MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plymul0or Unicode version

Theorem plymul0or 19677
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 19631 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
2 dgrcl 19631 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3 nn0addcl 10015 . . . . . . 7  |-  ( ( (deg `  F )  e.  NN0  /\  (deg `  G )  e.  NN0 )  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
41, 2, 3syl2an 463 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
5 c0ex 8848 . . . . . . 7  |-  0  e.  _V
65fvconst2 5745 . . . . . 6  |-  ( ( (deg `  F )  +  (deg `  G )
)  e.  NN0  ->  ( ( NN0  X.  {
0 } ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 )
74, 6syl 15 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( NN0  X.  { 0 } ) `  ( (deg
`  F )  +  (deg `  G )
) )  =  0 )
8 fveq2 5541 . . . . . . . 8  |-  ( ( F  o F  x.  G )  =  0 p  ->  (coeff `  ( F  o F  x.  G
) )  =  (coeff `  0 p ) )
9 coe0 19653 . . . . . . . 8  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
108, 9syl6eq 2344 . . . . . . 7  |-  ( ( F  o F  x.  G )  =  0 p  ->  (coeff `  ( F  o F  x.  G
) )  =  ( NN0  X.  { 0 } ) )
1110fveq1d 5543 . . . . . 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 2304 . . . . 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 213 . . . 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 2296 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2296 . . . . . . 7  |-  (coeff `  G )  =  (coeff `  G )
16 eqid 2296 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
17 eqid 2296 . . . . . . 7  |-  (deg `  G )  =  (deg
`  G )
1814, 15, 16, 17coemulhi 19651 . . . . . 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 2304 . . . . 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 19630 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
2120adantr 451 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  F
) : NN0 --> CC )
221adantr 451 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  NN0 )
23 ffvelrn 5679 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  (deg `  F )  e.  NN0 )  ->  ( (coeff `  F ) `  (deg `  F ) )  e.  CC )
2421, 22, 23syl2anc 642 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  F ) `  (deg `  F ) )  e.  CC )
2515coef3 19630 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
2625adantl 452 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  G
) : NN0 --> CC )
272adantl 452 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  NN0 )
28 ffvelrn 5679 . . . . . . 7  |-  ( ( (coeff `  G ) : NN0 --> CC  /\  (deg `  G )  e.  NN0 )  ->  ( (coeff `  G ) `  (deg `  G ) )  e.  CC )
2926, 27, 28syl2anc 642 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  G ) `  (deg `  G ) )  e.  CC )
3024, 29mul0ord 9434 . . . . 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 ) ) )
3119, 30bitrd 244 . . . 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 ) ) )
3213, 31sylibd 205 . . 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 ) ) )
3316, 14dgreq0 19662 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3433adantr 451 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0 p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3517, 15dgreq0 19662 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3635adantl 452 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3734, 36orbi12d 690 . . 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 ) ) )
3832, 37sylibrd 225 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  ->  ( F  =  0 p  \/  G  =  0 p ) ) )
39 cnex 8834 . . . . . . 7  |-  CC  e.  _V
4039a1i 10 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
41 plyf 19596 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
4241adantl 452 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
43 0cn 8847 . . . . . . 7  |-  0  e.  CC
4443a1i 10 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  e.  CC )
45 mul02 9006 . . . . . . 7  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
4645adantl 452 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
4740, 42, 44, 44, 46caofid2 6124 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( CC  X.  { 0 } )  o F  x.  G )  =  ( CC  X.  { 0 } ) )
48 id 19 . . . . . . . 8  |-  ( F  =  0 p  ->  F  =  0 p
)
49 df-0p 19041 . . . . . . . 8  |-  0 p  =  ( CC  X.  { 0 } )
5048, 49syl6eq 2344 . . . . . . 7  |-  ( F  =  0 p  ->  F  =  ( CC  X.  { 0 } ) )
5150oveq1d 5889 . . . . . 6  |-  ( F  =  0 p  -> 
( F  o F  x.  G )  =  ( ( CC  X.  { 0 } )  o F  x.  G
) )
5251eqeq1d 2304 . . . . 5  |-  ( F  =  0 p  -> 
( ( F  o F  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( ( CC  X.  { 0 } )  o F  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
5347, 52syl5ibrcom 213 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0 p  -> 
( F  o F  x.  G )  =  ( CC  X.  {
0 } ) ) )
54 plyf 19596 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
5554adantr 451 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
56 mul01 9007 . . . . . . 7  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5756adantl 452 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5840, 55, 44, 44, 57caofid1 6123 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC  X.  { 0 } ) )
59 id 19 . . . . . . . 8  |-  ( G  =  0 p  ->  G  =  0 p
)
6059, 49syl6eq 2344 . . . . . . 7  |-  ( G  =  0 p  ->  G  =  ( CC  X.  { 0 } ) )
6160oveq2d 5890 . . . . . 6  |-  ( G  =  0 p  -> 
( F  o F  x.  G )  =  ( F  o F  x.  ( CC  X.  { 0 } ) ) )
6261eqeq1d 2304 . . . . 5  |-  ( G  =  0 p  -> 
( ( F  o F  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( F  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) ) )
6358, 62syl5ibrcom 213 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0 p  -> 
( F  o F  x.  G )  =  ( CC  X.  {
0 } ) ) )
6453, 63jaod 369 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0 p  \/  G  =  0 p )  ->  ( F  o F  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
6549eqeq2i 2306 . . 3  |-  ( ( F  o F  x.  G )  =  0 p  <->  ( F  o F  x.  G )  =  ( CC  X.  { 0 } ) )
6664, 65syl6ibr 218 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0 p  \/  G  =  0 p )  ->  ( F  o F  x.  G
)  =  0 p ) )
6738, 66impbid 183 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753    + caddc 8756    x. cmul 8758   NN0cn0 9981   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  plydiveu  19694  quotcan  19705  vieta1lem1  19706  vieta1lem2  19707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
  Copyright terms: Public domain W3C validator