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Theorem dgrmulc 24027
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 6658 . . . 4 |- ( F = 0p -> ( ( CC X. { A } ) oF x. F ) = ( ( CC X. { A } ) oF x. 0p ) )
21fveq2d 6195 . . 3 |- ( F = 0p -> (deg ` ( ( CC X. { A } ) oF x. F ) ) = (deg ` ( ( CC X. { A } ) oF x. 0p ) ) )
3 fveq2 6191 . . . 4 |- ( F = 0p -> (deg ` F ) = (deg ` 0p ) )
4 dgr0 24018 . . . 4 |- (deg ` 0p ) = 0
53, 4syl6eq 2672 . . 3 |- ( F = 0p -> (deg ` F ) = 0 )
62, 5eqeq12d 2637 . 2 |- ( F = 0p -> ( (deg ` ( ( CC X. { A } ) oF x. F ) ) = (deg ` F ) <-> (deg ` ( ( CC X. { A } ) oF x. 0p ) ) = 0 ) )
7 ssid 3624 . . . . 5 |- CC C_ CC
8 simpl1 1064 . . . . 5 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> A e. CC )
9 plyconst 23962 . . . . 5 |- ( ( CC C_ CC /\ A e. CC ) -> ( CC X. { A } ) e. (Poly ` CC ) )
107, 8, 9sylancr 695 . . . 4 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> ( CC X. { A } ) e. (Poly ` CC ) )
11 0cn 10032 . . . . 5 |- 0 e. CC
12 fvconst2g 6467 . . . . . . 7 |- ( ( A e. CC /\ 0 e. CC ) -> ( ( CC X. { A } ) ` 0 ) = A )
138, 11, 12sylancl 694 . . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> ( ( CC X. { A } ) ` 0 ) = A )
14 simpl2 1065 . . . . . 6 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> A =/= 0 )
1513, 14eqnetrd 2861 . . . . 5 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> ( ( CC X. { A } ) ` 0 ) =/= 0 )
16 ne0p 23963 . . . . 5 |- ( ( 0 e. CC /\ ( ( CC X. { A } ) ` 0 ) =/= 0 ) -> ( CC X. { A } ) =/= 0p )
1711, 15, 16sylancr 695 . . . 4 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> ( CC X. { A } ) =/= 0p )
18 plyssc 23956 . . . . 5 |- (Poly ` S ) C_ (Poly ` CC )
19 simpl3 1066 . . . . 5 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> F e. (Poly ` S ) )
2018, 19sseldi 3601 . . . 4 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> F e. (Poly ` CC ) )
21 simpr 477 . . . 4 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> F =/= 0p )
22 eqid 2622 . . . . 5 |- (deg ` ( CC X. { A } ) ) = (deg ` ( CC X. { A } ) )
23 eqid 2622 . . . . 5 |- (deg ` F ) = (deg ` F )
2422, 23dgrmul 24026 . . . 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 1327 . . 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 24001 . . . . 5 |- ( A e. CC -> (deg ` ( CC X. { A } ) ) = 0 )
278, 26syl 17 . . . 4 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> (deg ` ( CC X. { A } ) ) = 0 )
2827oveq1d 6665 . . 3 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> ( (deg ` ( CC X. { A } ) ) + (deg ` F ) ) = ( 0 + (deg ` F ) ) )
29 dgrcl 23989 . . . . . 6 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
3019, 29syl 17 . . . . 5 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> (deg ` F ) e. NN0 )
3130nn0cnd 11353 . . . 4 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> (deg ` F ) e. CC )
3231addid2d 10237 . . 3 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> ( 0 + (deg ` F ) ) = (deg ` F ) )
3325, 28, 323eqtrd 2660 . 2 |- ( ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) /\ F =/= 0p ) -> (deg ` ( ( CC X. { A } ) oF x. F ) ) = (deg ` F ) )
34 cnex 10017 . . . . . . . 8 |- CC e. _V
3534a1i 11 . . . . . . 7 |- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> CC e. _V )
36 simp1 1061 . . . . . . 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 6922 . . . . . 6 |- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> ( ( CC X. { A } ) oF x. ( CC X. { 0 } ) ) = ( CC X. { ( A x. 0 ) } ) )
3936mul01d 10235 . . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> ( A x. 0 ) = 0 )
4039sneqd 4189 . . . . . . 7 |- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> { ( A x. 0 ) } = { 0 } )
4140xpeq2d 5139 . . . . . 6 |- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> ( CC X. { ( A x. 0 ) } ) = ( CC X. { 0 } ) )
4238, 41eqtrd 2656 . . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> ( ( CC X. { A } ) oF x. ( CC X. { 0 } ) ) = ( CC X. { 0 } ) )
43 df-0p 23437 . . . . . 6 |- 0p = ( CC X. { 0 } )
4443oveq2i 6661 . . . . 5 |- ( ( CC X. { A } ) oF x. 0p ) = ( ( CC X. { A } ) oF x. ( CC X. { 0 } ) )
4542, 44, 433eqtr4g 2681 . . . 4 |- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> ( ( CC X. { A } ) oF x. 0p ) = 0p )
4645fveq2d 6195 . . 3 |- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> (deg ` ( ( CC X. { A } ) oF x. 0p ) ) = (deg ` 0p ) )
4746, 4syl6eq 2672 . 2 |- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> (deg ` ( ( CC X. { A } ) oF x. 0p ) ) = 0 )
486, 33, 47pm2.61ne 2879 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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   + caddc 9939   x. cmul 9941   NN0cn0 11292   0pc0p 23436  Polycply 23940  degcdgr 23943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947
This theorem is referenced by:  dgrsub  24028  dgrcolem2  24030  mpaaeu  37720
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