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Theorem vieta1 23265
 Description: The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree has distinct roots, then the sum over these roots can be calculated as . (If the roots are not distinct, then this formula is still true but must double-count some of the roots according to their multiplicities.) (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
vieta1.1 coeff
vieta1.2 deg
vieta1.3
vieta1.4 Poly
vieta1.5
vieta1.6
Assertion
Ref Expression
vieta1
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem vieta1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 23154 . . 3 Poly Poly
2 vieta1.4 . . 3 Poly
31, 2sseldi 3430 . 2 Poly
4 vieta1.6 . . 3
5 eqeq1 2455 . . . . . . 7 deg deg
65anbi1d 711 . . . . . 6 deg deg deg deg
76imbi1d 319 . . . . 5 deg deg coeffdeg coeffdeg deg deg coeffdeg coeffdeg
87ralbidv 2827 . . . 4 Poly deg deg coeffdeg coeffdeg Poly deg deg coeffdeg coeffdeg
9 eqeq1 2455 . . . . . . 7 deg deg
109anbi1d 711 . . . . . 6 deg deg deg deg
1110imbi1d 319 . . . . 5 deg deg coeffdeg coeffdeg deg deg coeffdeg coeffdeg
1211ralbidv 2827 . . . 4 Poly deg deg coeffdeg coeffdeg Poly deg deg coeffdeg coeffdeg
13 eqeq1 2455 . . . . . . 7 deg deg
1413anbi1d 711 . . . . . 6 deg deg deg deg
1514imbi1d 319 . . . . 5 deg deg coeffdeg coeffdeg deg deg coeffdeg coeffdeg
1615ralbidv 2827 . . . 4 Poly deg deg coeffdeg coeffdeg Poly deg deg coeffdeg coeffdeg
17 eqeq1 2455 . . . . . . 7 deg deg
1817anbi1d 711 . . . . . 6 deg deg deg deg
1918imbi1d 319 . . . . 5 deg deg coeffdeg coeffdeg deg deg coeffdeg coeffdeg
2019ralbidv 2827 . . . 4 Poly deg deg coeffdeg coeffdeg Poly deg deg coeffdeg coeffdeg
21 eqid 2451 . . . . . . . . . . . . . 14 coeff coeff
2221coef3 23186 . . . . . . . . . . . . 13 Poly coeff
2322adantr 467 . . . . . . . . . . . 12 Poly deg coeff
24 0nn0 10884 . . . . . . . . . . . 12
25 ffvelrn 6020 . . . . . . . . . . . 12 coeff coeff
2623, 24, 25sylancl 668 . . . . . . . . . . 11 Poly deg coeff
27 1nn0 10885 . . . . . . . . . . . 12
28 ffvelrn 6020 . . . . . . . . . . . 12 coeff coeff
2923, 27, 28sylancl 668 . . . . . . . . . . 11 Poly deg coeff
30 simpr 463 . . . . . . . . . . . . 13 Poly deg deg
3130fveq2d 5869 . . . . . . . . . . . 12 Poly deg coeff coeffdeg
32 ax-1ne0 9608 . . . . . . . . . . . . . . . 16
3332a1i 11 . . . . . . . . . . . . . . 15 Poly deg
3430, 33eqnetrrd 2692 . . . . . . . . . . . . . 14 Poly deg deg
35 fveq2 5865 . . . . . . . . . . . . . . . 16 deg deg
36 dgr0 23216 . . . . . . . . . . . . . . . 16 deg
3735, 36syl6eq 2501 . . . . . . . . . . . . . . 15 deg
3837necon3i 2656 . . . . . . . . . . . . . 14 deg
3934, 38syl 17 . . . . . . . . . . . . 13 Poly deg
