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Theorem plymulx0 29508
 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 3544 . . . . 5 Poly Poly
2 ax-resscn 9614 . . . . . . 7
3 1re 9660 . . . . . . 7
4 plyid 23242 . . . . . . 7 Poly
52, 3, 4mp2an 686 . . . . . 6 Poly
65a1i 11 . . . . 5 Poly Poly
7 simprl 772 . . . . . 6 Poly
8 simprr 774 . . . . . 6 Poly
97, 8readdcld 9688 . . . . 5 Poly
107, 8remulcld 9689 . . . . 5 Poly
111, 6, 9, 10plymul 23251 . . . 4 Poly Poly
12 0re 9661 . . . 4
13 eqid 2471 . . . . 5 coeff coeff
1413coef2 23264 . . . 4 Poly coeff
1511, 12, 14sylancl 675 . . 3 Poly coeff
1615feqmptd 5932 . 2 Poly coeff coeff
17 cnex 9638 . . . . . . . . 9
1817a1i 11 . . . . . . . 8 Poly
19 plyf 23231 . . . . . . . . 9 Poly
201, 19syl 17 . . . . . . . 8 Poly
21 plyf 23231 . . . . . . . . . 10 Poly
225, 21ax-mp 5 . . . . . . . . 9
2322a1i 11 . . . . . . . 8 Poly
24 simprl 772 . . . . . . . . 9 Poly
25 simprr 774 . . . . . . . . 9 Poly
2624, 25mulcomd 9682 . . . . . . . 8 Poly
2718, 20, 23, 26caofcom 6582 . . . . . . 7 Poly
2827fveq2d 5883 . . . . . 6 Poly coeff coeff
2928fveq1d 5881 . . . . 5 Poly coeff coeff
3029adantr 472 . . . 4 Poly coeff coeff
315a1i 11 . . . . . 6 Poly Poly
321adantr 472 . . . . . 6 Poly Poly
33 simpr 468 . . . . . 6 Poly
34 eqid 2471 . . . . . . 7 coeff coeff
35 eqid 2471 . . . . . . 7 coeff coeff
3634, 35coemul 23285 . . . . . 6 Poly Poly coeff coeff coeff
3731, 32, 33, 36syl3anc 1292 . . . . 5 Poly coeff coeff coeff
38 elfznn0 11913 . . . . . . . . . 10
39 coeidp 23296 . . . . . . . . . 10 coeff
4038, 39syl 17 . . . . . . . . 9 coeff
4140oveq1d 6323 . . . . . . . 8 coeff coeff coeff
42 ovif 6392 . . . . . . . 8 coeff coeff coeff
4341, 42syl6eq 2521 . . . . . . 7 coeff coeff coeff coeff
4443adantl 473 . . . . . 6 Poly coeff coeff coeff coeff
4544sumeq2dv 13846 . . . . 5 Poly coeff coeff coeff coeff
46 elsn 3973 . . . . . . . . . 10
4746bicomi 207 . . . . . . . . 9
4847a1i 11 . . . . . . . 8 Poly
4935coef2 23264 . . . . . . . . . . . . 13 Poly coeff
501, 12, 49sylancl 675 . . . . . . . . . . . 12 Poly coeff
5150ad2antrr 740 . . . . . . . . . . 11 Poly coeff
52 fznn0sub 11857 . . . . . . . . . . . 12
5352adantl 473 . . . . . . . . . . 11 Poly
5451, 53ffvelrnd 6038 . . . . . . . . . 10 Poly coeff
5554recnd 9687 . . . . . . . . 9 Poly coeff
5655mulid2d 9679 . . . . . . . 8 Poly coeff coeff
5755mul02d 9849 . . . . . . . 8 Poly coeff
5848, 56, 57ifbieq12d 3899 . . . . . . 7 Poly coeff coeff coeff
5958sumeq2dv 13846 . . . . . 6 Poly coeff coeff coeff
60 eqeq2 2482 . . . . . . 7 coeff coeff coeff coeff
61 eqeq2 2482 . . . . . . 7 coeff coeff coeff coeff coeff coeff
62 oveq2 6316 . . . . . . . . . . 11
63 0z 10972 . . . . . . . . . . . 12
64 fzsn 11866 . . . . . . . . . . . 12
6563, 64ax-mp 5 . . . . . . . . . . 11
6662, 65syl6eq 2521 . . . . . . . . . 10
67 elsni 3985 . . . . . . . . . . . . 13
6867adantl 473 . . . . . . . . . . . 12
69 ax-1ne0 9626 . . . . . . . . . . . . . 14
7069nesymi 2700 . . . . . . . . . . . . 13
71 eqeq1 2475 . . . . . . . . . . . . 13
7270, 71mtbiri 310 . . . . . . . . . . . 12
7368, 72syl 17 . . . . . . . . . . 11
7447notbii 303 . . . . . . . . . . . 12
7574biimpi 199 . . . . . . . . . . 11
