Theorem discrlem3 7908

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	ltp1i 6991	. . . . . . . . 9 |- C < (C + 1)
3		df-ne 2019	. . . . . . . . . . 11 |- (B =/= 0 <-> -. B = 0)
4		discrlem.2	. . . . . . . . . . . . 13 |- B e. RR
5	4	recni 6467	. . . . . . . . . . . 12 |- B e. CC
6	5	negne0bi 6985	. . . . . . . . . . 11 |- (B =/= 0 <-> -uB =/= 0)
7	3, 6	bitr3i 192	. . . . . . . . . 10 |- (-. B = 0 <-> -uB =/= 0)
8		1re 6598	. . . . . . . . . . . . . . . . . . . 20 |- 1 e. RR
9	1, 8	readdcli 6487	. . . . . . . . . . . . . . . . . . 19 |- (C + 1) e. RR
10	4	renegcli 6576	. . . . . . . . . . . . . . . . . . 19 |- -uB e. RR
11	9, 10	redivclzi 6977	. . . . . . . . . . . . . . . . . 18 |- (-uB =/= 0 -> ((C + 1) / -uB) e. RR)
12		discrlem3.4	. . . . . . . . . . . . . . . . . 18 |- D = ((C + 1) / -uB)
13	11, 12	syl5eqel 1975	. . . . . . . . . . . . . . . . 17 |- (-uB =/= 0 -> D e. RR)
14		discrlem3.5	. . . . . . . . . . . . . . . . 17 |- (D e. RR -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
15	13, 14	syl 12	. . . . . . . . . . . . . . . 16 |- (-uB =/= 0 -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
16	15	adantr 425	. . . . . . . . . . . . . . 15 |- ((-uB =/= 0 /\ 0 = A) -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
17		opreq1 4889	. . . . . . . . . . . . . . . . . . . 20 |- (0 = A -> (0 x. (D^2)) = (A x. (D^2)))
18	17	eqcomd 1889	. . . . . . . . . . . . . . . . . . 19 |- (0 = A -> (A x. (D^2)) = (0 x. (D^2)))
19	13	recnd 6468	. . . . . . . . . . . . . . . . . . . 20 |- (-uB =/= 0 -> D e. CC)
20		sqcl 7856	. . . . . . . . . . . . . . . . . . . 20 |- (D e. CC -> (D^2) e. CC)
21		mul02 6607	. . . . . . . . . . . . . . . . . . . 20 |- ((D^2) e. CC -> (0 x. (D^2)) = 0)
22	19, 20, 21	3syl 24	. . . . . . . . . . . . . . . . . . 19 |- (-uB =/= 0 -> (0 x. (D^2)) = 0)
23	18, 22	sylan9eqr 1951	. . . . . . . . . . . . . . . . . 18 |- ((-uB =/= 0 /\ 0 = A) -> (A x. (D^2)) = 0)
24	23	opreq1d 4897	. . . . . . . . . . . . . . . . 17 |- ((-uB =/= 0 /\ 0 = A) -> ((A x. (D^2)) + (B x. D)) = (0 + (B x. D)))
25		remulcl 6457	. . . . . . . . . . . . . . . . . . . . 21 |- ((B e. RR /\ D e. RR) -> (B x. D) e. RR)
26	25, 4, 13	sylancr 526	. . . . . . . . . . . . . . . . . . . 20 |- (-uB =/= 0 -> (B x. D) e. RR)
27	26	recnd 6468	. . . . . . . . . . . . . . . . . . 19 |- (-uB =/= 0 -> (B x. D) e. CC)
28		addid2 6482	. . . . . . . . . . . . . . . . . . 19 |- ((B x. D) e. CC -> (0 + (B x. D)) = (B x. D))