40 eqid 2451 . . . . . . . . . . . . . . . 16 deg deg
4140, 21dgreq0 23219 . . . . . . . . . . . . . . 15 Poly coeffdeg
4241necon3bid 2668 . . . . . . . . . . . . . 14 Poly coeffdeg
4342adantr 467 . . . . . . . . . . . . 13 Poly deg coeffdeg
4439, 43mpbid 214 . . . . . . . . . . . 12 Poly deg coeffdeg
4531, 44eqnetrd 2691 . . . . . . . . . . 11 Poly deg coeff
4626, 29, 45divcld 10383 . . . . . . . . . 10 Poly deg coeff coeff
4746negcld 9973 . . . . . . . . 9 Poly deg coeff coeff
48 id 22 . . . . . . . . . 10 coeff coeff coeff coeff
4948sumsn 13807 . . . . . . . . 9 coeff coeff coeff coeff coeff coeff coeff coeff
5047, 47, 49syl2anc 667 . . . . . . . 8 Poly deg coeff coeff coeff coeff
5150adantrr 723 . . . . . . 7 Poly deg deg coeff coeff coeff coeff
52 eqid 2451 . . . . . . . . . . . . 13
5352fta1 23261 . . . . . . . . . . . 12 Poly deg
5439, 53syldan 473 . . . . . . . . . . 11 Poly deg deg
5554simpld 461 . . . . . . . . . 10 Poly deg
5655adantrr 723 . . . . . . . . 9 Poly deg deg
5721, 40coeid2 23193 . . . . . . . . . . . . . 14 Poly coeff coeff coeff coeff degcoeff coeff coeff
5847, 57syldan 473 . . . . . . . . . . . . 13 Poly deg coeff coeff degcoeff coeff coeff
5930oveq2d 6306 . . . . . . . . . . . . . 14 Poly deg deg
6059sumeq1d 13767 . . . . . . . . . . . . 13 Poly deg coeff coeff coeff degcoeff coeff coeff
61 nn0uz 11193 . . . . . . . . . . . . . . 15
62 1e0p1 11079 . . . . . . . . . . . . . . 15
63 fveq2 5865 . . . . . . . . . . . . . . . 16 coeff coeff
64 oveq2 6298 . . . . . . . . . . . . . . . 16 coeff coeff coeff coeff
6563, 64oveq12d 6308 . . . . . . . . . . . . . . 15 coeff coeff coeff coeff coeff coeff
6623ffvelrnda 6022 . . . . . . . . . . . . . . . 16 Poly deg coeff
67 expcl 12290 . . . . . . . . . . . . . . . . 17 coeff coeff coeff coeff
6847, 67sylan 474 . . . . . . . . . . . . . . . 16 Poly deg coeff coeff
6966, 68mulcld 9663 . . . . . . . . . . . . . . 15 Poly deg coeff coeff coeff
70 0z 10948 . . . . . . . . . . . . . . . . . 18
7147exp0d 12410 . . . . . . . . . . . . . . . . . . . . 21 Poly deg coeff coeff
7271oveq2d 6306 . . . . . . . . . . . . . . . . . . . 20 Poly deg coeff coeff coeff coeff
7326mulid1d 9660 . . . . . . . . . . . . . . . . . . . 20 Poly deg coeff coeff
7472, 73eqtrd 2485 . . . . . . . . . . . . . . . . . . 19 Poly deg coeff coeff coeff coeff
7574, 26eqeltrd 2529 . . . . . . . . . . . . . . . . . 18 Poly deg coeff coeff coeff
76 fveq2 5865 . . . . . . . . . . . . . . . . . . . 20 coeff coeff
77 oveq2 6298 . . . . . . . . . . . . . . . . . . . 20 coeff coeff coeff coeff
7876, 77oveq12d 6308 . . . . . . . . . . . . . . . . . . 19 coeff coeff coeff coeff coeff coeff
