discr1 - Metamath Proof Explorer

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Theorem discr1 13000

Description: A nonnegative quadratic form has nonnegative leading coefficient. (Contributed by Mario Carneiro, 4-Jun-2014.)

**Hypotheses**
Ref	Expression
discr.1
discr.2
discr.3
discr.4	$/\$
discr1.5

**Assertion**
Ref	Expression
discr1

Distinct variable groups:

**Proof of Theorem discr1**
Step	Hyp	Ref	Expression
1		discr1.5	. . . . 5
2		discr.2	. . . . . . . . . 10
3	2	adantr 481	. . . . . . . . 9 $/\$
4		discr.3	. . . . . . . . . . 11
5	4	adantr 481	. . . . . . . . . 10 $/\$
6		0re 10040	. . . . . . . . . 10
7		ifcl 4130	. . . . . . . . . 10 $/\$
8	5, 6, 7	sylancl 694	. . . . . . . . 9 $/\$
9	3, 8	readdcld 10069	. . . . . . . 8 $/\$
10		peano2re 10209	. . . . . . . 8
11	9, 10	syl 17	. . . . . . 7 $/\$
12		discr.1	. . . . . . . . 9
13	12	adantr 481	. . . . . . . 8 $/\$
14	13	renegcld 10457	. . . . . . 7 $/\$
15	12	lt0neg1d 10597	. . . . . . . . 9
16	15	biimpa 501	. . . . . . . 8 $/\$
17	16	gt0ne0d 10592	. . . . . . 7 $/\$
18	11, 14, 17	redivcld 10853	. . . . . 6 $/\$
19		1re 10039	. . . . . 6
20		ifcl 4130	. . . . . 6 $/\$
21	18, 19, 20	sylancl 694	. . . . 5 $/\$
22	1, 21	syl5eqel 2705	. . . 4 $/\$
23		discr.4	. . . . . 6 $/\$
24	23	ralrimiva 2966	. . . . 5
25	24	adantr 481	. . . 4 $/\$
26		oveq1 6657	. . . . . . . . 9
27	26	oveq2d 6666	. . . . . . . 8
28		oveq2 6658	. . . . . . . 8
29	27, 28	oveq12d 6668	. . . . . . 7
30	29	oveq1d 6665	. . . . . 6
31	30	breq2d 4665	. . . . 5
32	31	rspcv 3305	. . . 4
33	22, 25, 32	sylc 65	. . 3 $/\$
34		resqcl 12931	. . . . . . . . 9
35	22, 34	syl 17	. . . . . . . 8 $/\$
36	13, 35	remulcld 10070	. . . . . . 7 $/\$
37	3, 22	remulcld 10070	. . . . . . 7 $/\$
38	36, 37	readdcld 10069	. . . . . 6 $/\$
39	38, 5	readdcld 10069	. . . . 5 $/\$
40	13, 22	remulcld 10070	. . . . . . 7 $/\$
41	40, 9	readdcld 10069	. . . . . 6 $/\$
42	41, 22	remulcld 10070	. . . . 5 $/\$
43	6	a1i 11	. . . . 5 $/\$
44	8, 22	remulcld 10070	. . . . . . 7 $/\$
45		max2 12018	. . . . . . . . 9 $/\$
46	6, 5, 45	sylancr 695	. . . . . . . 8 $/\$
47		max1 12016	. . . . . . . . . 10 $/\$
48	6, 5, 47	sylancr 695	. . . . . . . . 9 $/\$
49		max1 12016	. . . . . . . . . . 11 $/\$
50	19, 18, 49	sylancr 695	. . . . . . . . . 10 $/\$
51	50, 1	syl6breqr 4695	. . . . . . . . 9 $/\$
52	8, 22, 48, 51	lemulge11d 10961	. . . . . . . 8 $/\$
53	5, 8, 44, 46, 52	letrd 10194	. . . . . . 7 $/\$
54	5, 44, 38, 53	leadd2dd 10642	. . . . . 6 $/\$
55	40, 3	readdcld 10069	. . . . . . . . 9 $/\$
56	55	recnd 10068	. . . . . . . 8 $/\$
57	8	recnd 10068	. . . . . . . 8 $/\$
