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Theorem plymulx0 28160
 Description: Coefficients of a polynomial multiplyed by . (Contributed by Thierry Arnoux, 25-Sep-2018.)
Assertion
Ref Expression
plymulx0 Poly coeff coeff
Distinct variable group:   ,

Proof of Theorem plymulx0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 3626 . . . . . 6 Poly Poly
2 ax-resscn 9548 . . . . . . . 8
3 1re 9594 . . . . . . . 8
4 plyid 22357 . . . . . . . 8 Poly
52, 3, 4mp2an 672 . . . . . . 7 Poly
65a1i 11 . . . . . 6 Poly Poly
7 simprl 755 . . . . . . 7 Poly
8 simprr 756 . . . . . . 7 Poly
97, 8readdcld 9622 . . . . . 6 Poly
107, 8remulcld 9623 . . . . . 6 Poly
111, 6, 9, 10plymul 22366 . . . . 5 Poly Poly
12 0re 9595 . . . . 5
13 eqid 2467 . . . . . 6 coeff coeff
1413coef2 22379 . . . . 5 Poly coeff
1511, 12, 14sylancl 662 . . . 4 Poly coeff
16 ffn 5730 . . . 4 coeff coeff
1715, 16syl 16 . . 3 Poly coeff
18 dffn5 5912 . . 3 coeff coeff coeff
1917, 18sylib 196 . 2 Poly coeff coeff
20 cnex 9572 . . . . . . . . 9
2120a1i 11 . . . . . . . 8 Poly
22 plyf 22346 . . . . . . . . 9 Poly
231, 22syl 16 . . . . . . . 8 Poly
24 plyf 22346 . . . . . . . . . 10 Poly
255, 24ax-mp 5 . . . . . . . . 9
2625a1i 11 . . . . . . . 8 Poly
27 simprl 755 . . . . . . . . 9 Poly
28 simprr 756 . . . . . . . . 9 Poly
2927, 28mulcomd 9616 . . . . . . . 8 Poly
3021, 23, 26, 29caofcom 6555 . . . . . . 7 Poly
3130fveq2d 5869 . . . . . 6 Poly coeff coeff
3231fveq1d 5867 . . . . 5 Poly coeff coeff
3332adantr 465 . . . 4 Poly coeff coeff
345a1i 11 . . . . . 6 Poly Poly
351adantr 465 . . . . . 6 Poly Poly
36 simpr 461 . . . . . 6 Poly
37 eqid 2467 . . . . . . 7 coeff coeff
38 eqid 2467 . . . . . . 7 coeff coeff
3937, 38coemul 22399 . . . . . 6 Poly Poly coeff coeff coeff
4034, 35, 36, 39syl3anc 1228 . . . . 5 Poly coeff coeff coeff
41 elfznn0 11769 . . . . . . . . . 10
42 coeidp 22410 . . . . . . . . . 10 coeff
4341, 42syl 16 . . . . . . . . 9 coeff
4443oveq1d 6298 . . . . . . . 8 coeff coeff coeff
45 ovif 6362 . . . . . . . 8 coeff coeff coeff
4644, 45syl6eq 2524 . . . . . . 7 coeff coeff coeff coeff
4746adantl 466 . . . . . 6 Poly coeff coeff coeff coeff
4847sumeq2dv 13487 . . . . 5 Poly coeff coeff coeff coeff
49 elsn 4041 . . . . . . . . . 10
5049bicomi 202 . . . . . . . . 9
5150a1i 11 . . . . . . . 8 Poly
5238coef2 22379 . . . . . . . . . . . . 13 Poly coeff
531, 12, 52sylancl 662 . . . . . . . . . . . 12 Poly coeff
5453ad2antrr 725 . . . . . . . . . . 11 Poly coeff
55 fznn0sub 11715 . . . . . . . . . . . 12
5655adantl 466 . . . . . . . . . . 11 Poly
5754, 56ffvelrnd 6021 . . . . . . . . . 10 Poly coeff
582, 57sseldi 3502 . . . . . . . . 9 Poly coeff
5958mulid2d 9613 . . . . . . . 8 Poly coeff coeff
6058mul02d 9776 . . . . . . . 8 Poly coeff
6151, 59, 60ifbieq12d 3966 . . . . . . 7 Poly coeff coeff coeff
6261sumeq2dv 13487 . . . . . 6 Poly coeff coeff coeff
63 eqeq2 2482 . . . . . . 7 coeff coeff coeff coeff
64 eqeq2 2482 . . . . . . 7 coeff coeff coeff coeff coeff coeff
65 oveq2 6291 . . . . . . . . . . 11
66 0z 10874 . . . . . . . . . . . 12
67 fzsn 11724 . . . . . . . . . . . 12
6866, 67ax-mp 5 . . . . . . . . . . 11
6965, 68syl6eq 2524 . . . . . . . . . 10
70 elsni 4052 . . . . . . . . . . . . 13
7170adantl 466 . . . . . . . . . . . 12
72 ax-1ne0 9560 . . . . . . . . . . . . . 14
7372nesymi 2740 . . . . . . . . . . . . 13
74 eqeq1 2471 . . . . . . . . . . . . 13
7573, 74mtbiri 303 . . . . . . . . . . . 12
7671, 75syl 16 . . . . . . . . . . 11
7750notbii 296 . . . . . . . . . . . 12
