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Theorem dgrmulc 22395
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 6283 . . . 4  |-  ( F  =  0p  -> 
( ( CC  X.  { A } )  oF  x.  F )  =  ( ( CC 
X.  { A }
)  oF  x.  0p ) )
21fveq2d 5861 . . 3  |-  ( F  =  0p  -> 
(deg `  ( ( CC  X.  { A }
)  oF  x.  F ) )  =  (deg `  ( ( CC  X.  { A }
)  oF  x.  0p ) ) )
3 fveq2 5857 . . . 4  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
4 dgr0 22386 . . . 4  |-  (deg ` 
0p )  =  0
53, 4syl6eq 2517 . . 3  |-  ( F  =  0p  -> 
(deg `  F )  =  0 )
62, 5eqeq12d 2482 . 2  |-  ( F  =  0p  -> 
( (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  (deg `  F
)  <->  (deg `  ( ( CC  X.  { A }
)  oF  x.  0p ) )  =  0 ) )
7 ssid 3516 . . . . 5  |-  CC  C_  CC
8 simpl1 994 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  A  e.  CC )
9 plyconst 22331 . . . . 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 9577 . . . . 5  |-  0  e.  CC
12 fvconst2g 6105 . . . . . . 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 995 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  A  =/=  0
)
1513, 14eqnetrd 2753 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( ( CC 
X.  { A }
) `  0 )  =/=  0 )
16 ne0p 22332 . . . . 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 22325 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
19 simpl3 996 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  F  e.  (Poly `  S ) )
2018, 19sseldi 3495 . . . 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 2460 . . . . 5  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
23 eqid 2460 . . . . 5  |-  (deg `  F )  =  (deg
`  F )
2422, 23dgrmul 22394 . . . 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 1224 . . 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 22370 . . . . 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 6290 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( (deg `  ( CC  X.  { A } ) )  +  (deg `  F )
)  =  ( 0  +  (deg `  F
) ) )
29 dgrcl 22358 . . . . . 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 10843 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  F
)  e.  CC )
3231addid2d 9769 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  ( 0  +  (deg `  F )
)  =  (deg `  F ) )
3325, 28, 323eqtrd 2505 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S ) )  /\  F  =/=  0p )  ->  (deg `  (
( CC  X.  { A } )  oF  x.  F ) )  =  (deg `  F
) )
34 cnex 9562 . . . . . . . 8  |-  CC  e.  _V
3534a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  CC  e.  _V )
36 simp1 991 . . . . . . 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 6540 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )  =  ( CC  X.  { ( A  x.  0 ) } ) )
3936mul01d 9767 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( A  x.  0 )  =  0 )
4039sneqd 4032 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  { ( A  x.  0 ) }  =  { 0 } )
4140xpeq2d 5016 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( CC  X.  { ( A  x.  0 ) } )  =  ( CC  X.  { 0 } ) )
4238, 41eqtrd 2501 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )  =  ( CC  X.  { 0 } ) )
43 df-0p 21805 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
4443oveq2i 6286 . . . . 5  |-  ( ( CC  X.  { A } )  oF  x.  0p )  =  ( ( CC 
X.  { A }
)  oF  x.  ( CC  X.  {
0 } ) )
4542, 44, 433eqtr4g 2526 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  X.  { A }
)  oF  x.  0p )  =  0p )
4645fveq2d 5861 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  oF  x.  0p ) )  =  (deg ` 
0p ) )
4746, 4syl6eq 2517 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  F  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { A } )  oF  x.  0p ) )  =  0 )
486, 33, 47pm2.61ne 2775 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106    C_ wss 3469   {csn 4020    X. cxp 4990   ` cfv 5579  (class class class)co 6275    oFcof 6513   CCcc 9479   0cc0 9481    + caddc 9484    x. cmul 9486   NN0cn0 10784   0pc0p 21804  Polycply 22309  degcdgr 22312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-rlim 13261  df-sum 13458  df-0p 21805  df-ply 22313  df-coe 22315  df-dgr 22316
This theorem is referenced by:  dgrsub  22396  dgrcolem2  22398  mpaaeu  30693
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