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Theorem plyco 23274
 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 23231 . . . . 5 Poly
31, 2syl 17 . . . 4
43ffvelrnda 6037 . . 3
53feqmptd 5932 . . 3
6 plyco.1 . . . 4 Poly
7 eqid 2471 . . . . 5 coeff coeff
8 eqid 2471 . . . . 5 deg deg
97, 8coeid 23271 . . . 4 Poly degcoeff
106, 9syl 17 . . 3 degcoeff
11 oveq1 6315 . . . . 5
1211oveq2d 6324 . . . 4 coeff coeff
1312sumeq2sdv 13847 . . 3 degcoeff degcoeff
144, 5, 10, 13fmptco 6072 . 2 degcoeff
15 dgrcl 23266 . . . 4 Poly deg
166, 15syl 17 . . 3 deg
17 oveq2 6316 . . . . . . . 8
1817sumeq1d 13844 . . . . . . 7 coeff coeff
1918mpteq2dv 4483 . . . . . 6 coeff coeff
2019eleq1d 2533 . . . . 5 coeff Poly coeff Poly
2120imbi2d 323 . . . 4 coeff Poly coeff Poly
22 oveq2 6316 . . . . . . . 8
2322sumeq1d 13844 . . . . . . 7 coeff coeff
2423mpteq2dv 4483 . . . . . 6 coeff coeff
2524eleq1d 2533 . . . . 5 coeff Poly coeff Poly
2625imbi2d 323 . . . 4 coeff Poly coeff Poly
27 oveq2 6316 . . . . . . . 8
2827sumeq1d 13844 . . . . . . 7 coeff coeff
2928mpteq2dv 4483 . . . . . 6 coeff coeff
3029eleq1d 2533 . . . . 5 coeff Poly coeff Poly
3130imbi2d 323 . . . 4 coeff Poly coeff Poly
32 oveq2 6316 . . . . . . . 8 deg deg
3332sumeq1d 13844 . . . . . . 7 deg coeff degcoeff
3433mpteq2dv 4483 . . . . . 6 deg coeff degcoeff
3534eleq1d 2533 . . . . 5 deg coeff Poly degcoeff Poly
3635imbi2d 323 . . . 4 deg coeff Poly degcoeff Poly
37 0z 10972 . . . . . . . . 9
384exp0d 12448 . . . . . . . . . . . 12
3938oveq2d 6324 . . . . . . . . . . 11 coeff coeff
40 plybss 23227 . . . . . . . . . . . . . . . 16 Poly
416, 40syl 17 . . . . . . . . . . . . . . 15
42 0cnd 9654 . . . . . . . . . . . . . . . 16
4342snssd 4108 . . . . . . . . . . . . . . 15
4441, 43unssd 3601 . . . . . . . . . . . . . 14
457coef 23263 . . . . . . . . . . . . . . . 16 Poly coeff
466, 45syl 17 . . . . . . . . . . . . . . 15 coeff
47 0nn0 10908 . . . . . . . . . . . . . . 15
48 ffvelrn 6035 . . . . . . . . . . . . . . 15 coeff coeff
4946, 47, 48sylancl 675 . . . . . . . . . . . . . 14 coeff
5044, 49sseldd 3419 . . . . . . . . . . . . 13 coeff
5150adantr 472 . . . . . . . . . . . 12 coeff
5251mulid1d 9678 . . . . . . . . . . 11 coeff coeff
5339, 52eqtrd 2505 . . . . . . . . . 10 coeff coeff
5453, 51eqeltrd 2549 . . . . . . . . 9 coeff
55 fveq2 5879 . . . . . . . . . . 11 coeff coeff
56 oveq2 6316 . . . . . . . . . . 11
5755, 56oveq12d 6326 . . . . . . . . . 10 coeff coeff
5857fsum1 13885 . . . . . . . . 9 coeff coeff coeff
5937, 54, 58sylancr 676 . . . . . . . 8 coeff coeff
6059, 53eqtrd 2505 . . . . . . 7 coeff coeff
6160mpteq2dva 4482 . . . . . 6 coeff coeff
62 fconstmpt 4883 . . . . . 6 coeff coeff
6361, 62syl6eqr 2523 . . . . 5 coeff coeff
64 plyconst 23239 . . . . . . 7 coeff coeff Poly
6544, 49, 64syl2anc 673 . . . . . 6 coeff Poly
66 plyun0 23230 . . . . . 6 Poly Poly
6765, 66syl6eleq 2559 . . . . 5 coeff Poly
6863, 67eqeltrd 2549 . . . 4 coeff Poly
