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Theorem dgrmulc 20182
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 6082 . . . 4  |-  ( F  =  0 p  -> 
( ( CC  X.  { A } )  o F  x.  F )  =  ( ( CC 
X.  { A }
)  o F  x.  0 p ) )
21fveq2d 5725 . . 3  |-  ( F  =  0 p  -> 
(deg `  ( ( CC  X.  { A }
)  o F  x.  F ) )  =  (deg `  ( ( CC  X.  { A }
)  o F  x.  0 p ) ) )
3 fveq2 5721 . . . 4  |-  ( F  =  0 p  -> 
(deg `  F )  =  (deg `  0 p
) )
4 dgr0 20173 . . . 4  |-  (deg ` 
0 p )  =  0
53, 4syl6eq 2484 . . 3  |-  ( F  =  0 p  -> 
(deg `  F )  =  0 )
62, 5eqeq12d 2450 . 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 3360 . . . . 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 20118 . . . . 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 9077 . . . . 5  |-  0  e.  CC
12 fvconst2g 5938 . . . . . . 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 2617 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  ( ( CC 
X.  { A }
) `  0 )  =/=  0 )
16 ne0p 20119 . . . . 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 20112 . . . . 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 3339 . . . 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 2436 . . . . 5  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
23 eqid 2436 . . . . 5  |-  (deg `  F )  =  (deg
`  F )
2422, 23dgrmul 20181 . . . 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 20157 . . . . 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 6089 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  ( (deg `  ( CC  X.  { A } ) )  +  (deg `  F )
)  =  ( 0  +  (deg `  F
) ) )
29 dgrcl 20145 . . . . . 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 10269 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  (deg `  F
)  e.  CC )
3231addid2d 9260 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0 p )  ->  ( 0  +  (deg `  F )
)  =  (deg `  F ) )
3325, 28, 323eqtrd 2472 . 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 9064 . . . . . . . 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 6322 . . . . . 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 9258 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( A  x.  0 )  =  0 )
4039sneqd 3820 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  { ( A  x.  0 ) }  =  { 0 } )
4140xpeq2d 4895 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( CC  X.  { ( A  x.  0 ) } )  =  ( CC  X.  { 0 } ) )
4238, 41eqtrd 2468 . . . . 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 19555 . . . . . 6  |-  0 p  =  ( CC  X.  { 0 } )
4443oveq2i 6085 . . . . 5  |-  ( ( CC  X.  { A } )  o F  x.  0 p )  =  ( ( CC 
X.  { A }
)  o F  x.  ( CC  X.  { 0 } ) )
4542, 44, 433eqtr4g 2493 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  o F  x.  0 p )  =  0 p )
4645fveq2d 5725 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  o F  x.  0 p ) )  =  (deg ` 
0 p ) )
4746, 4syl6eq 2484 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  o F  x.  0 p ) )  =  0 )
486, 33, 47pm2.61ne 2674 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 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2949    C_ wss 3313   {csn 3807    X. cxp 4869   ` cfv 5447  (class class class)co 6074    o Fcof 6296   CCcc 8981   0cc0 8983    + caddc 8986    x. cmul 8988   NN0cn0 10214   0 pc0p 19554  Polycply 20096  degcdgr 20099
This theorem is referenced by:  dgrsub  20183  dgrcolem2  20185  mpaaeu  27324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061  ax-addf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-of 6298  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-map 7013  df-pm 7014  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-sup 7439  df-oi 7472  df-card 7819  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-n0 10215  df-z 10276  df-uz 10482  df-rp 10606  df-fz 11037  df-fzo 11129  df-fl 11195  df-seq 11317  df-exp 11376  df-hash 11612  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-clim 12275  df-rlim 12276  df-sum 12473  df-0p 19555  df-ply 20100  df-coe 20102  df-dgr 20103
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