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Theorem dgrmulc 22753
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 } )  oF  x.  F ) )  =  (deg `  F
) )

Proof of Theorem dgrmulc
StepHypRef Expression
1 oveq2 6204 . . . 4  |-  ( F  =  0p  -> 
( ( CC  X.  { A } )  oF  x.  F )  =  ( ( CC 
X.  { A }
)  oF  x.  0p ) )
21fveq2d 5778 . . 3  |-  ( F  =  0p  -> 
(deg `  ( ( CC  X.  { A }
)  oF  x.  F ) )  =  (deg `  ( ( CC  X.  { A }
)  oF  x.  0p ) ) )
3 fveq2 5774 . . . 4  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
4 dgr0 22744 . . . 4  |-  (deg ` 
0p )  =  0
53, 4syl6eq 2439 . . 3  |-  ( F  =  0p  -> 
(deg `  F )  =  0 )
62, 5eqeq12d 2404 . 2  |-  ( F  =  0p  -> 
( (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  (deg `  F
)  <->  (deg `  ( ( CC  X.  { A }
)  oF  x.  0p ) )  =  0 ) )
7 ssid 3436 . . . . 5  |-  CC  C_  CC
8 simpl1 997 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  A  e.  CC )
9 plyconst 22688 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
107, 8, 9sylancr 661 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( CC  X.  { A } )  e.  (Poly `  CC )
)
11 0cn 9499 . . . . 5  |-  0  e.  CC
12 fvconst2g 6027 . . . . . . 7  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  ( ( CC  X.  { A } ) ` 
0 )  =  A )
138, 11, 12sylancl 660 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( ( CC 
X.  { A }
) `  0 )  =  A )
14 simpl2 998 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  A  =/=  0
)
1513, 14eqnetrd 2675 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( ( CC 
X.  { A }
) `  0 )  =/=  0 )
16 ne0p 22689 . . . . 5  |-  ( ( 0  e.  CC  /\  ( ( CC  X.  { A } ) ` 
0 )  =/=  0
)  ->  ( CC  X.  { A } )  =/=  0p )
1711, 15, 16sylancr 661 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( CC  X.  { A } )  =/=  0p )
18 plyssc 22682 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
19 simpl3 999 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  F  e.  (Poly `  S ) )
2018, 19sseldi 3415 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  F  e.  (Poly `  CC ) )
21 simpr 459 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  F  =/=  0p )
22 eqid 2382 . . . . 5  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
23 eqid 2382 . . . . 5  |-  (deg `  F )  =  (deg
`  F )
2422, 23dgrmul 22752 . . . 4  |-  ( ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  ( CC  X.  { A } )  =/=  0p )  /\  ( F  e.  (Poly `  CC )  /\  F  =/=  0p ) )  -> 
(deg `  ( ( CC  X.  { A }
)  oF  x.  F ) )  =  ( (deg `  ( CC  X.  { A }
) )  +  (deg
`  F ) ) )
2510, 17, 20, 21, 24syl22anc 1227 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  ( (deg `  ( CC  X.  { A } ) )  +  (deg `  F )
) )
26 0dgr 22727 . . . . 5  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { A } ) )  =  0 )
278, 26syl 16 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  ( CC  X.  { A }
) )  =  0 )
2827oveq1d 6211 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( (deg `  ( CC  X.  { A } ) )  +  (deg `  F )
)  =  ( 0  +  (deg `  F
) ) )
29 dgrcl 22715 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
3019, 29syl 16 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  F
)  e.  NN0 )
3130nn0cnd 10771 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  F
)  e.  CC )
3231addid2d 9692 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( 0  +  (deg `  F )
)  =  (deg `  F ) )
3325, 28, 323eqtrd 2427 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  (deg `  F
) )
34 cnex 9484 . . . . . . . 8  |-  CC  e.  _V
3534a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  CC  e.  _V )
36 simp1 994 . . . . . . 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 6464 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )  =  ( CC  X.  { ( A  x.  0 ) } ) )
3936mul01d 9690 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( A  x.  0 )  =  0 )
4039sneqd 3956 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  { ( A  x.  0 ) }  =  { 0 } )
4140xpeq2d 4937 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( CC  X.  { ( A  x.  0 ) } )  =  ( CC  X.  { 0 } ) )
4238, 41eqtrd 2423 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )  =  ( CC  X.  { 0 } ) )
43 df-0p 22162 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
4443oveq2i 6207 . . . . 5  |-  ( ( CC  X.  { A } )  oF  x.  0p )  =  ( ( CC 
X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )
4542, 44, 433eqtr4g 2448 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  0p )  =  0p )
4645fveq2d 5778 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  oF  x.  0p ) )  =  (deg ` 
0p ) )
4746, 4syl6eq 2439 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  oF  x.  0p ) )  =  0 )
486, 33, 47pm2.61ne 2697 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  (deg `  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   _Vcvv 3034    C_ wss 3389   {csn 3944    X. cxp 4911   ` cfv 5496  (class class class)co 6196    oFcof 6437   CCcc 9401   0cc0 9403    + caddc 9406    x. cmul 9408   NN0cn0 10712   0pc0p 22161  Polycply 22666  degcdgr 22669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-rlim 13314  df-sum 13511  df-0p 22162  df-ply 22670  df-coe 22672  df-dgr 22673
This theorem is referenced by:  dgrsub  22754  dgrcolem2  22756  mpaaeu  31267
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