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Theorem dgrmulc 22646
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 6289 . . . 4  |-  ( F  =  0p  -> 
( ( CC  X.  { A } )  oF  x.  F )  =  ( ( CC 
X.  { A }
)  oF  x.  0p ) )
21fveq2d 5860 . . 3  |-  ( F  =  0p  -> 
(deg `  ( ( CC  X.  { A }
)  oF  x.  F ) )  =  (deg `  ( ( CC  X.  { A }
)  oF  x.  0p ) ) )
3 fveq2 5856 . . . 4  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
4 dgr0 22637 . . . 4  |-  (deg ` 
0p )  =  0
53, 4syl6eq 2500 . . 3  |-  ( F  =  0p  -> 
(deg `  F )  =  0 )
62, 5eqeq12d 2465 . 2  |-  ( F  =  0p  -> 
( (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  (deg `  F
)  <->  (deg `  ( ( CC  X.  { A }
)  oF  x.  0p ) )  =  0 ) )
7 ssid 3508 . . . . 5  |-  CC  C_  CC
8 simpl1 1000 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  A  e.  CC )
9 plyconst 22581 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
107, 8, 9sylancr 663 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( CC  X.  { A } )  e.  (Poly `  CC )
)
11 0cn 9591 . . . . 5  |-  0  e.  CC
12 fvconst2g 6109 . . . . . . 7  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  ( ( CC  X.  { A } ) ` 
0 )  =  A )
138, 11, 12sylancl 662 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( ( CC 
X.  { A }
) `  0 )  =  A )
14 simpl2 1001 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  A  =/=  0
)
1513, 14eqnetrd 2736 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( ( CC 
X.  { A }
) `  0 )  =/=  0 )
16 ne0p 22582 . . . . 5  |-  ( ( 0  e.  CC  /\  ( ( CC  X.  { A } ) ` 
0 )  =/=  0
)  ->  ( CC  X.  { A } )  =/=  0p )
1711, 15, 16sylancr 663 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( CC  X.  { A } )  =/=  0p )
18 plyssc 22575 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
19 simpl3 1002 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  F  e.  (Poly `  S ) )
2018, 19sseldi 3487 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  F  e.  (Poly `  CC ) )
21 simpr 461 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  F  =/=  0p )
22 eqid 2443 . . . . 5  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
23 eqid 2443 . . . . 5  |-  (deg `  F )  =  (deg
`  F )
2422, 23dgrmul 22645 . . . 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 1230 . . 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 22620 . . . . 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 6296 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( (deg `  ( CC  X.  { A } ) )  +  (deg `  F )
)  =  ( 0  +  (deg `  F
) ) )
29 dgrcl 22608 . . . . . 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 10861 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  F
)  e.  CC )
3231addid2d 9784 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( 0  +  (deg `  F )
)  =  (deg `  F ) )
3325, 28, 323eqtrd 2488 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  (deg `  F
) )
34 cnex 9576 . . . . . . . 8  |-  CC  e.  _V
3534a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  CC  e.  _V )
36 simp1 997 . . . . . . 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 6550 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )  =  ( CC  X.  { ( A  x.  0 ) } ) )
3936mul01d 9782 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( A  x.  0 )  =  0 )
4039sneqd 4026 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  { ( A  x.  0 ) }  =  { 0 } )
4140xpeq2d 5013 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( CC  X.  { ( A  x.  0 ) } )  =  ( CC  X.  { 0 } ) )
4238, 41eqtrd 2484 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )  =  ( CC  X.  { 0 } ) )
43 df-0p 22055 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
4443oveq2i 6292 . . . . 5  |-  ( ( CC  X.  { A } )  oF  x.  0p )  =  ( ( CC 
X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )
4542, 44, 433eqtr4g 2509 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  0p )  =  0p )
4645fveq2d 5860 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  oF  x.  0p ) )  =  (deg ` 
0p ) )
4746, 4syl6eq 2500 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  oF  x.  0p ) )  =  0 )
486, 33, 47pm2.61ne 2758 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 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095    C_ wss 3461   {csn 4014    X. cxp 4987   ` cfv 5578  (class class class)co 6281    oFcof 6523   CCcc 9493   0cc0 9495    + caddc 9498    x. cmul 9500   NN0cn0 10802   0pc0p 22054  Polycply 22559  degcdgr 22562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-rlim 13294  df-sum 13491  df-0p 22055  df-ply 22563  df-coe 22565  df-dgr 22566
This theorem is referenced by:  dgrsub  22647  dgrcolem2  22649  mpaaeu  31075
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