29	27, 28	syl 12	. . . . . . . . . . . . . . . . . 18 |- (-uB =/= 0 -> (0 + (B x. D)) = (B x. D))
30	29	adantr 425	. . . . . . . . . . . . . . . . 17 |- ((-uB =/= 0 /\ 0 = A) -> (0 + (B x. D)) = (B x. D))
31	24, 30	eqtrd 1925	. . . . . . . . . . . . . . . 16 |- ((-uB =/= 0 /\ 0 = A) -> ((A x. (D^2)) + (B x. D)) = (B x. D))
32	31	opreq1d 4897	. . . . . . . . . . . . . . 15 |- ((-uB =/= 0 /\ 0 = A) -> (((A x. (D^2)) + (B x. D)) + C) = ((B x. D) + C))
33	16, 32	breqtrd 3361	. . . . . . . . . . . . . 14 |- ((-uB =/= 0 /\ 0 = A) -> 0 <_ ((B x. D) + C))
34	13, 4	jctil 316	. . . . . . . . . . . . . . . 16 |- (-uB =/= 0 -> (B e. RR /\ D e. RR))
35		0re 6603	. . . . . . . . . . . . . . . . 17 |- 0 e. RR
36		lesubadd2 6813	. . . . . . . . . . . . . . . . 17 |- ((0 e. RR /\ (B x. D) e. RR /\ C e. RR) -> ((0 - (B x. D)) <_ C <-> 0 <_ ((B x. D) + C)))
37	35, 1, 36	mp3an13 1182	. . . . . . . . . . . . . . . 16 |- ((B x. D) e. RR -> ((0 - (B x. D)) <_ C <-> 0 <_ ((B x. D) + C)))
38	34, 25, 37	3syl 24	. . . . . . . . . . . . . . 15 |- (-uB =/= 0 -> ((0 - (B x. D)) <_ C <-> 0 <_ ((B x. D) + C)))
39	38	adantr 425	. . . . . . . . . . . . . 14 |- ((-uB =/= 0 /\ 0 = A) -> ((0 - (B x. D)) <_ C <-> 0 <_ ((B x. D) + C)))
40	33, 39	mpbird 213	. . . . . . . . . . . . 13 |- ((-uB =/= 0 /\ 0 = A) -> (0 - (B x. D)) <_ C)
41		recn 6466	. . . . . . . . . . . . . . . . . . . 20 |- (B e. RR -> B e. CC)
42		recn 6466	. . . . . . . . . . . . . . . . . . . 20 |- (D e. RR -> D e. CC)
43	41, 42	anim12i 360	. . . . . . . . . . . . . . . . . . 19 |- ((B e. RR /\ D e. RR) -> (B e. CC /\ D e. CC))
44		mulneg1 6615	. . . . . . . . . . . . . . . . . . 19 |- ((B e. CC /\ D e. CC) -> (-uB x. D) = -u(B x. D))
45	34, 43, 44	3syl 24	. . . . . . . . . . . . . . . . . 18 |- (-uB =/= 0 -> (-uB x. D) = -u(B x. D))
46	45	eqcomd 1889	. . . . . . . . . . . . . . . . 17 |- (-uB =/= 0 -> -u(B x. D) = (-uB x. D))
47		df-neg 6513	. . . . . . . . . . . . . . . . 17 |- -u(B x. D) = (0 - (B x. D))
48	12	opreq2i 4893	. . . . . . . . . . . . . . . . 17 |- (-uB x. D) = (-uB x. ((C + 1) / -uB))
49	46, 47, 48	3eqtr3g 1952	. . . . . . . . . . . . . . . 16 |- (-uB =/= 0 -> (0 - (B x. D)) = (-uB x. ((C + 1) / -uB)))
50	9	recni 6467	. . . . . . . . . . . . . . . . 17 |- (C + 1) e. CC
51	5	negcli 6526	. . . . . . . . . . . . . . . . 17 |- -uB e. CC
52	50, 51	divcan2zi 6908	. . . . . . . . . . . . . . . 16 |- (-uB =/= 0 -> (-uB x. ((C + 1) / -uB)) = (C + 1))
