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Theorem dgrmul 22532
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  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (deg `  ( F  oF  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 22531 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  x.  G
) )  <_  ( M  +  N )
)
43ad2ant2r 746 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (deg `  ( F  oF  x.  G
) )  <_  ( M  +  N )
)
5 plymulcl 22484 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  x.  G
)  e.  (Poly `  CC ) )
65ad2ant2r 746 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( F  oF  x.  G )  e.  (Poly `  CC )
)
7 dgrcl 22496 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
81, 7syl5eqel 2533 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
98ad2antrr 725 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  M  e.  NN0 )
10 dgrcl 22496 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
112, 10syl5eqel 2533 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
1211ad2antrl 727 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  N  e.  NN0 )
139, 12nn0addcld 10857 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( M  +  N )  e.  NN0 )
14 eqid 2441 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2441 . . . . . 6  |-  (coeff `  G )  =  (coeff `  G )
1614, 15, 1, 2coemulhi 22516 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( M  +  N ) )  =  ( ( (coeff `  F ) `  M
)  x.  ( (coeff `  G ) `  N
) ) )
1716ad2ant2r 746 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( (coeff `  ( F  oF  x.  G ) ) `  ( M  +  N
) )  =  ( ( (coeff `  F
) `  M )  x.  ( (coeff `  G
) `  N )
) )
1814coef3 22495 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1918ad2antrr 725 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (coeff `  F
) : NN0 --> CC )
2019, 9ffvelrnd 6013 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( (coeff `  F ) `  M
)  e.  CC )
2115coef3 22495 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
2221ad2antrl 727 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (coeff `  G
) : NN0 --> CC )
2322, 12ffvelrnd 6013 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( (coeff `  G ) `  N
)  e.  CC )
241, 14dgreq0 22527 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( (coeff `  F ) `  M
)  =  0 ) )
2524necon3bid 2699 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( F  =/=  0p  <->  ( (coeff `  F ) `  M
)  =/=  0 ) )
2625biimpa 484 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
( (coeff `  F
) `  M )  =/=  0 )
2726adantr 465 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( (coeff `  F ) `  M
)  =/=  0 )
282, 15dgreq0 22527 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  N
)  =  0 ) )
2928necon3bid 2699 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  ( G  =/=  0p  <->  ( (coeff `  G ) `  N
)  =/=  0 ) )
3029biimpa 484 . . . . . 6  |-  ( ( G  e.  (Poly `  S )  /\  G  =/=  0p )  -> 
( (coeff `  G
) `  N )  =/=  0 )
3130adantl 466 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( (coeff `  G ) `  N
)  =/=  0 )
3220, 23, 27, 31mulne0d 10202 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( ( (coeff `  F ) `  M
)  x.  ( (coeff `  G ) `  N
) )  =/=  0
)
3317, 32eqnetrd 2734 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( (coeff `  ( F  oF  x.  G ) ) `  ( M  +  N
) )  =/=  0
)
34 eqid 2441 . . . 4  |-  (coeff `  ( F  oF  x.  G ) )  =  (coeff `  ( F  oF  x.  G
) )
35 eqid 2441 . . . 4  |-  (deg `  ( F  oF  x.  G ) )  =  (deg `  ( F  oF  x.  G
) )
3634, 35dgrub 22497 . . 3  |-  ( ( ( F  oF  x.  G )  e.  (Poly `  CC )  /\  ( M  +  N
)  e.  NN0  /\  ( (coeff `  ( F  oF  x.  G
) ) `  ( M  +  N )
)  =/=  0 )  ->  ( M  +  N )  <_  (deg `  ( F  oF  x.  G ) ) )
376, 13, 33, 36syl3anc 1227 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( M  +  N )  <_  (deg `  ( F  oF  x.  G ) ) )
38 dgrcl 22496 . . . . 5  |-  ( ( F  oF  x.  G )  e.  (Poly `  CC )  ->  (deg `  ( F  oF  x.  G ) )  e.  NN0 )
396, 38syl 16 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (deg `  ( F  oF  x.  G
) )  e.  NN0 )
4039nn0red 10854 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (deg `  ( F  oF  x.  G
) )  e.  RR )
4113nn0red 10854 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( M  +  N )  e.  RR )
4240, 41letri3d 9725 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( (deg `  ( F  oF  x.  G ) )  =  ( M  +  N
)  <->  ( (deg `  ( F  oF  x.  G ) )  <_ 
( M  +  N
)  /\  ( M  +  N )  <_  (deg `  ( F  oF  x.  G ) ) ) ) )
434, 37, 42mpbir2and 920 1  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (deg `  ( F  oF  x.  G
) )  =  ( M  +  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433   -->wf 5570   ` cfv 5574  (class class class)co 6277    oFcof 6519   CCcc 9488   0cc0 9490    + caddc 9493    x. cmul 9495    <_ cle 9627   NN0cn0 10796   0pc0p 21942  Polycply 22447  coeffccoe 22449  degcdgr 22450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-fl 11903  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-rlim 13286  df-sum 13483  df-0p 21943  df-ply 22451  df-coe 22453  df-dgr 22454
This theorem is referenced by:  dgrmulc  22533  dgrcolem1  22535  plydivlem4  22557  plydiveu  22559  fta1lem  22568  quotcan  22570  vieta1lem1  22571  vieta1lem2  22572
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