Theorem discrlem2 6587

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 5936	. . . . . . 7 |- 0 < 2
2		2re 5926	. . . . . . . 8 |- 2 e. RR
3		discrlem.1	. . . . . . . 8 |- A e. RR
4	2, 3	mulgt0 5580	. . . . . . 7 |- ((0 < 2 /\ 0 < A) -> 0 < (2 x. A))
5	1, 4	mpan 693	. . . . . 6 |- (0 < A -> 0 < (2 x. A))
6	2, 3	remulcl 5307	. . . . . . 7 |- (2 x. A) e. RR
7	6	gt0ne0 5585	. . . . . 6 |- (0 < (2 x. A) -> (2 x. A) =/= 0)
8	5, 7	syl 10	. . . . 5 |- (0 < A -> (2 x. A) =/= 0)
9		discrlem.2	. . . . . 6 |- B e. RR
10	9, 6	redivclz 5755	. . . . 5 |- ((2 x. A) =/= 0 -> (B / (2 x. A)) e. RR)
11		renegclt 5409	. . . . 5 |- ((B / (2 x. A)) e. RR -> -u(B / (2 x. A)) e. RR)
12	8, 10, 11	3syl 20	. . . 4 |- (0 < A -> -u(B / (2 x. A)) e. RR)
13		discrlem1.4	. . . 4 |- D = -u(B / (2 x. A))
14	12, 13	syl5eqel 1544	. . 3 |- (0 < A -> D e. RR)
15		discrlem2.5	. . 3 |- (D e. RR -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
16	14, 15	syl 10	. 2 |- (0 < A -> 0 <_ (((A x. (D^2)) + (B x. D)) + C))
17		id 59	. . . . . . . 8 |- (A = if(0 < A, A, 1) -> A = if(0 < A, A, 1))
18		opreq2 3954	. . . . . . . . . . . 12 |- (A = if(0 < A, A, 1) -> (2 x. A) = (2 x. if(0 < A, A, 1)))
19	18	opreq2d 3961	. . . . . . . . . . 11 |- (A = if(0 < A, A, 1) -> (B / (2 x. A)) = (B / (2 x. if(0 < A, A, 1))))
20	19	negeqd 5333	. . . . . . . . . 10 |- (A = if(0 < A, A, 1) -> -u(B / (2 x. A)) = -u(B / (2 x. if(0 < A, A, 1))))
21	20, 13	syl5eq 1511	. . . . . . . . 9 |- (A = if(0 < A, A, 1) -> D = -u(B / (2 x. if(0 < A, A, 1))))
22	21	opreq1d 3960	. . . . . . . 8 |- (A = if(0 < A, A, 1) -> (D^2) = (-u(B / (2 x. if(0 < A, A, 1)))^2))
23	17, 22	opreq12d 3963	. . . . . . 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)))
24	21	opreq2d 3961	. . . . . . 7 |- (A = if(0 < A, A, 1) -> (B x. D) = (B x. -u(B / (2 x. if(0 < A, A, 1)))))
25	23, 24	opreq12d 3963	. . . . . 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))))))
26	25	opreq1d 3960	. . . . 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))
27	26	breq2d 2620	. . . 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)))
28		opreq1 3953	. . . . . . 7 |- (A = if(0 < A, A, 1) -> (A x. C) = (if(0 < A, A, 1) x. C))
29	28	opreq2d 3961	. . . . . 6 |- (A = if(0 < A, A, 1) -> (4 x. (A x. C)) = (4 x. (if(0 < A, A, 1) x. C)))
30	29	opreq2d 3961	. . . . 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))))
31	30	breq1d 2619	. . . 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))
32	27, 31	bibi12d 627	. . 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)))
33		1re 5407	. . . . 5 |- 1 e. RR
34	3, 33	keepel 2389	. . . 4 |- if(0 < A, A, 1) e. RR
35		discrlem.3	. . . 4 |- C e. RR
36		eqid 1468	. . . 4 |- -u(B / (2 x. if(0 < A, A, 1))) = -u(B / (2 x. if(0 < A, A, 1)))
37		elimgt0 5765	. . . 4 |- 0 < if(0 < A, A, 1)
38	34, 9, 35, 36, 37	discrlem1 6586	. . 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)
39	32, 38	dedth 2373	. 2 |- (0 < A -> (0 <_ (((A x. (D^2)) + (B x. D)) + C) <-> ((B^2) - (4 x. (A x. C))) <_ 0))
40	16, 39	mpbid 195	1 |- (0 < A -> ((B^2) - (4 x. (A x. C))) <_ 0)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955   =/= wne 1577  ifcif 2351   class class class wbr 2609  (class class class)co 3948  RRcr 5205  0cc0 5206  1c1 5207   + caddc 5209   x. cmul 5211   - cmin 5264  -ucneg 5265   / cdiv 5266   <_ cle 5267   < clt 5458  2c2 5908  4c4 5910  ^cexp 6500

This theorem is referenced by:  discrlem 6589

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-3 5918  df-4 5919  df-n0 6047  df-z 6083  df-seq1 6245  df-exp 6501

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