) <_ 0)
89	82, 88	syl 12	1 |- (0 = A -> ((B^2) - (4 x. (A x. C))) <_ 0)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092   =/= wne 1190   class class class wbr 2054  (class class class)co 3001  CCcc 4026  RRcr 4027  0cc0 4028  1c1 4029   + caddc 4031   x. cmulc 4032   < clt 4033   - cmin 4089  -ucneg 4090   / cdiv 4091   <_ cle 4092  2c2 4454  4c4 4456  ^cexp 4675

This theorem is referenced by:  discrlem 4716

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f</