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Theorem plymul0or 19661
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 19615 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
2 dgrcl 19615 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3 nn0addcl 9999 . . . . . . 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 8832 . . . . . . 7  |-  0  e.  _V
65fvconst2 5729 . . . . . 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 5525 . . . . . . . 8  |-  ( ( F  o F  x.  G )  =  0 p  ->  (coeff `  ( F  o F  x.  G
) )  =  (coeff `  0 p ) )
9 coe0 19637 . . . . . . . 8  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
108, 9syl6eq 2331 . . . . . . 7  |-  ( ( F  o F  x.  G )  =  0 p  ->  (coeff `  ( F  o F  x.  G
) )  =  ( NN0  X.  { 0 } ) )
1110fveq1d 5527 . . . . . 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 2291 . . . . 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 2283 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2283 . . . . . . 7  |-  (coeff `  G )  =  (coeff `  G )
16 eqid 2283 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
17 eqid 2283 . . . . . . 7  |-  (deg `  G )  =  (deg
`  G )
1814, 15, 16, 17coemulhi 19635 . . . . . 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 2291 . . . . 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 19614 . . . . . . . 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 5663 . . . . . . 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 19614 . . . . . . . 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 5663 . . . . . . 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 9418 . . . . 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 19646 . . . . 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 19646 . . . . 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 8818 . . . . . . 7  |-  CC  e.  _V
4039a1i 10 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
41 plyf 19580 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
4241adantl 452 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
43 0cn 8831 . . . . . . 7  |-  0  e.  CC
4443a1i 10 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  e.  CC )
45 mul02 8990 . . . . . . 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 6108 . . . . 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 19025 . . . . . . . 8  |-  0 p  =  ( CC  X.  { 0 } )
5048, 49syl6eq 2331 . . . . . . 7  |-  ( F  =  0 p  ->  F  =  ( CC  X.  { 0 } ) )
5150oveq1d 5873 . . . . . 6  |-  ( F  =  0 p  -> 
( F  o F  x.  G )  =  ( ( CC  X.  { 0 } )  o F  x.  G
) )
5251eqeq1d 2291 . . . . 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 19580 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
5554adantr 451 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
56 mul01 8991 . . . . . . 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 6107 . . . . 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 2331 . . . . . . 7  |-  ( G  =  0 p  ->  G  =  ( CC  X.  { 0 } ) )
6160oveq2d 5874 . . . . . 6  |-  ( G  =  0 p  -> 
( F  o F  x.  G )  =  ( F  o F  x.  ( CC  X.  { 0 } ) ) )
6261eqeq1d 2291 . . . . 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 2293 . . 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 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   0cc0 8737    + caddc 8740    x. cmul 8742   NN0cn0 9965   0 pc0p 19024  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  plydiveu  19678  quotcan  19689  vieta1lem1  19690  vieta1lem2  19691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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