69 simprr 774 . . . . . . . . 9 coeff Poly coeff Poly
7044adantr 472 . . . . . . . . . . . . 13
71 peano2nn0 10934 . . . . . . . . . . . . . 14
72 ffvelrn 6035 . . . . . . . . . . . . . 14 coeff coeff
7346, 71, 72syl2an 485 . . . . . . . . . . . . 13 coeff
74 plyconst 23239 . . . . . . . . . . . . 13 coeff coeff Poly
7570, 73, 74syl2anc 673 . . . . . . . . . . . 12 coeff Poly
7675, 66syl6eleq 2559 . . . . . . . . . . 11 coeff Poly
77 nn0p1nn 10933 . . . . . . . . . . . . 13
78 oveq2 6316 . . . . . . . . . . . . . . . . 17
7978mpteq2dv 4483 . . . . . . . . . . . . . . . 16
8079eleq1d 2533 . . . . . . . . . . . . . . 15 Poly Poly
8180imbi2d 323 . . . . . . . . . . . . . 14 Poly Poly
82 oveq2 6316 . . . . . . . . . . . . . . . . 17
8382mpteq2dv 4483 . . . . . . . . . . . . . . . 16
8483eleq1d 2533 . . . . . . . . . . . . . . 15 Poly Poly
8584imbi2d 323 . . . . . . . . . . . . . 14 Poly Poly
86 oveq2 6316 . . . . . . . . . . . . . . . . 17
8786mpteq2dv 4483 . . . . . . . . . . . . . . . 16
8887eleq1d 2533 . . . . . . . . . . . . . . 15 Poly Poly
8988imbi2d 323 . . . . . . . . . . . . . 14 Poly Poly
904exp1d 12449 . . . . . . . . . . . . . . . . 17
9190mpteq2dva 4482 . . . . . . . . . . . . . . . 16
9291, 5eqtr4d 2508 . . . . . . . . . . . . . . 15
9392, 1eqeltrd 2549 . . . . . . . . . . . . . 14 Poly
94 simprr 774 . . . . . . . . . . . . . . . . . . 19 Poly Poly
951adantr 472 . . . . . . . . . . . . . . . . . . 19 Poly Poly
96 plyco.3 . . . . . . . . . . . . . . . . . . . 20
9796adantlr 729 . . . . . . . . . . . . . . . . . . 19 Poly
98 plyco.4 . . . . . . . . . . . . . . . . . . . 20
9998adantlr 729 . . . . . . . . . . . . . . . . . . 19 Poly
10094, 95, 97, 99plymul 23251 . . . . . . . . . . . . . . . . . 18 Poly Poly
101100expr 626 . . . . . . . . . . . . . . . . 17 Poly Poly
102 cnex 9638 . . . . . . . . . . . . . . . . . . . . 21
103102a1i 11 . . . . . . . . . . . . . . . . . . . 20
104 ovex 6336 . . . . . . . . . . . . . . . . . . . . 21
105104a1i 11 . . . . . . . . . . . . . . . . . . . 20
1064adantlr 729 . . . . . . . . . . . . . . . . . . . 20
107 eqidd 2472 . . . . . . . . . . . . . . . . . . . 20
1085adantr 472 . . . . . . . . . . . . . . . . . . . 20
109103, 105, 106, 107, 108offval2 6567 . . . . . . . . . . . . . . . . . . 19
110 nnnn0 10900 . . . . . . . . . . . . . . . . . . . . . 22
111110ad2antlr 741 . . . . . . . . . . . . . . . . . . . . 21
112106, 111expp1d 12455 . . . . . . . . . . . . . . . . . . . 20
113112mpteq2dva 4482 . . . . . . . . . . . . . . . . . . 19
114109, 113eqtr4d 2508 . . . . . . . . . . . . . . . . . 18
115114eleq1d 2533 . . . . . . . . . . . . . . . . 17 Poly Poly
116101, 115sylibd 222 . . . . . . . . . . . . . . . 16 Poly Poly
117116expcom 442 . . . . . . . . . . . . . . 15 Poly Poly
118117a2d 28 . . . . . . . . . . . . . 14 Poly Poly
11981, 85, 89, 89, 93, 118nnind 10649 . . . . . . . . . . . . 13 Poly
12077, 119syl 17 . . . . . . . . . . . 12 Poly
121120impcom 437 . . . . . . . . . . 11 Poly