76 iffalse 3881 . . . . . . . . . . 11 coeff
7773, 75, 763syl 18 . . . . . . . . . 10 coeff
7866, 77sumeq12rdv 13850 . . . . . . . . 9 coeff
79 snfi 7668 . . . . . . . . . . 11
8079olci 398 . . . . . . . . . 10
81 sumz 13865 . . . . . . . . . 10
8280, 81ax-mp 5 . . . . . . . . 9
8378, 82syl6eq 2521 . . . . . . . 8 coeff
8483adantl 473 . . . . . . 7 Poly coeff
85 simpll 768 . . . . . . . 8 Poly Poly
8633adantr 472 . . . . . . . . 9 Poly
87 simpr 468 . . . . . . . . . 10 Poly
8887neqned 2650 . . . . . . . . 9 Poly
89 elnnne0 10907 . . . . . . . . 9
9086, 88, 89sylanbrc 677 . . . . . . . 8 Poly
91 1nn0 10909 . . . . . . . . . . . . 13
9291a1i 11 . . . . . . . . . . . 12 Poly
93 simpr 468 . . . . . . . . . . . . 13 Poly
9493nnnn0d 10949 . . . . . . . . . . . 12 Poly
9593nnge1d 10674 . . . . . . . . . . . 12 Poly
96 elfz2nn0 11911 . . . . . . . . . . . 12
9792, 94, 95, 96syl3anbrc 1214 . . . . . . . . . . 11 Poly
9897snssd 4108 . . . . . . . . . 10 Poly
9950ad2antrr 740 . . . . . . . . . . . . 13 Poly coeff
100 oveq2 6316 . . . . . . . . . . . . . . . 16
10146, 100sylbi 200 . . . . . . . . . . . . . . 15
102101adantl 473 . . . . . . . . . . . . . 14 Poly
103 nnm1nn0 10935 . . . . . . . . . . . . . . 15
104103ad2antlr 741 . . . . . . . . . . . . . 14 Poly
105102, 104eqeltrd 2549 . . . . . . . . . . . . 13 Poly
10699, 105ffvelrnd 6038 . . . . . . . . . . . 12 Poly coeff
107106recnd 9687 . . . . . . . . . . 11 Poly coeff
108107ralrimiva 2809 . . . . . . . . . 10 Poly coeff
109 fzfi 12223 . . . . . . . . . . . 12
110109olci 398 . . . . . . . . . . 11
111110a1i 11 . . . . . . . . . 10 Poly
112 sumss2 13869 . . . . . . . . . 10 coeff coeff coeff
11398, 108, 111, 112syl21anc 1291 . . . . . . . . 9 Poly coeff coeff
11450adantr 472 . . . . . . . . . . . 12 Poly coeff
115103adantl 473 . . . . . . . . . . . 12 Poly
116114, 115ffvelrnd 6038 . . . . . . . . . . 11 Poly coeff
117116recnd 9687 . . . . . . . . . 10 Poly coeff
118100fveq2d 5883 . . . . . . . . . . 11 coeff coeff
119118sumsn 13884 . . . . . . . . . 10 coeff coeff coeff
1203, 117, 119sylancr 676 . . . . . . . . 9 Poly coeff coeff
121113, 120eqtr3d 2507 . . . . . . . 8 Poly coeff coeff
12285, 90, 121syl2anc 673 . . . . . . 7 Poly coeff coeff
12360, 61, 84, 122ifbothda 3907 . . . . . 6 Poly coeff coeff
12459, 123eqtrd 2505 . . . . 5 Poly coeff coeff coeff
12537, 45, 1243eqtrd 2509 . . . 4 Poly coeff coeff
12630, 125eqtrd 2505 . . 3 Poly coeff coeff
127126mpteq2dva 4482 . 2 Poly coeff coeff
12816, 127eqtrd 2505 1 Poly coeff coeff
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wo 375   wa 376   wceq 1452   wcel 1904   wne 2641  wral 2756  cvv 3031   cdif 3387   wss 3390  cif 3872  csn 3959   class class class wbr 4395   cmpt 4454  wf 5585  cfv 5589  (class class class)co 6308   cof 6548  cfn 7587  cc 9555  cr 9556  cc0 9557  c1 9558   cmul 9562   cle 9694   cmin 9880  cn 10631  cn0 10893  cz 10961  cuz 11182  cfz 11810  csu 13829  c0p 22706  Polycply 23217  cidp 23218  coeffccoe 23219 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-idp 23222  df-coe 23223  df-dgr 23224 This theorem is referenced by:  plymulx  29509
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