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Theorem dgrmulc 21757
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 6118 . . . 4  |-  ( F  =  0p  -> 
( ( CC  X.  { A } )  oF  x.  F )  =  ( ( CC 
X.  { A }
)  oF  x.  0p ) )
21fveq2d 5714 . . 3  |-  ( F  =  0p  -> 
(deg `  ( ( CC  X.  { A }
)  oF  x.  F ) )  =  (deg `  ( ( CC  X.  { A }
)  oF  x.  0p ) ) )
3 fveq2 5710 . . . 4  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
4 dgr0 21748 . . . 4  |-  (deg ` 
0p )  =  0
53, 4syl6eq 2491 . . 3  |-  ( F  =  0p  -> 
(deg `  F )  =  0 )
62, 5eqeq12d 2457 . 2  |-  ( F  =  0p  -> 
( (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  (deg `  F
)  <->  (deg `  ( ( CC  X.  { A }
)  oF  x.  0p ) )  =  0 ) )
7 ssid 3394 . . . . 5  |-  CC  C_  CC
8 simpl1 991 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  A  e.  CC )
9 plyconst 21693 . . . . 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 9397 . . . . 5  |-  0  e.  CC
12 fvconst2g 5950 . . . . . . 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 992 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  A  =/=  0
)
1513, 14eqnetrd 2651 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( ( CC 
X.  { A }
) `  0 )  =/=  0 )
16 ne0p 21694 . . . . 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 21687 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
19 simpl3 993 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  F  e.  (Poly `  S ) )
2018, 19sseldi 3373 . . . 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 21756 . . . 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 1219 . . 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 21732 . . . . 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 6125 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( (deg `  ( CC  X.  { A } ) )  +  (deg `  F )
)  =  ( 0  +  (deg `  F
) ) )
29 dgrcl 21720 . . . . . 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 10657 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  F
)  e.  CC )
3231addid2d 9589 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( 0  +  (deg `  F )
)  =  (deg `  F ) )
3325, 28, 323eqtrd 2479 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  (deg `  F
) )
34 cnex 9382 . . . . . . . 8  |-  CC  e.  _V
3534a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  CC  e.  _V )
36 simp1 988 . . . . . . 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 6364 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )  =  ( CC  X.  { ( A  x.  0 ) } ) )
3936mul01d 9587 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( A  x.  0 )  =  0 )
4039sneqd 3908 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  { ( A  x.  0 ) }  =  { 0 } )
4140xpeq2d 4883 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( CC  X.  { ( A  x.  0 ) } )  =  ( CC  X.  { 0 } ) )
4238, 41eqtrd 2475 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )  =  ( CC  X.  { 0 } ) )
43 df-0p 21167 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
4443oveq2i 6121 . . . . 5  |-  ( ( CC  X.  { A } )  oF  x.  0p )  =  ( ( CC 
X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )
4542, 44, 433eqtr4g 2500 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  0p )  =  0p )
4645fveq2d 5714 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  oF  x.  0p ) )  =  (deg ` 
0p ) )
4746, 4syl6eq 2491 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  oF  x.  0p ) )  =  0 )
486, 33, 47pm2.61ne 2708 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2991    C_ wss 3347   {csn 3896    X. cxp 4857   ` cfv 5437  (class class class)co 6110    oFcof 6337   CCcc 9299   0cc0 9301    + caddc 9304    x. cmul 9306   NN0cn0 10598   0pc0p 21166  Polycply 21671  degcdgr 21674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379  ax-addf 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6339  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-map 7235  df-pm 7236  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-sup 7710  df-oi 7743  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-n0 10599  df-z 10666  df-uz 10881  df-rp 11011  df-fz 11457  df-fzo 11568  df-fl 11661  df-seq 11826  df-exp 11885  df-hash 12123  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-clim 12985  df-rlim 12986  df-sum 13183  df-0p 21167  df-ply 21675  df-coe 21677  df-dgr 21678
This theorem is referenced by:  dgrsub  21758  dgrcolem2  21760  mpaaeu  29530
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