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Theorem dgrmul 20190
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 20189 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  x.  G
) )  <_  ( M  +  N )
)
43ad2ant2r 729 . 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 20142 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  x.  G
)  e.  (Poly `  CC ) )
65ad2ant2r 729 . . 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 20154 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
81, 7syl5eqel 2522 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
98ad2antrr 708 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  M  e.  NN0 )
10 dgrcl 20154 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
112, 10syl5eqel 2522 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
1211ad2antrl 710 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  N  e.  NN0 )
139, 12nn0addcld 10280 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( M  +  N )  e.  NN0 )
14 eqid 2438 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2438 . . . . . 6  |-  (coeff `  G )  =  (coeff `  G )
1614, 15, 1, 2coemulhi 20174 . . . . 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 729 . . . 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 20153 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1918ad2antrr 708 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (coeff `  F
) : NN0 --> CC )
2019, 9ffvelrnd 5873 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  F ) `  M
)  e.  CC )
2115coef3 20153 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
2221ad2antrl 710 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (coeff `  G
) : NN0 --> CC )
2322, 12ffvelrnd 5873 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  G ) `  N
)  e.  CC )
241, 14dgreq0 20185 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( (coeff `  F ) `  M
)  =  0 ) )
2524necon3bid 2638 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( F  =/=  0 p  <->  ( (coeff `  F ) `  M
)  =/=  0 ) )
2625biimpa 472 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  -> 
( (coeff `  F
) `  M )  =/=  0 )
2726adantr 453 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  F ) `  M
)  =/=  0 )
282, 15dgreq0 20185 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  N
)  =  0 ) )
2928necon3bid 2638 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  ( G  =/=  0 p  <->  ( (coeff `  G ) `  N
)  =/=  0 ) )
3029biimpa 472 . . . . . 6  |-  ( ( G  e.  (Poly `  S )  /\  G  =/=  0 p )  -> 
( (coeff `  G
) `  N )  =/=  0 )
3130adantl 454 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  G ) `  N
)  =/=  0 )
3220, 23, 27, 31mulne0d 9676 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( ( (coeff `  F ) `  M
)  x.  ( (coeff `  G ) `  N
) )  =/=  0
)
3317, 32eqnetrd 2621 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( (coeff `  ( F  o F  x.  G ) ) `  ( M  +  N
) )  =/=  0
)
34 eqid 2438 . . . 4  |-  (coeff `  ( F  o F  x.  G ) )  =  (coeff `  ( F  o F  x.  G
) )
35 eqid 2438 . . . 4  |-  (deg `  ( F  o F  x.  G ) )  =  (deg `  ( F  o F  x.  G
) )
3634, 35dgrub 20155 . . 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 ) ) )
376, 13, 33, 36syl3anc 1185 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( M  +  N )  <_  (deg `  ( F  o F  x.  G ) ) )
38 dgrcl 20154 . . . . 5  |-  ( ( F  o F  x.  G )  e.  (Poly `  CC )  ->  (deg `  ( F  o F  x.  G ) )  e.  NN0 )
396, 38syl 16 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( F  o F  x.  G
) )  e.  NN0 )
4039nn0red 10277 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( F  o F  x.  G
) )  e.  RR )
4113nn0red 10277 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  ( M  +  N )  e.  RR )
4240, 41letri3d 9217 . 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 ) ) ) ) )
434, 37, 42mpbir2and 890 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 360    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   -->wf 5452   ` cfv 5456  (class class class)co 6083    o Fcof 6305   CCcc 8990   0cc0 8992    + caddc 8995    x. cmul 8997    <_ cle 9123   NN0cn0 10223   0 pc0p 19563  Polycply 20105  coeffccoe 20107  degcdgr 20108
This theorem is referenced by:  dgrmulc  20191  dgrcolem1  20193  plydivlem4  20215  plydiveu  20217  fta1lem  20226  quotcan  20228  vieta1lem1  20229  vieta1lem2  20230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071
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 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482  df-0p 19564  df-ply 20109  df-coe 20111  df-dgr 20112
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