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Theorem dgrmulc 20056
Description: Scalar multiplication by a nonzero constant does not change the degree of a function. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
dgrmulc  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  o F  x.  F ) )  =  (deg `  F
) )

Proof of Theorem dgrmulc
StepHypRef Expression
1 oveq2 6028 . . . 4  |-  ( F  =  0 p  -> 
( ( CC  X.  { A } )  o F  x.  F )  =  ( ( CC 
X.  { A }
)  o F  x.  0 p ) )
21fveq2d 5672 . . 3  |-  ( F  =  0 p  -> 
(deg `  ( ( CC  X.  { A }
)  o F  x.  F ) )  =  (deg `  ( ( CC  X.  { A }
)  o F  x.  0 p ) ) )
3 fveq2 5668 . . . 4  |-  ( F  =  0 p  -> 
(deg `  F )  =  (deg `  0 p
) )
4 dgr0 20047 . . . 4  |-  (deg ` 
0 p )  =  0
53, 4syl6eq 2435 . . 3  |-  ( F  =  0 p  -> 
(deg `  F )  =  0 )
62, 5eqeq12d 2401 . 2  |-  ( F  =  0 p  -> 
( (deg `  (
( CC  X.  { A } )  o F  x.  F ) )  =  (deg `  F
)  <->  (deg `  ( ( CC  X.  { A }
)  o F  x.  0 p ) )  =  0 ) )
7 ssid 3310 . . . . 5  |-  CC  C_  CC
8 simpl1 960 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  A  e.  CC )
9 plyconst 19992 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
107, 8, 9sylancr 645 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  ( CC  X.  { A } )  e.  (Poly `  CC )
)
11 0cn 9017 . . . . 5  |-  0  e.  CC
12 fvconst2g 5884 . . . . . . 7  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  ( ( CC  X.  { A } ) ` 
0 )  =  A )
138, 11, 12sylancl 644 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  ( ( CC 
X.  { A }
) `  0 )  =  A )
14 simpl2 961 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  A  =/=  0
)
1513, 14eqnetrd 2568 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  ( ( CC 
X.  { A }
) `  0 )  =/=  0 )
16 ne0p 19993 . . . . 5  |-  ( ( 0  e.  CC  /\  ( ( CC  X.  { A } ) ` 
0 )  =/=  0
)  ->  ( CC  X.  { A } )  =/=  0 p )
1711, 15, 16sylancr 645 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  ( CC  X.  { A } )  =/=  0 p )
18 plyssc 19986 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
19 simpl3 962 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  F  e.  (Poly `  S ) )
2018, 19sseldi 3289 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  F  e.  (Poly `  CC ) )
21 simpr 448 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  F  =/=  0 p )
22 eqid 2387 . . . . 5  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
23 eqid 2387 . . . . 5  |-  (deg `  F )  =  (deg
`  F )
2422, 23dgrmul 20055 . . . 4  |-  ( ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  ( CC  X.  { A } )  =/=  0 p )  /\  ( F  e.  (Poly `  CC )  /\  F  =/=  0 p ) )  -> 
(deg `  ( ( CC  X.  { A }
)  o F  x.  F ) )  =  ( (deg `  ( CC  X.  { A }
) )  +  (deg
`  F ) ) )
2510, 17, 20, 21, 24syl22anc 1185 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  (deg `  (
( CC  X.  { A } )  o F  x.  F ) )  =  ( (deg `  ( CC  X.  { A } ) )  +  (deg `  F )
) )
26 0dgr 20031 . . . . 5  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { A } ) )  =  0 )
278, 26syl 16 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  (deg `  ( CC  X.  { A }
) )  =  0 )
2827oveq1d 6035 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  ( (deg `  ( CC  X.  { A } ) )  +  (deg `  F )
)  =  ( 0  +  (deg `  F
) ) )
29 dgrcl 20019 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
3019, 29syl 16 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  (deg `  F
)  e.  NN0 )
3130nn0cnd 10208 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  (deg `  F
)  e.  CC )
3231addid2d 9199 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  ( 0  +  (deg `  F )
)  =  (deg `  F ) )
3325, 28, 323eqtrd 2423 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  (deg `  (
( CC  X.  { A } )  o F  x.  F ) )  =  (deg `  F
) )
34 cnex 9004 . . . . . . . 8  |-  CC  e.  _V
3534a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  CC  e.  _V )
36 simp1 957 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  A  e.  CC )
3711a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  0  e.  CC )
3835, 36, 37ofc12 6268 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC  X.  {
( A  x.  0 ) } ) )
3936mul01d 9197 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( A  x.  0 )  =  0 )
4039sneqd 3770 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  { ( A  x.  0 ) }  =  { 0 } )
4140xpeq2d 4842 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( CC  X.  { ( A  x.  0 ) } )  =  ( CC  X.  { 0 } ) )
4238, 41eqtrd 2419 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC  X.  {
0 } ) )
43 df-0p 19429 . . . . . 6  |-  0 p  =  ( CC  X.  { 0 } )
4443oveq2i 6031 . . . . 5  |-  ( ( CC  X.  { A } )  o F  x.  0 p )  =  ( ( CC 
X.  { A }
)  o F  x.  ( CC  X.  { 0 } ) )
4542, 44, 433eqtr4g 2444 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  o F  x.  0 p )  =  0 p )
4645fveq2d 5672 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  o F  x.  0 p ) )  =  (deg ` 
0 p ) )
4746, 4syl6eq 2435 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  o F  x.  0 p ) )  =  0 )
486, 33, 47pm2.61ne 2625 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  o F  x.  F ) )  =  (deg `  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899    C_ wss 3263   {csn 3757    X. cxp 4816   ` cfv 5394  (class class class)co 6020    o Fcof 6242   CCcc 8921   0cc0 8923    + caddc 8926    x. cmul 8928   NN0cn0 10153   0 pc0p 19428  Polycply 19970  degcdgr 19973
This theorem is referenced by:  dgrsub  20057  dgrcolem2  20059  mpaaeu  27024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407  df-0p 19429  df-ply 19974  df-coe 19976  df-dgr 19977
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