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Theorem coemulc 20178
 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 3369 . . . . 5
2 plyconst 20130 . . . . 5 Poly
31, 2mpan 653 . . . 4 Poly
4 plyssc 20124 . . . . 5 Poly Poly
54sseli 3346 . . . 4 Poly Poly
6 plymulcl 20145 . . . 4 Poly Poly Poly
73, 5, 6syl2an 465 . . 3 Poly Poly
8 eqid 2438 . . . 4 coeff coeff
98coef3 20156 . . 3 Poly coeff
10 ffn 5594 . . 3 coeff coeff
117, 9, 103syl 19 . 2 Poly coeff
12 fconstg 5633 . . . . 5
1312adantr 453 . . . 4 Poly
14 ffn 5594 . . . 4
1513, 14syl 16 . . 3 Poly
16 eqid 2438 . . . . . 6 coeff coeff
1716coef3 20156 . . . . 5 Poly coeff
1817adantl 454 . . . 4 Poly coeff
19 ffn 5594 . . . 4 coeff coeff
2018, 19syl 16 . . 3 Poly coeff
21 nn0ex 10232 . . . 4
2221a1i 11 . . 3 Poly
23 inidm 3552 . . 3
2415, 20, 22, 22, 23offn 6319 . 2 Poly coeff
253ad2antrr 708 . . . . . 6 Poly Poly
26 eqid 2438 . . . . . . 7 coeff coeff
2726coefv0 20171 . . . . . 6 Poly coeff
2825, 27syl 16 . . . . 5 Poly coeff
29 simpll 732 . . . . . 6 Poly
30 0cn 9089 . . . . . 6
31 fvconst2g 5948 . . . . . 6
3229, 30, 31sylancl 645 . . . . 5 Poly
3328, 32eqtr3d 2472 . . . 4 Poly coeff
34 simpr 449 . . . . . . 7 Poly
3534nn0cnd 10281 . . . . . 6 Poly
3635subid1d 9405 . . . . 5 Poly
3736fveq2d 5735 . . . 4 Poly coeff coeff
3833, 37oveq12d 6102 . . 3 Poly coeff coeff coeff
395ad2antlr 709 . . . . 5 Poly Poly
4026, 16coemul 20175 . . . . 5 Poly Poly coeff coeff coeff
4125, 39, 34, 40syl3anc 1185 . . . 4 Poly coeff coeff coeff
42 nn0uz 10525 . . . . . . 7
4334, 42syl6eleq 2528 . . . . . 6 Poly
44 fzss2 11097 . . . . . 6
4543, 44syl 16 . . . . 5 Poly
46 elfz1eq 11073 . . . . . . . 8
4746adantl 454 . . . . . . 7 Poly
48 fveq2 5731 . . . . . . . 8 coeff coeff
49 oveq2 6092 . . . . . . . . 9
5049fveq2d 5735 . . . . . . . 8 coeff coeff
5148, 50oveq12d 6102 . . . . . . 7 coeff coeff coeff coeff
5247, 51syl 16 . . . . . 6 Poly coeff coeff coeff coeff
5318ffvelrnda 5873 . . . . . . . . 9 Poly coeff
5429, 53mulcld 9113 . . . . . . . 8 Poly coeff
5538, 54eqeltrd 2512 . . . . . . 7 Poly coeff coeff
5655adantr 453 . . . . . 6 Poly coeff coeff
5752, 56eqeltrd 2512 . . . . 5 Poly coeff coeff
58 eldifn 3472 . . . . . . . . 9
5958adantl 454 . . . . . . . 8 Poly
60 eldifi 3471 . . . . . . . . . . . . 13
61 elfznn0 11088 . . . . . . . . . . . . 13
6260, 61syl 16 . . . . . . . . . . . 12
63 eqid 2438 . . . . . . . . . . . . . 14 deg deg
6426, 63dgrub 20158 . . . . . . . . . . . . 13 Poly coeff deg
65643expia 1156 . . . . . . . . . . . 12 Poly coeff deg
6625, 62, 65syl2an 465 . . . . . . . . . . 11 Poly coeff deg
67 0dgr 20169 . . . . . . . . . . . . . 14 deg
6867ad3antrrr 712 . . . . . . . . . . . . 13 Poly deg
6968breq2d 4227 . . . . . . . . . . . 12 Poly deg
7062adantl 454 . . . . . . . . . . . . 13 Poly
71 nn0le0eq0 10255 . . . . . . . . . . . . 13
7270, 71syl 16 . . . . . . . . . . . 12 Poly
7369, 72bitrd 246 . . . . . . . . . . 11 Poly deg
7466, 73sylibd 207 . . . . . . . . . 10 Poly coeff
75 id 21 . . . . . . . . . . 11
76 0z 10298 . . . . . . . . . . . 12
77 elfz3 11072 . . . . . . . . . . . 12
7876, 77ax-mp 5 . . . . . . . . . . 11
7975, 78syl6eqel 2526 . . . . . . . . . 10
8074, 79syl6 32 . . . . . . . . 9 Poly coeff
8180necon1bd 2674 . . . . . . . 8 Poly coeff
8259, 81mpd 15 . . . . . . 7 Poly coeff
8382oveq1d 6099 . . . . . 6 Poly coeff coeff coeff
8418adantr 453 . . . . . . . 8 Poly coeff
85 fznn0sub 11090 . . . . . . . . 9
8660, 85syl 16 . . . . . . . 8
87 ffvelrn 5871 . . . . . . . 8 coeff coeff
8884, 86, 87syl2an 465 . . . . . . 7 Poly coeff
8988mul02d 9269 . . . . . 6 Poly coeff
9083, 89eqtrd 2470 . . . . 5 Poly coeff coeff
91 fzfid 11317 . . . . 5 Poly
9245, 57, 90, 91fsumss 12524 . . . 4 Poly coeff coeff coeff coeff
9351fsum1 12540 . . . . 5 coeff coeff coeff coeff coeff coeff
9476, 55, 93sylancr 646 . . . 4 Poly coeff coeff coeff coeff
9541, 92, 943eqtr2d 2476 . . 3 Poly coeff coeff coeff
96 simpl 445 . . . 4 Poly
97 eqidd 2439 . . . 4 Poly coeff coeff
9822, 96, 20, 97ofc1 6330 . . 3 Poly coeff coeff
9938, 95, 983eqtr4d 2480 . 2 Poly coeff coeff
10011, 24, 99eqfnfvd 5833 1 Poly coeff coeff
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360   wceq 1653   wcel 1726   wne 2601  cvv 2958   cdif 3319   wss 3322  csn 3816   class class class wbr 4215   cxp 4879   wfn 5452  wf 5453  cfv 5457  (class class class)co 6084   cof 6306  cc 8993  cc0 8995   cmul 9000   cle 9126   cmin 9296  cn0 10226  cz 10287  cuz 10493  cfz 11048  csu 12484  Polycply 20108  coeffccoe 20110  degcdgr 20111 This theorem is referenced by:  coe0  20179  coesub  20180  mpaaeu  27346 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-0p 19565  df-ply 20112  df-coe 20114  df-dgr 20115
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