12296adantlr 729 . . . . . . . . . . 11
12398adantlr 729 . . . . . . . . . . 11
12476, 121, 122, 123plymul 23251 . . . . . . . . . 10 coeff Poly
125124adantrr 731 . . . . . . . . 9 coeff Poly coeff Poly
12696adantlr 729 . . . . . . . . 9 coeff Poly
12769, 125, 126plyadd 23250 . . . . . . . 8 coeff Poly coeff coeff Poly
128127expr 626 . . . . . . 7 coeff Poly coeff coeff Poly
129102a1i 11 . . . . . . . . . 10
130 sumex 13831 . . . . . . . . . . 11 coeff
131130a1i 11 . . . . . . . . . 10 coeff
132 ovex 6336 . . . . . . . . . . 11 coeff
133132a1i 11 . . . . . . . . . 10 coeff
134 eqidd 2472 . . . . . . . . . 10 coeff coeff
135 fvex 5889 . . . . . . . . . . . 12 coeff
136135a1i 11 . . . . . . . . . . 11 coeff
137 ovex 6336 . . . . . . . . . . . 12
138137a1i 11 . . . . . . . . . . 11
139 fconstmpt 4883 . . . . . . . . . . . 12 coeff coeff
140139a1i 11 . . . . . . . . . . 11 coeff coeff
141 eqidd 2472 . . . . . . . . . . 11
142129, 136, 138, 140, 141offval2 6567 . . . . . . . . . 10 coeff coeff
143129, 131, 133, 134, 142offval2 6567 . . . . . . . . 9 coeff coeff coeff coeff
144 simplr 770 . . . . . . . . . . . 12
145 nn0uz 11217 . . . . . . . . . . . 12
146144, 145syl6eleq 2559 . . . . . . . . . . 11
1477coef3 23265 . . . . . . . . . . . . . . 15 Poly coeff
1486, 147syl 17 . . . . . . . . . . . . . 14 coeff
149148ad2antrr 740 . . . . . . . . . . . . 13 coeff
150 elfznn0 11913 . . . . . . . . . . . . 13
151 ffvelrn 6035 . . . . . . . . . . . . 13 coeff coeff
152149, 150, 151syl2an 485 . . . . . . . . . . . 12 coeff
1534adantlr 729 . . . . . . . . . . . . 13
154 expcl 12328 . . . . . . . . . . . . 13
155153, 150, 154syl2an 485 . . . . . . . . . . . 12
156152, 155mulcld 9681 . . . . . . . . . . 11 coeff
157 fveq2 5879 . . . . . . . . . . . 12 coeff coeff
158 oveq2 6316 . . . . . . . . . . . 12
159157, 158oveq12d 6326 . . . . . . . . . . 11 coeff coeff
160146, 156, 159fsump1 13894 . . . . . . . . . 10 coeff coeff coeff
161160mpteq2dva 4482 . . . . . . . . 9 coeff coeff coeff
162143, 161eqtr4d 2508 . . . . . . . 8 coeff coeff coeff
163162eleq1d 2533 . . . . . . 7 coeff coeff Poly coeff Poly
164128, 163sylibd 222 . . . . . 6 coeff Poly coeff Poly
165164expcom 442 . . . . 5 coeff Poly coeff Poly
166165a2d 28 . . . 4 coeff Poly coeff Poly
16721, 26, 31, 36, 68, 166nn0ind 11053 . . 3 deg degcoeff Poly
16816, 167mpcom 36 . 2 degcoeff Poly
16914, 168eqeltrd 2549 1 Poly
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452   wcel 1904  cvv 3031   cun 3388   wss 3390  csn 3959   cmpt 4454   cxp 4837   ccom 4843  wf 5585  cfv 5589  (class class class)co 6308   cof 6548  cc 9555  cc0 9557  c1 9558   caddc 9560   cmul 9562  cn 10631  cn0 10893  cz 10961  cuz 11182  cfz 11810  cexp 12310  csu 13829  Polycply 23217  coeffccoe 23219  degcdgr 23220 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224 This theorem is referenced by:  dgrcolem1  23306  dgrcolem2  23307  taylply2  23402  ftalem7  24084
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