Theorem discrlem2 4714

Description: Lemma for discriminant theorem.

Hypotheses

Ref	Expression
discrlem.1	|- A e. RR
discrlem.2	|- B e. RR
discrlem.3	|- C e. RR
discrlem1.4	|- D = -u(B / (2 x. A))
discrlem2.5	|- (D e. RR -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))

Assertion

Ref	Expression
discrlem2	|- (0 < A -> ((B^2) - (4 x. (A x. C))) <_ 0)

Proof of Theorem discrlem2

Step	Hyp	Ref	Expression
1		2pos 4479	. . . . . . 7 |- 0 < 2
2		2re 4470	. . . . . . . 8 |- 2 e. RR
3		discrlem.1	. . . . . . . 8 |- A e. RR
4	2, 3	mulgt0 4334	. . . . . . 7 |- ((0 < 2 /\ 0 < A) -> 0 < (2 x. A))
5	1, 4	mpan 518	. . . . . 6 |- (0 < A -> 0 < (2 x. A))
6	2, 3	remulcl 4119	. . . . . . 7 |- (2 x. A) e. RR
7	6	gt0ne0 4340	. . . . . 6 |- (0 < (2 x. A) -> (2 x. A) =/= 0)
8	5, 7	syl 12	. . . . 5 |- (0 < A -> (2 x. A) =/= 0)
9		discrlem.2	. . . . . 6 |- B e. RR
10	9, 6	redivclz 4275	. . . . 5 |- ((2 x. A) =/= 0 -> (B / (2 x. A)) e. RR)
11		renegclt 4172	. . . . 5 |- ((B / (2 x. A)) e. RR -> -u(B / (2 x. A)) e. RR)
12	8, 10, 11	3syl 21	. . . 4 |- (0 < A -> -u(B / (2 x. A)) e. RR)
13		discrlem1.4	. . . . 5 |- D = -u(B / (2 x. A))
14	13	eleq1i 1152	. . . 4 |- (D e. RR <-> -u(B / (2 x. A)) e. RR)
15	12, 14	sylibr 175	. . 3 |- (0 < A -> D e. RR)
16		discrlem2.5	. . 3 |- (D e. RR -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
17	15, 16	syl 12	. 2 |- (0 < A -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
18		id 9	. . . . . . . 8 |- (A = if(0 < A, A, 1) -> A = if(0 < A, A, 1))
19		opreq2 3007	. . . . . . . . . . . 12 |- (A = if(0 < A, A, 1) -> (2 x. A) = (2 x. if(0 < A, A, 1)))
20	19	opreq2d 3013	. . . . . . . . . . 11 |- (A = if(0 < A, A, 1) -> (B / (2 x. A)) = (B / (2 x. if(0 < A, A, 1))))
21	20	negeqd 4138	. . . . . . . . . 10 |- (A = if(0 < A, A, 1) -> -u(B / (2 x. A)) = -u(B / (2 x. if(0 < A, A, 1))))
22	21, 13	syl5eq 1136	. . . . . . . . 9 |- (A = if(0 < A, A, 1) -> D = -u(B / (2 x. if(0 < A, A, 1))))
23	22	opreq1d 3012	. . . . . . . 8 |- (A = if(0 < A, A, 1) -> (D^2) = (-u(B / (2 x. if(0 < A, A, 1)))^2))
24	18, 23	opreq12d 3014	. . . . . . 7 |- (A = if(0 < A, A, 1) -> (A x. (D^2)) = (if(0 < A, A, 1) x. (-u(B / (2 x. if(0 < A, A, 1)))^2)))
25	22	opreq2d 3013	. . . . . . 7 |- (A = if(0 < A, A, 1) -> (B x. D) = (B x. -u(B / (2 x. if(0 < A, A, 1)))))
26	24, 25	opreq12d 3014	. . . . . 6 |- (A = if(0 < A, A, 1) -> ((A x. (D^2)) + (B x. D)) = ((if(0 < A, A, 1) x. (-u(B / (2 x. if(0 < A, A, 1)))^2)) + (B x. -u(B / (2 x. if(0 < A, A, 1))))))
27	26	opreq1d 3012	. . . . 5 |- (A = if(0 < A, A, 1) -> (((A x. (D^2)) + (B x. D)) + C) = (((if(0 < A, A, 1) x. (-u(B / (2 x. if(0 < A, A, 1)))^2)) + (B x. -u(B / (2 x. if(0 < A, A, 1))))) + C))
28	27	breq2d 2072	. . . 4 |- (A = if(0 < A, A, 1) -> (0 <_ (((A x. (D^2)) + (B x. D)) + C) <-> 0 <_ (((if(0 < A, A, 1) x. (-u(B / (2 x. if(0 < A, A, 1)))^2)) + (B x. -u(B / (2 x. if(0 < A, A, 1))))) + C)))
29		opreq1 3006	. . . . . . 7 |- (A = if(0 < A, A, 1) -> (A x. C) = (if(0 < A, A, 1) x. C))
30	29	opreq2d 3013	. . . . . 6 |- (A = if(0 < A, A, 1) -> (4 x. (A x. C)) = (4 x. (if(0 < A, A, 1) x. C)))
31	30	opreq2d 3013	. . . . 5 |- (A = if(0 < A, A, 1) -> ((B^2) - (4 x. (A x. C))) = ((B^2) - (4 x. (if(0 < A, A, 1) x. C))))
32	31	breq1d 2071	. . . 4 |- (A = if(0 < A, A, 1) -> (((B^2) - (4 x. (A x. C))) <_ 0 <-> ((B^2) - (4 x. (if(0 < A, A, 1) x. C))) <_ 0))
33	28, 32	bibi12d 477	. . 3 |- (A = if(0 < A, A, 1) -> ((0 <_ (((A x. (D^2)) + (B x. D)) + C) <-> ((B^2) - (4 x. (A x. C))) <_ 0) <-> (0 <_ (((if(0 < A, A, 1) x. (-u(B / (2 x. if(0 < A, A, 1)))^2)) + (B x. -u(B / (2 x. if(0 < A, A, 1))))) + C) <-> ((B^2) - (4 x. (if(0 < A, A, 1) x. C))) <_ 0)))
34		ax1re 4064	. . . . 5 |- 1 e. RR
35	3, 34	keepel 1796	. . . 4 |- if(0 < A, A, 1) e. RR
36		discrlem.3	. . . 4 |- C e. RR
37		cleqid 1102	. . . 4 |- -u(B / (2 x. if(0 < A, A, 1))) = -u(B / (2 x. if(0 < A, A, 1)))
38		elimgt0 4381	. . . 4 |- 0 < if(0 < A, A, 1)
39	35, 9, 36, 37, 38	discrlem1 4713	. . 3 |- (0 <_ (((if(0 < A, A, 1) x. (-u(B / (2 x. if(0 < A, A, 1)))^2)) + (B x. -u(B / (2 x. if(0 < A, A, 1))))) + C) <-> ((B^2) - (4 x. (if(0 < A, A, 1) x. C))) <_ 0)
40	33, 39	dedth 1784	. 2 |- (0 < A -> (0 <_ (((A x. (D^2)) + (B x. D)) + C) <-> ((B^2) - (4 x. (A x. C))) <_ 0))
41	17, 40	mpbid 170	1 |- (0 < A -> ((B^2) - (4 x. (A x. C))) <_ 0)

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091   e. wcel 1092   =/= wne 1190  ifcif 1776   class class class wbr 2054  (class class class)co 3001  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 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-4 4464  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676

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