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Theorem dgrmul 19667
Description: The degree of a product of nonzero polynomials is the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
dgradd.1  |-  M  =  (deg `  F )
dgradd.2  |-  N  =  (deg `  G )
Assertion
Ref Expression
dgrmul  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( F  o F  x.  G
) )  =  ( M  +  N ) )

Proof of Theorem dgrmul
StepHypRef Expression
1 dgradd.1 . . . 4  |-  M  =  (deg `  F )
2 dgradd.2 . . . 4  |-  N  =  (deg `  G )
31, 2dgrmul2 19666 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  x.  G
) )  <_  ( M  +  N )
)
43ad2ant2r 727 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( F  o F  x.  G
) )  <_  ( M  +  N )
)
5 plymulcl 19619 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  x.  G
)  e.  (Poly `  CC ) )
65ad2ant2r 727 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( F  o F  x.  G )  e.  (Poly `  CC )
)
7 dgrcl 19631 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
81, 7syl5eqel 2380 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
98ad2antrr 706 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  M  e.  NN0 )
10 dgrcl 19631 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
112, 10syl5eqel 2380 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
1211ad2antrl 708 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  N  e.  NN0 )
139, 12nn0addcld 10038 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( M  +  N )  e.  NN0 )
14 eqid 2296 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2296 . . . . . 6  |-  (coeff `  G )  =  (coeff `  G )
1614, 15, 1, 2coemulhi 19651 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  o F  x.  G ) ) `
 ( M  +  N ) )  =  ( ( (coeff `  F ) `  M
)  x.  ( (coeff `  G ) `  N
) ) )
1716ad2ant2r 727 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  ( F  o F  x.  G ) ) `  ( M  +  N
) )  =  ( ( (coeff `  F
) `  M )  x.  ( (coeff `  G
) `  N )
) )
1814coef3 19630 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1918ad2antrr 706 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (coeff `  F
) : NN0 --> CC )
20 ffvelrn 5679 . . . . . 6  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  M  e.  NN0 )  ->  (
(coeff `  F ) `  M )  e.  CC )
2119, 9, 20syl2anc 642 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  F ) `  M
)  e.  CC )
2215coef3 19630 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
2322ad2antrl 708 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (coeff `  G
) : NN0 --> CC )
24 ffvelrn 5679 . . . . . 6  |-  ( ( (coeff `  G ) : NN0 --> CC  /\  N  e.  NN0 )  ->  (
(coeff `  G ) `  N )  e.  CC )
2523, 12, 24syl2anc 642 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  G ) `  N
)  e.  CC )
261, 14dgreq0 19662 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( (coeff `  F ) `  M
)  =  0 ) )
2726necon3bid 2494 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( F  =/=  0 p  <->  ( (coeff `  F ) `  M
)  =/=  0 ) )
2827biimpa 470 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  -> 
( (coeff `  F
) `  M )  =/=  0 )
2928adantr 451 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  F ) `  M
)  =/=  0 )
302, 15dgreq0 19662 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  N
)  =  0 ) )
3130necon3bid 2494 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  ( G  =/=  0 p  <->  ( (coeff `  G ) `  N
)  =/=  0 ) )
3231biimpa 470 . . . . . 6  |-  ( ( G  e.  (Poly `  S )  /\  G  =/=  0 p )  -> 
( (coeff `  G
) `  N )  =/=  0 )
3332adantl 452 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  G ) `  N
)  =/=  0 )
3421, 25, 29, 33mulne0d 9436 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( ( (coeff `  F ) `  M
)  x.  ( (coeff `  G ) `  N
) )  =/=  0
)
3517, 34eqnetrd 2477 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  ( F  o F  x.  G ) ) `  ( M  +  N
) )  =/=  0
)
36 eqid 2296 . . . 4  |-  (coeff `  ( F  o F  x.  G ) )  =  (coeff `  ( F  o F  x.  G
) )
37 eqid 2296 . . . 4  |-  (deg `  ( F  o F  x.  G ) )  =  (deg `  ( F  o F  x.  G
) )
3836, 37dgrub 19632 . . 3  |-  ( ( ( F  o F  x.  G )  e.  (Poly `  CC )  /\  ( M  +  N
)  e.  NN0  /\  ( (coeff `  ( F  o F  x.  G
) ) `  ( M  +  N )
)  =/=  0 )  ->  ( M  +  N )  <_  (deg `  ( F  o F  x.  G ) ) )
396, 13, 35, 38syl3anc 1182 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( M  +  N )  <_  (deg `  ( F  o F  x.  G ) ) )
40 dgrcl 19631 . . . . 5  |-  ( ( F  o F  x.  G )  e.  (Poly `  CC )  ->  (deg `  ( F  o F  x.  G ) )  e.  NN0 )
416, 40syl 15 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( F  o F  x.  G
) )  e.  NN0 )
4241nn0red 10035 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( F  o F  x.  G
) )  e.  RR )
4313nn0red 10035 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( M  +  N )  e.  RR )
4442, 43letri3d 8977 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (deg `  ( F  o F  x.  G ) )  =  ( M  +  N
)  <->  ( (deg `  ( F  o F  x.  G ) )  <_ 
( M  +  N
)  /\  ( M  +  N )  <_  (deg `  ( F  o F  x.  G ) ) ) ) )
454, 39, 44mpbir2and 888 1  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( F  o F  x.  G
) )  =  ( M  +  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753    + caddc 8756    x. cmul 8758    <_ cle 8884   NN0cn0 9981   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  dgrmulc  19668  dgrcolem1  19670  plydivlem4  19692  plydiveu  19694  fta1lem  19703  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
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