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Theorem coemulc 22524
 Description: The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
coemulc Poly coeff coeff

Proof of Theorem coemulc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3508 . . . . 5
2 plyconst 22476 . . . . 5 Poly
31, 2mpan 670 . . . 4 Poly
4 plyssc 22470 . . . . 5 Poly Poly
54sseli 3485 . . . 4 Poly Poly
6 plymulcl 22491 . . . 4 Poly Poly Poly
73, 5, 6syl2an 477 . . 3 Poly Poly
8 eqid 2443 . . . 4 coeff coeff
98coef3 22502 . . 3 Poly coeff
10 ffn 5721 . . 3 coeff coeff
117, 9, 103syl 20 . 2 Poly coeff
12 fconstg 5762 . . . . 5
1312adantr 465 . . . 4 Poly
14 ffn 5721 . . . 4
1513, 14syl 16 . . 3 Poly
16 eqid 2443 . . . . . 6 coeff coeff
1716coef3 22502 . . . . 5 Poly coeff
1817adantl 466 . . . 4 Poly coeff
19 ffn 5721 . . . 4 coeff coeff
2018, 19syl 16 . . 3 Poly coeff
21 nn0ex 10807 . . . 4
2221a1i 11 . . 3 Poly
23 inidm 3692 . . 3
2415, 20, 22, 22, 23offn 6536 . 2 Poly coeff
253ad2antrr 725 . . . . . 6 Poly Poly
26 eqid 2443 . . . . . . 7 coeff coeff
2726coefv0 22517 . . . . . 6 Poly coeff
2825, 27syl 16 . . . . 5 Poly coeff
29 simpll 753 . . . . . 6 Poly
30 0cn 9591 . . . . . 6
31 fvconst2g 6109 . . . . . 6
3229, 30, 31sylancl 662 . . . . 5 Poly
3328, 32eqtr3d 2486 . . . 4 Poly coeff
34 simpr 461 . . . . . . 7 Poly
3534nn0cnd 10860 . . . . . 6 Poly
3635subid1d 9925 . . . . 5 Poly
3736fveq2d 5860 . . . 4 Poly coeff coeff
3833, 37oveq12d 6299 . . 3 Poly coeff coeff coeff
395ad2antlr 726 . . . . 5 Poly Poly
4026, 16coemul 22521 . . . . 5 Poly Poly coeff coeff coeff
4125, 39, 34, 40syl3anc 1229 . . . 4 Poly coeff coeff coeff
42 nn0uz 11124 . . . . . . 7
4334, 42syl6eleq 2541 . . . . . 6 Poly
44 fzss2 11732 . . . . . 6
4543, 44syl 16 . . . . 5 Poly
46 elfz1eq 11706 . . . . . . . 8
4746adantl 466 . . . . . . 7 Poly
48 fveq2 5856 . . . . . . . 8 coeff coeff
49 oveq2 6289 . . . . . . . . 9
5049fveq2d 5860 . . . . . . . 8 coeff coeff
5148, 50oveq12d 6299 . . . . . . 7 coeff coeff coeff coeff
5247, 51syl 16 . . . . . 6 Poly coeff coeff coeff coeff
5318ffvelrnda 6016 . . . . . . . . 9 Poly coeff
5429, 53mulcld 9619 . . . . . . . 8 Poly coeff
5538, 54eqeltrd 2531 . . . . . . 7 Poly coeff coeff
5655adantr 465 . . . . . 6 Poly coeff coeff
5752, 56eqeltrd 2531 . . . . 5 Poly coeff coeff
58 eldifn 3612 . . . . . . . . 9
5958adantl 466 . . . . . . . 8 Poly
60 eldifi 3611 . . . . . . . . . . . . 13
61 elfznn0 11779 . . . . . . . . . . . . 13
6260, 61syl 16 . . . . . . . . . . . 12
63 eqid 2443 . . . . . . . . . . . . . 14 deg deg
6426, 63dgrub 22504 . . . . . . . . . . . . 13 Poly coeff deg
65643expia 1199 . . . . . . . . . . . 12 Poly coeff deg
6625, 62, 65syl2an 477 . . . . . . . . . . 11 Poly coeff deg
67 0dgr 22515 . . . . . . . . . . . . . 14 deg
6867ad3antrrr 729 . . . . . . . . . . . . 13 Poly deg
6968breq2d 4449 . . . . . . . . . . . 12 Poly deg
7062adantl 466 . . . . . . . . . . . . 13 Poly
71 nn0le0eq0 10830 . . . . . . . . . . . . 13
7270, 71syl 16 . . . . . . . . . . . 12 Poly
7369, 72bitrd 253 . . . . . . . . . . 11 Poly deg
7466, 73sylibd 214 . . . . . . . . . 10 Poly coeff
75 id 22 . . . . . . . . . . 11
76 0z 10881 . . . . . . . . . . . 12
77 elfz3 11705 . . . . . . . . . . . 12
7876, 77ax-mp 5 . . . . . . . . . . 11
7975, 78syl6eqel 2539 . . . . . . . . . 10
8074, 79syl6 33 . . . . . . . . 9 Poly coeff
8180necon1bd 2661 . . . . . . . 8 Poly coeff
8259, 81mpd 15 . . . . . . 7 Poly coeff
8382oveq1d 6296 . . . . . 6 Poly coeff coeff coeff
8418adantr 465 . . . . . . . 8 Poly coeff
85 fznn0sub 11725 . . . . . . . . 9
8660, 85syl 16 . . . . . . . 8
87 ffvelrn 6014 . . . . . . . 8 coeff coeff
8884, 86, 87syl2an 477 . . . . . . 7 Poly coeff
8988mul02d 9781 . . . . . 6 Poly coeff
9083, 89eqtrd 2484 . . . . 5 Poly coeff coeff
91 fzfid 12062 . . . . 5 Poly
9245, 57, 90, 91fsumss 13526 . . . 4 Poly coeff coeff coeff coeff
9351fsum1 13543 . . . . 5 coeff coeff coeff coeff coeff coeff
9476, 55, 93sylancr 663 . . . 4 Poly coeff coeff coeff coeff
9541, 92, 943eqtr2d 2490 . . 3 Poly coeff coeff coeff
96 simpl 457 . . . 4 Poly
97 eqidd 2444 . . . 4 Poly coeff coeff
9822, 96, 20, 97ofc1 6548 . . 3 Poly coeff coeff
9938, 95, 983eqtr4d 2494 . 2 Poly coeff coeff
10011, 24, 99eqfnfvd 5969 1 Poly coeff coeff
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369   wceq 1383   wcel 1804   wne 2638  cvv 3095   cdif 3458   wss 3461  csn 4014   class class class wbr 4437   cxp 4987   wfn 5573  wf 5574  cfv 5578  (class class class)co 6281   cof 6523  cc 9493  cc0 9495   cmul 9500   cle 9632   cmin 9810  cn0 10801  cz 10870  cuz 11090  cfz 11681  csu 13487  Polycply 22454  coeffccoe 22456  degcdgr 22457 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-fz 11682  df-fzo 11804  df-fl 11908  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-rlim 13291  df-sum 13488  df-0p 21950  df-ply 22458  df-coe 22460  df-dgr 22461 This theorem is referenced by:  coe0  22525  coesub  22526  mpaaeu  31075
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