7978fsum1 13808 . . . . . . . . . . . . . . . . . 18 coeff coeff coeff coeff coeff coeff coeff coeff coeff
8070, 75, 79sylancr 669 . . . . . . . . . . . . . . . . 17 Poly deg coeff coeff coeff coeff coeff coeff
8180, 74eqtrd 2485 . . . . . . . . . . . . . . . 16 Poly deg coeff coeff coeff coeff
8281, 24jctil 540 . . . . . . . . . . . . . . 15 Poly deg coeff coeff coeff coeff
8347exp1d 12411 . . . . . . . . . . . . . . . . . . 19 Poly deg coeff coeff coeff coeff
8483oveq2d 6306 . . . . . . . . . . . . . . . . . 18 Poly deg coeff coeff coeff coeff coeff coeff
8529, 46mulneg2d 10072 . . . . . . . . . . . . . . . . . 18 Poly deg coeff coeff coeff coeff coeff coeff
8626, 29, 45divcan2d 10385 . . . . . . . . . . . . . . . . . . 19 Poly deg coeff coeff coeff coeff
8786negeqd 9869 . . . . . . . . . . . . . . . . . 18 Poly deg coeff coeff coeff coeff
8884, 85, 873eqtrd 2489 . . . . . . . . . . . . . . . . 17 Poly deg coeff coeff coeff coeff
8988oveq2d 6306 . . . . . . . . . . . . . . . 16 Poly deg coeff coeff coeff coeff coeff coeff
9026negidd 9976 . . . . . . . . . . . . . . . 16 Poly deg coeff coeff
9189, 90eqtrd 2485 . . . . . . . . . . . . . . 15 Poly deg coeff coeff coeff coeff
9261, 62, 65, 69, 82, 91fsump1i 13830 . . . . . . . . . . . . . 14 Poly deg coeff coeff coeff
9392simprd 465 . . . . . . . . . . . . 13 Poly deg coeff coeff coeff
9458, 60, 933eqtr2d 2491 . . . . . . . . . . . 12 Poly deg coeff coeff
95 plyf 23152 . . . . . . . . . . . . . . 15 Poly
96 ffn 5728 . . . . . . . . . . . . . . 15
9795, 96syl 17 . . . . . . . . . . . . . 14 Poly
9897adantr 467 . . . . . . . . . . . . 13 Poly deg
99 fniniseg 6003 . . . . . . . . . . . . 13 coeff coeff coeff coeff coeff coeff
10098, 99syl 17 . . . . . . . . . . . 12 Poly deg coeff coeff coeff coeff coeff coeff
10147, 94, 100mpbir2and 933 . . . . . . . . . . 11 Poly deg coeff coeff
102101snssd 4117 . . . . . . . . . 10 Poly deg coeff coeff
103102adantrr 723 . . . . . . . . 9 Poly deg deg coeff coeff
104 hashsng 12549 . . . . . . . . . . . . . 14 coeff coeff coeff coeff
10547, 104syl 17 . . . . . . . . . . . . 13 Poly deg coeff coeff
106105, 30eqtrd 2485 . . . . . . . . . . . 12 Poly deg coeff coeff deg
107106adantrr 723 . . . . . . . . . . 11 Poly deg deg coeff coeff deg
108 simprr 766 . . . . . . . . . . 11 Poly deg deg deg
109107, 108eqtr4d 2488 . . . . . . . . . 10 Poly deg deg coeff coeff
110 snfi 7650 . . . . . . . . . . . 12 coeff coeff
111 hashen 12530 . . . . . . . . . . . 12 coeff coeff coeff coeff coeff coeff
112110, 55, 111sylancr 669 . . . . . . . . . . 11 Poly deg coeff coeff coeff coeff
113112adantrr 723 . . . . . . . . . 10 Poly deg deg coeff coeff coeff coeff
114109, 113mpbid 214 . . . . . . . . 9 Poly deg deg coeff coeff
