Theorem discrlem2 6658

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 5991	. . . . . . 7 |- 0 < 2
2		2re 5981	. . . . . . . 8 |- 2 e. RR
3		discrlem.1	. . . . . . . 8 |- A e. RR
4	2, 3	mulgt0 5618	. . . . . . 7 |- ((0 < 2 /\ 0 < A) -> 0 < (2 x. A))
5	1, 4	mpan 697	. . . . . 6 |- (0 < A -> 0 < (2 x. A))
6	2, 3	remulcl 5347	. . . . . . 7 |- (2 x. A) e. RR
7	6	gt0ne0 5623	. . . . . 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 5801	. . . . 5 |- ((2 x. A) =/= 0 -> (B / (2 x. A)) e. RR)
11		renegclt 5449	. . . . 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 1555	. . 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 3975	. . . . . . . . . . . 12 |- (A = if(0 < A, A, 1) -> (2 x. A) = (2 x. if(0 < A, A, 1)))
19	18	opreq2d 3982	. . . . . . . . . . 11 |- (A = if(0 < A, A, 1) -> (B / (2 x. A)) = (B / (2 x. if(0 < A, A, 1))))
20	19	negeqd 5373	. . . . . . . . . 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 1522	. . . . . . . . 9 |- (A = if(0 < A, A, 1) -> D = -u(B / (2 x. if(0 < A, A, 1))))
22	21	opreq1d 3981	. . . . . . . 8 |- (A = if(0 < A, A, 1) -> (D^2) = (-u(B / (2 x. if(0 < A, A, 1)))^2))
23	17, 22	opreq12d 3984	. . . . . . 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 3982	. . . . . . 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 3984	. . . . . 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 3981	. . . . 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 2635	. . . 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 3974	. . . . . . 7 |- (A = if(0 < A, A, 1) -> (A x. C) = (if(0 < A, A, 1) x. C))
29	28	opreq2d 3982	. . . . . 6 |- (A = if(0 < A, A, 1) -> (4 x. (A x. C)) = (4 x. (if(0 < A, A, 1) x. C)))
30	29	opreq2d 3982	. . . . 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 2634	. . . 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 631	. . 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 5447	. . . . 5 |- 1 e. RR
34	3, 33	keepel 2403	. . . 4 |- if(0 < A, A, 1) e. RR
35		discrlem.3	. . . 4 |- C e. RR
36		eqid 1478	. . . 4 |- -u(B / (2 x. if(0 < A, A, 1))) = -u(B / (2 x. if(0 < A, A, 1)))
37		elimgt0 5811	. . . 4 |- 0 < if(0 < A, A, 1)
38	34, 9, 35, 36, 37	discrlem1 6657	. . 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 2387	. 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 958   e. wcel 960   =/= wne 1588  ifcif 2365   class class class wbr 2624  (class class class)co 3969  RRcr 5245  0cc0 5246  1c1 5247   + caddc 5249   x. cmul 5251   - cmin 5304  -ucneg 5305   / cdiv 5306   <_ cle 5307   < clt 5498  2c2 5963  4c4 5965  ^cexp 6569

This theorem is referenced by:  discrlem 6660

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-3 5973  df-4 5974  df-n0 6102  df-z 6138  df-seq1 6309  df-exp 6570

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