53	49, 52	eqtrd 1925	. . . . . . . . . . . . . . 15 |- (-uB =/= 0 -> (0 - (B x. D)) = (C + 1))
54	53	breq1d 3348	. . . . . . . . . . . . . 14 |- (-uB =/= 0 -> ((0 - (B x. D)) <_ C <-> (C + 1) <_ C))
55	54	adantr 425	. . . . . . . . . . . . 13 |- ((-uB =/= 0 /\ 0 = A) -> ((0 - (B x. D)) <_ C <-> (C + 1) <_ C))
56	40, 55	mpbid 212	. . . . . . . . . . . 12 |- ((-uB =/= 0 /\ 0 = A) -> (C + 1) <_ C)
57	9, 1	lenlti 6753	. . . . . . . . . . . 12 |- ((C + 1) <_ C <-> -. C < (C + 1))
58	56, 57	sylib 215	. . . . . . . . . . 11 |- ((-uB =/= 0 /\ 0 = A) -> -. C < (C + 1))
59	58	ex 402	. . . . . . . . . 10 |- (-uB =/= 0 -> (0 = A -> -. C < (C + 1)))
60	7, 59	sylbi 216	. . . . . . . . 9 |- (-. B = 0 -> (0 = A -> -. C < (C + 1)))
61	2, 60	mt2i 125	. . . . . . . 8 |- (-. B = 0 -> -. 0 = A)
62	61	con4i 90	. . . . . . 7 |- (0 = A -> B = 0)
63	62	opreq1d 4897	. . . . . 6 |- (0 = A -> (B x. B) = (0 x. B))
64	5	mul02i 6595	. . . . . 6 |- (0 x. B) = 0
65	63, 64	syl6eq 1944	. . . . 5 |- (0 = A -> (B x. B) = 0)
66	5	sqvali 7859	. . . . 5 |- (B^2) = (B x. B)
67	65, 66	syl5eq 1940	. . . 4 |- (0 = A -> (B^2) = 0)
68		opreq1 4889	. . . . . . 7 |- (0 = A -> (0 x. C) = (A x. C))
69	1	recni 6467	. . . . . . . 8 |- C e. CC
70	69	mul02i 6595	. . . . . . 7 |- (0 x. C) = 0
71	68, 70	syl5reqr 1943	. . . . . 6 |- (0 = A -> (A x. C) = 0)
72	71	opreq2d 4898	. . . . 5 |- (0 = A -> (4 x. (A x. C)) = (4 x. 0))
73		4re 7166	. . . . . . 7 |- 4 e. RR
74	73	recni 6467	. . . . . 6 |- 4 e. CC
75	74	mul01i 6594	. . . . 5 |- (4 x. 0) = 0
76	72, 75	syl6eq 1944	. . . 4 |- (0 = A -> (4 x. (A x. C)) = 0)
77	67, 76	opreq12d 4900	. . 3 |- (0 = A -> ((B^2) - (4 x. (A x. C))) = (0 - 0))
78		0cn 6481	. . . 4 |- 0 e. CC
79	78	subidi 6551	. . 3 |- (0 - 0) = 0
80	77, 79	syl6eq 1944	. 2 |- (0 = A -> ((B^2) - (4 x. (A x. C))) = 0)
81	4	resqcli 7868	. . . 4 |- (B^2) e. RR
82		discrlem.1	. . . . . 6 |- A e. RR
83	82, 1	remulcli 6488	. . . . 5 |- (A x. C) e. RR
84	73, 83	remulcli 6488	. . . 4 |- (4 x. (A x. C)) e. RR
85	81, 84	resubcli 6602	. . 3 |- ((B^2) - (4 x. (A x. C))) e. RR
86	85, 35	eqlei 6757	. 2 |- (((B^2) - (4 x. (A x. C))) = 0 -> ((B^2) - (4 x. (A x. C))) <_ 0)
87	80, 86	syl 12	1 |- (0 = A -> ((B^2) - (4 x. (A x. C))) <_ 0)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447   <_ cle 6448   < clt 6653  2c2 7145  4c4 7147  ^cexp 7811

This theorem is referenced by:  discrlem 7909

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

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