115 fisseneq 7783 . . . . . . . . 9 coeff coeff coeff coeff coeff coeff
11656, 103, 114, 115syl3anc 1268 . . . . . . . 8 Poly deg deg coeff coeff
117116sumeq1d 13767 . . . . . . 7 Poly deg deg coeff coeff
118 1m1e0 10678 . . . . . . . . . . . 12
11930oveq1d 6305 . . . . . . . . . . . 12 Poly deg deg
120118, 119syl5eqr 2499 . . . . . . . . . . 11 Poly deg deg
121120fveq2d 5869 . . . . . . . . . 10 Poly deg coeff coeffdeg
122121, 31oveq12d 6308 . . . . . . . . 9 Poly deg coeff coeff coeffdeg coeffdeg
123122negeqd 9869 . . . . . . . 8 Poly deg coeff coeff coeffdeg coeffdeg
124123adantrr 723 . . . . . . 7 Poly deg deg coeff coeff coeffdeg coeffdeg
12551, 117, 1243eqtr3d 2493 . . . . . 6 Poly deg deg coeffdeg coeffdeg
126125ex 436 . . . . 5 Poly deg deg coeffdeg coeffdeg
127126rgen 2747 . . . 4 Poly deg deg coeffdeg coeffdeg
128 id 22 . . . . . . . . . . 11
129128cbvsumv 13762 . . . . . . . . . 10
130129eqeq1i 2456 . . . . . . . . 9 coeffdeg coeffdeg coeffdeg coeffdeg
131130imbi2i 314 . . . . . . . 8 deg deg coeffdeg coeffdeg deg deg coeffdeg coeffdeg
132131ralbii 2819 . . . . . . 7 Poly deg deg coeffdeg coeffdeg Poly deg deg coeffdeg coeffdeg
133 eqid 2451 . . . . . . . . 9 coeff coeff
134 eqid 2451 . . . . . . . . 9 deg deg
135 eqid 2451 . . . . . . . . 9
136 simp1r 1033 . . . . . . . . 9 Poly Poly deg deg coeffdeg coeffdeg deg deg Poly
137 simp3r 1037 . . . . . . . . 9 Poly Poly deg deg coeffdeg coeffdeg deg deg deg
138 simp1l 1032 . . . . . . . . 9 Poly Poly deg deg coeffdeg coeffdeg deg deg
139 simp3l 1036 . . . . . . . . 9 Poly Poly deg deg coeffdeg coeffdeg deg deg deg
140 simp2 1009 . . . . . . . . . 10 Poly Poly deg deg coeffdeg coeffdeg deg deg Poly deg deg coeffdeg coeffdeg
141140, 132sylib 200 . . . . . . . . 9 Poly Poly deg deg coeffdeg coeffdeg deg deg Poly deg deg coeffdeg coeffdeg
142 eqid 2451 . . . . . . . . 9 quot quot
143133, 134, 135, 136, 137, 138, 139, 141, 142vieta1lem2 23264 . . . . . . . 8 Poly Poly deg deg coeffdeg coeffdeg deg deg coeffdeg coeffdeg
1441433exp 1207 . . . . . . 7 Poly Poly deg deg coeffdeg coeffdeg deg deg coeffdeg coeffdeg
145132, 144syl5bir 222 . . . . . 6 Poly Poly deg deg coeffdeg coeffdeg deg deg coeffdeg coeffdeg
146145ralrimdva 2806 . . . . 5 Poly deg deg coeffdeg coeffdeg Poly deg deg coeffdeg coeffdeg
147 fveq2 5865 . . . . . . . . 9 deg deg
148147eqeq2d 2461 . . . . . . . 8 deg deg
149 cnveq 5008 . . . . . . . . . . 11
150149imaeq1d 5167 . . . . . . . . . 10
151150fveq2d 5869 . . . . . . . . 9
152151, 147eqeq12d 2466 . . . . . . . 8 deg deg
153148, 152anbi12d 717 . . . . . . 7 deg deg deg deg
154150sumeq1d 13767 . . . . . . . 8
155 fveq2 5865 . . . . . . . . . . 11 coeff coeff
