Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  plyco Unicode version

Theorem plyco 20113
 Description: The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
plyco.1 Poly
plyco.2 Poly
plyco.3
plyco.4
Assertion
Ref Expression
plyco Poly
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem plyco
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyco.2 . . . . 5 Poly
2 plyf 20070 . . . . 5 Poly
31, 2syl 16 . . . 4
43ffvelrnda 5829 . . 3
53feqmptd 5738 . . 3
6 plyco.1 . . . 4 Poly
7 eqid 2404 . . . . 5 coeff coeff
8 eqid 2404 . . . . 5 deg deg
97, 8coeid 20110 . . . 4 Poly degcoeff
106, 9syl 16 . . 3 degcoeff
11 oveq1 6047 . . . . 5
1211oveq2d 6056 . . . 4 coeff coeff
1312sumeq2sdv 12453 . . 3 degcoeff degcoeff
144, 5, 10, 13fmptco 5860 . 2 degcoeff
15 dgrcl 20105 . . . 4 Poly deg
166, 15syl 16 . . 3 deg
17 oveq2 6048 . . . . . . . 8
1817sumeq1d 12450 . . . . . . 7 coeff coeff
1918mpteq2dv 4256 . . . . . 6 coeff coeff
2019eleq1d 2470 . . . . 5 coeff Poly coeff Poly
2120imbi2d 308 . . . 4 coeff Poly coeff Poly
22 oveq2 6048 . . . . . . . 8
2322sumeq1d 12450 . . . . . . 7 coeff coeff
2423mpteq2dv 4256 . . . . . 6 coeff coeff
2524eleq1d 2470 . . . . 5 coeff Poly coeff Poly
2625imbi2d 308 . . . 4 coeff Poly coeff Poly
27 oveq2 6048 . . . . . . . 8
2827sumeq1d 12450 . . . . . . 7 coeff coeff
2928mpteq2dv 4256 . . . . . 6 coeff coeff
3029eleq1d 2470 . . . . 5 coeff Poly coeff Poly
3130imbi2d 308 . . . 4 coeff Poly coeff Poly
32 oveq2 6048 . . . . . . . 8 deg deg
3332sumeq1d 12450 . . . . . . 7 deg coeff degcoeff
3433mpteq2dv 4256 . . . . . 6 deg coeff degcoeff
3534eleq1d 2470 . . . . 5 deg coeff Poly degcoeff Poly
3635imbi2d 308 . . . 4 deg coeff Poly degcoeff Poly
37 0z 10249 . . . . . . . . 9
384exp0d 11472 . . . . . . . . . . . 12
3938oveq2d 6056 . . . . . . . . . . 11 coeff coeff
40 plybss 20066 . . . . . . . . . . . . . . . 16 Poly
416, 40syl 16 . . . . . . . . . . . . . . 15
42 0cn 9040 . . . . . . . . . . . . . . . . 17
4342a1i 11 . . . . . . . . . . . . . . . 16
4443snssd 3903 . . . . . . . . . . . . . . 15
4541, 44unssd 3483 . . . . . . . . . . . . . 14
467coef 20102 . . . . . . . . . . . . . . . 16 Poly coeff
476, 46syl 16 . . . . . . . . . . . . . . 15 coeff
48 0nn0 10192 . . . . . . . . . . . . . . 15
49 ffvelrn 5827 . . . . . . . . . . . . . . 15 coeff coeff
5047, 48, 49sylancl 644 . . . . . . . . . . . . . 14 coeff
5145, 50sseldd 3309 . . . . . . . . . . . . 13 coeff
5251adantr 452 . . . . . . . . . . . 12 coeff
5352mulid1d 9061 . . . . . . . . . . 11 coeff coeff
5439, 53eqtrd 2436 . . . . . . . . . 10 coeff coeff
5554, 52eqeltrd 2478 . . . . . . . . 9 coeff
56 fveq2 5687 . . . . . . . . . . 11 coeff coeff
57 oveq2 6048 . . . . . . . . . . 11
5856, 57oveq12d 6058 . . . . . . . . . 10 coeff coeff
5958fsum1 12490 . . . . . . . . 9 coeff coeff coeff
6037, 55, 59sylancr 645 . . . . . . . 8 coeff coeff
6160, 54eqtrd 2436 . . . . . . 7 coeff coeff
6261mpteq2dva 4255 . . . . . 6 coeff coeff
63 fconstmpt 4880 . . . . . 6 coeff coeff
6462, 63syl6eqr 2454 . . . . 5 coeff coeff
65 plyconst 20078 . . . . . . 7 coeff coeff Poly
