Theorem discrlem2 6602

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 5946	. . . . . . 7 |- 0 < 2
2		2re 5936	. . . . . . . 8 |- 2 e. RR
3		discrlem.1	. . . . . . . 8 |- A e. RR
4	2, 3	mulgt0 5590	. . . . . . 7 |- ((0 < 2 /\ 0 < A) -> 0 < (2 x. A))
5	1, 4	mpan 694	. . . . . 6 |- (0 < A -> 0 < (2 x. A))
6	2, 3	remulcl 5318	. . . . . . 7 |- (2 x. A) e. RR
7	6	gt0ne0 5595	. . . . . 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 5765	. . . . 5 |- ((2 x. A) =/= 0 -> (B / (2 x. A)) e. RR)
11		renegclt 5420	. . . . 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 1550	. . 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 3964	. . . . . . . . . . . 12 |- (A = if(0 < A, A, 1) -> (2 x. A) = (2 x. if(0 < A, A, 1)))
19	18	opreq2d 3971	. . . . . . . . . . 11 |- (A = if(0 < A, A, 1) -> (B / (2 x. A)) = (B / (2 x. if(0 < A, A, 1))))
20	19	negeqd 5344	. . . . . . . . . 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 1517	. . . . . . . . 9 |- (A = if(0 < A, A, 1) -> D = -u(B / (2 x. if(0 < A, A, 1))))
22	21	opreq1d 3970	. . . . . . . 8 |- (A = if(0 < A, A, 1) -> (D^2) = (-u(B / (2 x. if(0 < A, A, 1)))^2))
23	17, 22	opreq12d 3973	. . . . . . 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 3971	. . . . . . 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 3973	. . . . . 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 3970	. . . . 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 2626	. . . 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 3963	. . . . . . 7 |- (A = if(0 < A, A, 1) -> (A x. C) = (if(0 < A, A, 1) x. C))
29	28	opreq2d 3971	. . . . . 6 |- (A = if(0 < A, A, 1) -> (4 x. (A x. C)) = (4 x. (if(0 < A, A, 1) x. C)))
30	29	opreq2d 3971	. . . . 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 2625	. . . 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 628	. . 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 5418	. . . . 5 |- 1 e. RR
34	3, 33	keepel 2396	. . . 4 |- if(0 < A, A, 1) e. RR
35		discrlem.3	. . . 4 |- C e. RR
36		eqid 1474	. . . 4 |- -u(B / (2 x. if(0 < A, A, 1))) = -u(B / (2 x. if(0 < A, A, 1)))
37		elimgt0 5775	. . . 4 |- 0 < if(0 < A, A, 1)
38	34, 9, 35, 36, 37	discrlem1 6601	. . 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 2380	. 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 955   e. wcel 957   =/= wne 1583  ifcif 2358   class class class wbr 2615  (class class class)co 3958  RRcr 5216  0cc0 5217  1c1 5218   + caddc 5220   x. cmul 5222   - cmin 5275  -ucneg 5276   / cdiv 5277   <_ cle 5278   < clt 5469  2c2 5918  4c4 5920  ^cexp 6513

This theorem is referenced by:  discrlem 6604

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-div 5682  df-n 5883  df-2 5927  df-3 5928  df-4 5929  df-n0 6057  df-z 6093  df-seq1 6258  df-exp 6514

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