156147oveq1d 6305 . . . . . . . . . . 11 deg deg
157155, 156fveq12d 5871 . . . . . . . . . 10 coeffdeg coeffdeg
158155, 147fveq12d 5871 . . . . . . . . . 10 coeffdeg coeffdeg
159157, 158oveq12d 6308 . . . . . . . . 9 coeffdeg coeffdeg coeffdeg coeffdeg
160159negeqd 9869 . . . . . . . 8 coeffdeg coeffdeg coeffdeg coeffdeg
161154, 160eqeq12d 2466 . . . . . . 7 coeffdeg coeffdeg coeffdeg coeffdeg
162153, 161imbi12d 322 . . . . . 6 deg deg coeffdeg coeffdeg deg deg coeffdeg coeffdeg
163162cbvralv 3019 . . . . 5 Poly deg deg coeffdeg coeffdeg Poly deg deg coeffdeg coeffdeg
164146, 163syl6ib 230 . . . 4 Poly deg deg coeffdeg coeffdeg Poly deg deg coeffdeg coeffdeg
1658, 12, 16, 20, 127, 164nnind 10627 . . 3 Poly deg deg coeffdeg coeffdeg
1664, 165syl 17 . 2 Poly deg deg coeffdeg coeffdeg
167 vieta1.5 . 2
168 fveq2 5865 . . . . . . 7 deg deg
169168eqeq2d 2461 . . . . . 6 deg deg
170 cnveq 5008 . . . . . . . . . 10
171170imaeq1d 5167 . . . . . . . . 9
172 vieta1.3 . . . . . . . . 9
173171, 172syl6eqr 2503 . . . . . . . 8
174173fveq2d 5869 . . . . . . 7
175 vieta1.2 . . . . . . . 8 deg
176168, 175syl6eqr 2503 . . . . . . 7 deg
177174, 176eqeq12d 2466 . . . . . 6 deg
178169, 177anbi12d 717 . . . . 5 deg deg deg
179175biantrur 509 . . . . 5 deg
180178, 179syl6bbr 267 . . . 4 deg deg
181173sumeq1d 13767 . . . . 5
182 fveq2 5865 . . . . . . . . 9 coeff coeff
183 vieta1.1 . . . . . . . . 9 coeff
184182, 183syl6eqr 2503 . . . . . . . 8 coeff
185176oveq1d 6305 . . . . . . . 8 deg
186184, 185fveq12d 5871 . . . . . . 7 coeffdeg
187184, 176fveq12d 5871 . . . . . . 7 coeffdeg
188186, 187oveq12d 6308 . . . . . 6 coeffdeg coeffdeg
189188negeqd 9869 . . . . 5 coeffdeg coeffdeg
190181, 189eqeq12d 2466 . . . 4 coeffdeg coeffdeg
191180, 190imbi12d 322 . . 3 deg deg coeffdeg coeffdeg
192191rspcv 3146 . 2 Poly Poly deg deg coeffdeg coeffdeg
1933, 166, 167, 192syl3c 63 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wa 371   w3a 985   wceq 1444   wcel 1887   wne 2622  wral 2737   wss 3404  csn 3968   class class class wbr 4402   cxp 4832  ccnv 4833  cima 4837   wfn 5577  wf 5578  cfv 5582  (class class class)co 6290   cof 6529   cen 7566  cfn 7569  cc 9537  cc0 9539  c1 9540   caddc 9542   cmul 9544   cle 9676   cmin 9860  cneg 9861   cdiv 10269  cn 10609  cn0 10869  cz 10937  cfz 11784  cexp 12272  chash 12515  csu 13752  c0p 22627  Polycply 23138  cidp 23139  coeffccoe 23140  degcdgr 23141   quot cquot 23243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-0p 22628  df-ply 23142  df-idp 23143  df-coe 23144  df-dgr 23145  df-quot 23244 This theorem is referenced by:  basellem5  24011
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