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Theorem plymul0or 20190
Description: Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
Assertion
Ref Expression
plymul0or  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  <-> 
( F  =  0 p  \/  G  =  0 p ) ) )

Proof of Theorem plymul0or
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dgrcl 20144 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
2 dgrcl 20144 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3 nn0addcl 10247 . . . . . . 7  |-  ( ( (deg `  F )  e.  NN0  /\  (deg `  G )  e.  NN0 )  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
41, 2, 3syl2an 464 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
5 c0ex 9077 . . . . . . 7  |-  0  e.  _V
65fvconst2 5939 . . . . . 6  |-  ( ( (deg `  F )  +  (deg `  G )
)  e.  NN0  ->  ( ( NN0  X.  {
0 } ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 )
74, 6syl 16 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( NN0  X.  { 0 } ) `  ( (deg
`  F )  +  (deg `  G )
) )  =  0 )
8 fveq2 5720 . . . . . . . 8  |-  ( ( F  o F  x.  G )  =  0 p  ->  (coeff `  ( F  o F  x.  G
) )  =  (coeff `  0 p ) )
9 coe0 20166 . . . . . . . 8  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
108, 9syl6eq 2483 . . . . . . 7  |-  ( ( F  o F  x.  G )  =  0 p  ->  (coeff `  ( F  o F  x.  G
) )  =  ( NN0  X.  { 0 } ) )
1110fveq1d 5722 . . . . . 6  |-  ( ( F  o F  x.  G )  =  0 p  ->  ( (coeff `  ( F  o F  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( NN0  X.  { 0 } ) `  (
(deg `  F )  +  (deg `  G )
) ) )
1211eqeq1d 2443 . . . . 5  |-  ( ( F  o F  x.  G )  =  0 p  ->  ( (
(coeff `  ( F  o F  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( NN0  X.  { 0 } ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  0 ) )
137, 12syl5ibrcom 214 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  ->  ( (coeff `  ( F  o F  x.  G ) ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 ) )
14 eqid 2435 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2435 . . . . . . 7  |-  (coeff `  G )  =  (coeff `  G )
16 eqid 2435 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
17 eqid 2435 . . . . . . 7  |-  (deg `  G )  =  (deg
`  G )
1814, 15, 16, 17coemulhi 20164 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  o F  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( (coeff `  F ) `  (deg `  F )
)  x.  ( (coeff `  G ) `  (deg `  G ) ) ) )
1918eqeq1d 2443 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  o F  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  x.  ( (coeff `  G
) `  (deg `  G
) ) )  =  0 ) )
2014coef3 20143 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
2120adantr 452 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  F
) : NN0 --> CC )
221adantr 452 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  NN0 )
2321, 22ffvelrnd 5863 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  F ) `  (deg `  F ) )  e.  CC )
2415coef3 20143 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
2524adantl 453 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  G
) : NN0 --> CC )
262adantl 453 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  NN0 )
2725, 26ffvelrnd 5863 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  G ) `  (deg `  G ) )  e.  CC )
2823, 27mul0ord 9664 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
( (coeff `  F
) `  (deg `  F
) )  x.  (
(coeff `  G ) `  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
2919, 28bitrd 245 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  o F  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3013, 29sylibd 206 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  ->  ( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3116, 14dgreq0 20175 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3231adantr 452 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0 p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3317, 15dgreq0 20175 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3433adantl 453 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3532, 34orbi12d 691 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0 p  \/  G  =  0 p )  <->  ( (
(coeff `  F ) `  (deg `  F )
)  =  0  \/  ( (coeff `  G
) `  (deg `  G
) )  =  0 ) ) )
3630, 35sylibrd 226 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  ->  ( F  =  0 p  \/  G  =  0 p ) ) )
37 cnex 9063 . . . . . . 7  |-  CC  e.  _V
3837a1i 11 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
39 plyf 20109 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
4039adantl 453 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
41 0cn 9076 . . . . . . 7  |-  0  e.  CC
4241a1i 11 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  e.  CC )
43 mul02 9236 . . . . . . 7  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
4443adantl 453 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
4538, 40, 42, 42, 44caofid2 6327 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( CC  X.  { 0 } )  o F  x.  G )  =  ( CC  X.  { 0 } ) )
46 id 20 . . . . . . . 8  |-  ( F  =  0 p  ->  F  =  0 p
)
47 df-0p 19554 . . . . . . . 8  |-  0 p  =  ( CC  X.  { 0 } )
4846, 47syl6eq 2483 . . . . . . 7  |-  ( F  =  0 p  ->  F  =  ( CC  X.  { 0 } ) )
4948oveq1d 6088 . . . . . 6  |-  ( F  =  0 p  -> 
( F  o F  x.  G )  =  ( ( CC  X.  { 0 } )  o F  x.  G
) )
5049eqeq1d 2443 . . . . 5  |-  ( F  =  0 p  -> 
( ( F  o F  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( ( CC  X.  { 0 } )  o F  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
5145, 50syl5ibrcom 214 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0 p  -> 
( F  o F  x.  G )  =  ( CC  X.  {
0 } ) ) )
52 plyf 20109 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
5352adantr 452 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
54 mul01 9237 . . . . . . 7  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5554adantl 453 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5638, 53, 42, 42, 55caofid1 6326 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC  X.  { 0 } ) )
57 id 20 . . . . . . . 8  |-  ( G  =  0 p  ->  G  =  0 p
)
5857, 47syl6eq 2483 . . . . . . 7  |-  ( G  =  0 p  ->  G  =  ( CC  X.  { 0 } ) )
5958oveq2d 6089 . . . . . 6  |-  ( G  =  0 p  -> 
( F  o F  x.  G )  =  ( F  o F  x.  ( CC  X.  { 0 } ) ) )
6059eqeq1d 2443 . . . . 5  |-  ( G  =  0 p  -> 
( ( F  o F  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( F  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) ) )
6156, 60syl5ibrcom 214 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0 p  -> 
( F  o F  x.  G )  =  ( CC  X.  {
0 } ) ) )
6251, 61jaod 370 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0 p  \/  G  =  0 p )  ->  ( F  o F  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
6347eqeq2i 2445 . . 3  |-  ( ( F  o F  x.  G )  =  0 p  <->  ( F  o F  x.  G )  =  ( CC  X.  { 0 } ) )
6462, 63syl6ibr 219 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0 p  \/  G  =  0 p )  ->  ( F  o F  x.  G
)  =  0 p ) )
6536, 64impbid 184 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  <-> 
( F  =  0 p  \/  G  =  0 p ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   CCcc 8980   0cc0 8982    + caddc 8985    x. cmul 8987   NN0cn0 10213   0 pc0p 19553  Polycply 20095  coeffccoe 20097  degcdgr 20098
This theorem is referenced by:  plydiveu  20207  quotcan  20218  vieta1lem1  20219  vieta1lem2  20220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102
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