6645, 50, 65syl2anc 643 . . . . . 6 coeff Poly
67 plyun0 20069 . . . . . 6 Poly Poly
6866, 67syl6eleq 2494 . . . . 5 coeff Poly
6964, 68eqeltrd 2478 . . . 4 coeff Poly
70 simprr 734 . . . . . . . . 9 coeff Poly coeff Poly
7145adantr 452 . . . . . . . . . . . . 13
72 peano2nn0 10216 . . . . . . . . . . . . . 14
73 ffvelrn 5827 . . . . . . . . . . . . . 14 coeff coeff
7447, 72, 73syl2an 464 . . . . . . . . . . . . 13 coeff
75 plyconst 20078 . . . . . . . . . . . . 13 coeff coeff Poly
7671, 74, 75syl2anc 643 . . . . . . . . . . . 12 coeff Poly
7776, 67syl6eleq 2494 . . . . . . . . . . 11 coeff Poly
78 nn0p1nn 10215 . . . . . . . . . . . . 13
79 oveq2 6048 . . . . . . . . . . . . . . . . 17
8079mpteq2dv 4256 . . . . . . . . . . . . . . . 16
8180eleq1d 2470 . . . . . . . . . . . . . . 15 Poly Poly
8281imbi2d 308 . . . . . . . . . . . . . 14 Poly Poly
83 oveq2 6048 . . . . . . . . . . . . . . . . 17
8483mpteq2dv 4256 . . . . . . . . . . . . . . . 16
8584eleq1d 2470 . . . . . . . . . . . . . . 15 Poly Poly
8685imbi2d 308 . . . . . . . . . . . . . 14 Poly Poly
87 oveq2 6048 . . . . . . . . . . . . . . . . 17
8887mpteq2dv 4256 . . . . . . . . . . . . . . . 16
8988eleq1d 2470 . . . . . . . . . . . . . . 15 Poly Poly
9089imbi2d 308 . . . . . . . . . . . . . 14 Poly Poly
914exp1d 11473 . . . . . . . . . . . . . . . . 17
9291mpteq2dva 4255 . . . . . . . . . . . . . . . 16
9392, 5eqtr4d 2439 . . . . . . . . . . . . . . 15
9493, 1eqeltrd 2478 . . . . . . . . . . . . . 14 Poly
95 simprr 734 . . . . . . . . . . . . . . . . . . 19 Poly Poly
961adantr 452 . . . . . . . . . . . . . . . . . . 19 Poly Poly
97 plyco.3 . . . . . . . . . . . . . . . . . . . 20
9897adantlr 696 . . . . . . . . . . . . . . . . . . 19 Poly
99 plyco.4 . . . . . . . . . . . . . . . . . . . 20
10099adantlr 696 . . . . . . . . . . . . . . . . . . 19 Poly
10195, 96, 98, 100plymul 20090 . . . . . . . . . . . . . . . . . 18 Poly Poly
102101expr 599 . . . . . . . . . . . . . . . . 17 Poly Poly
103 cnex 9027 . . . . . . . . . . . . . . . . . . . . 21
104103a1i 11 . . . . . . . . . . . . . . . . . . . 20
105 ovex 6065 . . . . . . . . . . . . . . . . . . . . 21
106105a1i 11 . . . . . . . . . . . . . . . . . . . 20
1074adantlr 696 . . . . . . . . . . . . . . . . . . . 20
108 eqidd 2405 . . . . . . . . . . . . . . . . . . . 20
1095adantr 452 . . . . . . . . . . . . . . . . . . . 20
110104, 106, 107, 108, 109offval2 6281 . . . . . . . . . . . . . . . . . . 19
111 nnnn0 10184 . . . . . . . . . . . . . . . . . . . . . 22
112111ad2antlr 708 . . . . . . . . . . . . . . . . . . . . 21
113107, 112expp1d 11479 . . . . . . . . . . . . . . . . . . . 20
114113mpteq2dva 4255 . . . . . . . . . . . . . . . . . . 19
115110, 114eqtr4d 2439 . . . . . . . . . . . . . . . . . 18
116115eleq1d 2470 . . . . . . . . . . . . . . . . 17 Poly Poly
117102, 116sylibd 206 . . . . . . . . . . . . . . . 16 Poly Poly
118117expcom 425 . . . . . . . . . . . . . . 15 Poly Poly
119118a2d 24 . . . . . . . . . . . . . 14 Poly Poly
12082, 86, 90, 90, 94, 119nnind 9974 . . . . . . . . . . . . 13 Poly