58	22	recnd 10068	. . . . . . . 8 $/\$
59	56, 57, 58	adddird 10065	. . . . . . 7 $/\$
60	40	recnd 10068	. . . . . . . . 9 $/\$
61	3	recnd 10068	. . . . . . . . 9 $/\$
62	60, 61, 57	addassd 10062	. . . . . . . 8 $/\$
63	62	oveq1d 6665	. . . . . . 7 $/\$
64	60, 61, 58	adddird 10065	. . . . . . . . 9 $/\$
65	13	recnd 10068	. . . . . . . . . . . 12 $/\$
66	65, 58, 58	mulassd 10063	. . . . . . . . . . 11 $/\$
67		sqval 12922	. . . . . . . . . . . . 13
68	58, 67	syl 17	. . . . . . . . . . . 12 $/\$
69	68	oveq2d 6666	. . . . . . . . . . 11 $/\$
70	66, 69	eqtr4d 2659	. . . . . . . . . 10 $/\$
71	70	oveq1d 6665	. . . . . . . . 9 $/\$
72	64, 71	eqtrd 2656	. . . . . . . 8 $/\$
73	72	oveq1d 6665	. . . . . . 7 $/\$
74	59, 63, 73	3eqtr3d 2664	. . . . . 6 $/\$
75	54, 74	breqtrrd 4681	. . . . 5 $/\$
76	14, 22	remulcld 10070	. . . . . . . . . 10 $/\$
77	9	ltp1d 10954	. . . . . . . . . 10 $/\$
78		max2 12018	. . . . . . . . . . . . 13 $/\$
79	19, 18, 78	sylancr 695	. . . . . . . . . . . 12 $/\$
80	79, 1	syl6breqr 4695	. . . . . . . . . . 11 $/\$
81		ledivmul 10899	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
82	11, 22, 14, 16, 81	syl112anc 1330	. . . . . . . . . . 11 $/\$
83	80, 82	mpbid 222	. . . . . . . . . 10 $/\$
84	9, 11, 76, 77, 83	ltletrd 10197	. . . . . . . . 9 $/\$
85	65, 58	mulneg1d 10483	. . . . . . . . . 10 $/\$
86		df-neg 10269	. . . . . . . . . 10
87	85, 86	syl6eq 2672	. . . . . . . . 9 $/\$
88	84, 87	breqtrd 4679	. . . . . . . 8 $/\$
89	40, 9, 43	ltaddsub2d 10628	. . . . . . . 8 $/\$
90	88, 89	mpbird 247	. . . . . . 7 $/\$
91	19	a1i 11	. . . . . . . . 9 $/\$
92		0lt1 10550	. . . . . . . . . 10
93	92	a1i 11	. . . . . . . . 9 $/\$
94	43, 91, 22, 93, 51	ltletrd 10197	. . . . . . . 8 $/\$
95		ltmul1 10873	. . . . . . . 8 $/\$ $/\$ $/\$
96	41, 43, 22, 94, 95	syl112anc 1330	. . . . . . 7 $/\$
97	90, 96	mpbid 222	. . . . . 6 $/\$
98	58	mul02d 10234	. . . . . 6 $/\$
99	97, 98	breqtrd 4679	. . . . 5 $/\$
100	39, 42, 43, 75, 99	lelttrd 10195	. . . 4 $/\$
101		ltnle 10117	. . . . 5 $/\$
102	39, 6, 101	sylancl 694	. . . 4 $/\$
103	100, 102	mpbid 222	. . 3 $/\$
104	33, 103	pm2.65da 600	. 2
105		lelttric 10144	. . . 4 $/\$ $\/$
106	6, 12, 105	sylancr 695	. . 3 $\/$
107	106	ord 392	. 2
108	104, 107	mt3d 140	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

cif 4086 class class class wbr 4653 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cmin 10266

cneg 10267

cdiv 10684

c2 11070

cexp 12860

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861

This theorem is referenced by: discr 13001

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