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Theorem dgrmul 22394
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 22393 . . 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 22346 . . . 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 22358 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
81, 7syl5eqel 2552 . . . . 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 22358 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
112, 10syl5eqel 2552 . . . . 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 10845 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( M  +  N )  e.  NN0 )
14 eqid 2460 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2460 . . . . . 6  |-  (coeff `  G )  =  (coeff `  G )
1614, 15, 1, 2coemulhi 22378 . . . . 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 22357 . . . . . . 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 22357 . . . . . . 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 22389 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( (coeff `  F ) `  M
)  =  0 ) )
2524necon3bid 2718 . . . . . . 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 22389 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  N
)  =  0 ) )
2928necon3bid 2718 . . . . . . 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 10190 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( ( (coeff `  F ) `  M
)  x.  ( (coeff `  G ) `  N
) )  =/=  0
)
3317, 32eqnetrd 2753 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( (coeff `  ( F  oF  x.  G ) ) `  ( M  +  N
) )  =/=  0
)
34 eqid 2460 . . . 4  |-  (coeff `  ( F  oF  x.  G ) )  =  (coeff `  ( F  oF  x.  G
) )
35 eqid 2460 . . . 4  |-  (deg `  ( F  oF  x.  G ) )  =  (deg `  ( F  oF  x.  G
) )
3634, 35dgrub 22359 . . 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 1223 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( M  +  N )  <_  (deg `  ( F  oF  x.  G ) ) )
38 dgrcl 22358 . . . . 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 10842 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (deg `  ( F  oF  x.  G
) )  e.  RR )
4113nn0red 10842 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  ( M  +  N )  e.  RR )
4240, 41letri3d 9715 . 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 915 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 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   -->wf 5575   ` cfv 5579  (class class class)co 6275    oFcof 6513   CCcc 9479   0cc0 9481    + caddc 9484    x. cmul 9486    <_ cle 9618   NN0cn0 10784   0pc0p 21804  Polycply 22309  coeffccoe 22311  degcdgr 22312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-rlim 13261  df-sum 13458  df-0p 21805  df-ply 22313  df-coe 22315  df-dgr 22316
This theorem is referenced by:  dgrmulc  22395  dgrcolem1  22397  plydivlem4  22419  plydiveu  22421  fta1lem  22430  quotcan  22432  vieta1lem1  22433  vieta1lem2  22434
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