12178, 120syl 16 . . . . . . . . . . . 12 Poly
122121impcom 420 . . . . . . . . . . 11 Poly
12397adantlr 696 . . . . . . . . . . 11
12499adantlr 696 . . . . . . . . . . 11
12577, 122, 123, 124plymul 20090 . . . . . . . . . 10 coeff Poly
126125adantrr 698 . . . . . . . . 9 coeff Poly coeff Poly
12797adantlr 696 . . . . . . . . 9 coeff Poly
12870, 126, 127plyadd 20089 . . . . . . . 8 coeff Poly coeff coeff Poly
129128expr 599 . . . . . . 7 coeff Poly coeff coeff Poly
130103a1i 11 . . . . . . . . . 10
131 sumex 12436 . . . . . . . . . . 11 coeff
132131a1i 11 . . . . . . . . . 10 coeff
133 ovex 6065 . . . . . . . . . . 11 coeff
134133a1i 11 . . . . . . . . . 10 coeff
135 eqidd 2405 . . . . . . . . . 10 coeff coeff
136 fvex 5701 . . . . . . . . . . . 12 coeff
137136a1i 11 . . . . . . . . . . 11 coeff
138 ovex 6065 . . . . . . . . . . . 12
139138a1i 11 . . . . . . . . . . 11
140 fconstmpt 4880 . . . . . . . . . . . 12 coeff coeff
141140a1i 11 . . . . . . . . . . 11 coeff coeff
142 eqidd 2405 . . . . . . . . . . 11
143130, 137, 139, 141, 142offval2 6281 . . . . . . . . . 10 coeff coeff
144130, 132, 134, 135, 143offval2 6281 . . . . . . . . 9 coeff coeff coeff coeff
145 simplr 732 . . . . . . . . . . . 12
146 nn0uz 10476 . . . . . . . . . . . 12
147145, 146syl6eleq 2494 . . . . . . . . . . 11
1487coef3 20104 . . . . . . . . . . . . . . 15 Poly coeff
1496, 148syl 16 . . . . . . . . . . . . . 14 coeff
150149ad2antrr 707 . . . . . . . . . . . . 13 coeff
151 elfznn0 11039 . . . . . . . . . . . . 13
152 ffvelrn 5827 . . . . . . . . . . . . 13 coeff coeff
153150, 151, 152syl2an 464 . . . . . . . . . . . 12 coeff
1544adantlr 696 . . . . . . . . . . . . 13
155 expcl 11354 . . . . . . . . . . . . 13
156154, 151, 155syl2an 464 . . . . . . . . . . . 12
157153, 156mulcld 9064 . . . . . . . . . . 11 coeff
158 fveq2 5687 . . . . . . . . . . . 12 coeff coeff
159 oveq2 6048 . . . . . . . . . . . 12
160158, 159oveq12d 6058 . . . . . . . . . . 11 coeff coeff
161147, 157, 160fsump1 12495 . . . . . . . . . 10 coeff coeff coeff
162161mpteq2dva 4255 . . . . . . . . 9 coeff coeff coeff
163144, 162eqtr4d 2439 . . . . . . . 8 coeff coeff coeff
164163eleq1d 2470 . . . . . . 7 coeff coeff Poly coeff Poly
165129, 164sylibd 206 . . . . . 6 coeff Poly coeff Poly
166165expcom 425 . . . . 5 coeff Poly coeff Poly
167166a2d 24 . . . 4 coeff Poly coeff Poly
16821, 26, 31, 36, 69, 167nn0ind 10322 . . 3 deg degcoeff Poly
16916, 168mpcom 34 . 2 degcoeff Poly
17014, 169eqeltrd 2478 1 Poly
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1649   wcel 1721  cvv 2916   cun 3278   wss 3280  csn 3774   cmpt 4226   cxp 4835   ccom 4841  wf 5409  cfv 5413  (class class class)co 6040   cof 6262  cc 8944  cc0 8946  c1 8947   caddc 8949   cmul 8951  cn 9956  cn0 10177  cz 10238  cuz 10444  cfz 10999  cexp 11337  csu 12434  Polycply 20056  coeffccoe 20058  degcdgr 20059 This theorem is referenced by:  dgrcolem1  20144  dgrcolem2  20145  taylply2  20237  ftalem7  20814 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063
 Copyright terms: Public domain W3C validator