7877biimpi 194 . . . . . . . . . . 11
79 iffalse 3948 . . . . . . . . . . 11 coeff
8076, 78, 793syl 20 . . . . . . . . . 10 coeff
8169, 80sumeq12rdv 13491 . . . . . . . . 9 coeff
82 snfi 7596 . . . . . . . . . . 11
8382olci 391 . . . . . . . . . 10
84 sumz 13506 . . . . . . . . . 10
8583, 84ax-mp 5 . . . . . . . . 9
8681, 85syl6eq 2524 . . . . . . . 8 coeff
8786adantl 466 . . . . . . 7 Poly coeff
88 simpll 753 . . . . . . . 8 Poly Poly
8936adantr 465 . . . . . . . . . 10 Poly
90 simpr 461 . . . . . . . . . . 11 Poly
9190neqned 2670 . . . . . . . . . 10 Poly
9289, 91jca 532 . . . . . . . . 9 Poly
93 elnnne0 10808 . . . . . . . . 9
9492, 93sylibr 212 . . . . . . . 8 Poly
95 1nn0 10810 . . . . . . . . . . . . 13
9695a1i 11 . . . . . . . . . . . 12 Poly
97 simpr 461 . . . . . . . . . . . . 13 Poly
9897nnnn0d 10851 . . . . . . . . . . . 12 Poly
99 nnge1 10561 . . . . . . . . . . . . 13
10097, 99syl 16 . . . . . . . . . . . 12 Poly
101 elfz2nn0 11767 . . . . . . . . . . . 12
10296, 98, 100, 101syl3anbrc 1180 . . . . . . . . . . 11 Poly
1033elexi 3123 . . . . . . . . . . . 12
104103snss 4151 . . . . . . . . . . 11
105102, 104sylib 196 . . . . . . . . . 10 Poly
10653ad2antrr 725 . . . . . . . . . . . . 13 Poly coeff
107 oveq2 6291 . . . . . . . . . . . . . . . 16
10849, 107sylbi 195 . . . . . . . . . . . . . . 15
109108adantl 466 . . . . . . . . . . . . . 14 Poly
110 nnm1nn0 10836 . . . . . . . . . . . . . . 15
111110ad2antlr 726 . . . . . . . . . . . . . 14 Poly
112109, 111eqeltrd 2555 . . . . . . . . . . . . 13 Poly
113106, 112ffvelrnd 6021 . . . . . . . . . . . 12 Poly coeff
1142, 113sseldi 3502 . . . . . . . . . . 11 Poly coeff
115114ralrimiva 2878 . . . . . . . . . 10 Poly coeff
116 fzfi 12049 . . . . . . . . . . . 12
117116olci 391 . . . . . . . . . . 11
118117a1i 11 . . . . . . . . . 10 Poly
119 sumss2 13510 . . . . . . . . . 10 coeff coeff coeff
120105, 115, 118, 119syl21anc 1227 . . . . . . . . 9 Poly coeff coeff
12153adantr 465 . . . . . . . . . . . 12 Poly coeff
12297, 110syl 16 . . . . . . . . . . . 12 Poly
123121, 122ffvelrnd 6021 . . . . . . . . . . 11 Poly coeff
1242, 123sseldi 3502 . . . . . . . . . 10 Poly coeff
125107fveq2d 5869 . . . . . . . . . . 11 coeff coeff
126125sumsn 13525 . . . . . . . . . 10 coeff coeff coeff
1273, 124, 126sylancr 663 . . . . . . . . 9 Poly coeff coeff
128120, 127eqtr3d 2510 . . . . . . . 8 Poly coeff coeff
12988, 94, 128syl2anc 661 . . . . . . 7 Poly coeff coeff
13063, 64, 87, 129ifbothda 3974 . . . . . 6 Poly coeff coeff
13162, 130eqtrd 2508 . . . . 5 Poly coeff coeff coeff
13240, 48, 1313eqtrd 2512 . . . 4 Poly coeff coeff
13333, 132eqtrd 2508 . . 3 Poly coeff coeff
134133mpteq2dva 4533 . 2 Poly coeff coeff
13519, 134eqtrd 2508 1 Poly coeff coeff
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wo 368   wa 369   wceq 1379   wcel 1767   wne 2662  wral 2814  cvv 3113   cdif 3473   wss 3476  cif 3939  csn 4027   class class class wbr 4447   cmpt 4505   wfn 5582  wf 5583  cfv 5587  (class class class)co 6283   cof 6521  cfn 7516  cc 9489  cr 9490  cc0 9491  c1 9492   cmul 9496   cle 9628   cmin 9804  cn 10535  cn0 10794  cz 10863  cuz 11081  cfz 11671  csu 13470  c0p 21827  Polycply 22332  cidp 22333  coeffccoe 22334 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-rp 11220  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-rlim 13274  df-sum 13471  df-0p 21828  df-ply 22336  df-idp 22337  df-coe 22338  df-dgr 22339 This theorem is referenced by:  plymulx  28161
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