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Theorem deg1mul3le 21588
Description: Degree of multiplication of a polynomial on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Hypotheses
Ref Expression
deg1mul3le.d  |-  D  =  ( deg1  `  R )
deg1mul3le.p  |-  P  =  (Poly1 `  R )
deg1mul3le.k  |-  K  =  ( Base `  R
)
deg1mul3le.b  |-  B  =  ( Base `  P
)
deg1mul3le.t  |-  .x.  =  ( .r `  P )
deg1mul3le.a  |-  A  =  (algSc `  P )
Assertion
Ref Expression
deg1mul3le  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  ( ( A `  F )  .x.  G ) )  <_ 
( D `  G
) )

Proof of Theorem deg1mul3le
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 deg1mul3le.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
21ply1rng 17703 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
323ad2ant1 1009 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  P  e.  Ring )
4 deg1mul3le.a . . . . . . . . 9  |-  A  =  (algSc `  P )
5 deg1mul3le.k . . . . . . . . 9  |-  K  =  ( Base `  R
)
6 deg1mul3le.b . . . . . . . . 9  |-  B  =  ( Base `  P
)
71, 4, 5, 6ply1sclf 17738 . . . . . . . 8  |-  ( R  e.  Ring  ->  A : K
--> B )
873ad2ant1 1009 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  A : K --> B )
9 simp2 989 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  F  e.  K )
108, 9ffvelrnd 5844 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( A `  F )  e.  B )
11 simp3 990 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  G  e.  B )
12 deg1mul3le.t . . . . . . 7  |-  .x.  =  ( .r `  P )
136, 12rngcl 16658 . . . . . 6  |-  ( ( P  e.  Ring  /\  ( A `  F )  e.  B  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
143, 10, 11, 13syl3anc 1218 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
15 eqid 2443 . . . . . 6  |-  (coe1 `  (
( A `  F
)  .x.  G )
)  =  (coe1 `  (
( A `  F
)  .x.  G )
)
1615, 6, 1, 5coe1f 17667 . . . . 5  |-  ( ( ( A `  F
)  .x.  G )  e.  B  ->  (coe1 `  (
( A `  F
)  .x.  G )
) : NN0 --> K )
1714, 16syl 16 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (coe1 `  ( ( A `  F )  .x.  G
) ) : NN0 --> K )
18 eldifi 3478 . . . . . 6  |-  ( a  e.  ( NN0  \  (
(coe1 `  G ) supp  ( 0g `  R ) ) )  ->  a  e.  NN0 )
19 simpl1 991 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  R  e.  Ring )
20 simpl2 992 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  F  e.  K
)
21 simpl3 993 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  G  e.  B
)
22 simpr 461 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  a  e.  NN0 )
23 eqid 2443 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
241, 6, 5, 4, 12, 23coe1sclmulfv 17736 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  ( (coe1 `  (
( A `  F
)  .x.  G )
) `  a )  =  ( F ( .r `  R ) ( (coe1 `  G ) `  a ) ) )
2519, 20, 21, 22, 24syl121anc 1223 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  ( (coe1 `  (
( A `  F
)  .x.  G )
) `  a )  =  ( F ( .r `  R ) ( (coe1 `  G ) `  a ) ) )
2618, 25sylan2 474 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( (coe1 `  G
) supp  ( 0g `  R ) ) ) )  ->  ( (coe1 `  ( ( A `  F )  .x.  G
) ) `  a
)  =  ( F ( .r `  R
) ( (coe1 `  G
) `  a )
) )
27 eqid 2443 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
2827, 6, 1, 5coe1f 17667 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> K )
29283ad2ant3 1011 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (coe1 `  G ) : NN0 --> K )
30 ssid 3375 . . . . . . . 8  |-  ( (coe1 `  G ) supp  ( 0g
`  R ) ) 
C_  ( (coe1 `  G
) supp  ( 0g `  R ) )
3130a1i 11 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
(coe1 `  G ) supp  ( 0g `  R ) ) 
C_  ( (coe1 `  G
) supp  ( 0g `  R ) ) )
32 nn0ex 10585 . . . . . . . 8  |-  NN0  e.  _V
3332a1i 11 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  NN0  e.  _V )
34 fvex 5701 . . . . . . . 8  |-  ( 0g
`  R )  e. 
_V
3534a1i 11 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( 0g `  R )  e. 
_V )
3629, 31, 33, 35suppssr 6720 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( (coe1 `  G
) supp  ( 0g `  R ) ) ) )  ->  ( (coe1 `  G ) `  a
)  =  ( 0g
`  R ) )
3736oveq2d 6107 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( (coe1 `  G
) supp  ( 0g `  R ) ) ) )  ->  ( F
( .r `  R
) ( (coe1 `  G
) `  a )
)  =  ( F ( .r `  R
) ( 0g `  R ) ) )
38 eqid 2443 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
395, 23, 38rngrz 16682 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
40393adant3 1008 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
4140adantr 465 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( (coe1 `  G
) supp  ( 0g `  R ) ) ) )  ->  ( F
( .r `  R
) ( 0g `  R ) )  =  ( 0g `  R
) )
4226, 37, 413eqtrd 2479 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( (coe1 `  G
) supp  ( 0g `  R ) ) ) )  ->  ( (coe1 `  ( ( A `  F )  .x.  G
) ) `  a
)  =  ( 0g
`  R ) )
4317, 42suppss 6719 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
(coe1 `  ( ( A `
 F )  .x.  G ) ) supp  ( 0g `  R ) ) 
C_  ( (coe1 `  G
) supp  ( 0g `  R ) ) )
44 suppssdm 6703 . . . . 5  |-  ( (coe1 `  G ) supp  ( 0g
`  R ) ) 
C_  dom  (coe1 `  G
)
45 fdm 5563 . . . . . 6  |-  ( (coe1 `  G ) : NN0 --> K  ->  dom  (coe1 `  G
)  =  NN0 )
4629, 45syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  dom  (coe1 `  G )  =  NN0 )
4744, 46syl5sseq 3404 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
(coe1 `  G ) supp  ( 0g `  R ) ) 
C_  NN0 )
48 nn0ssre 10583 . . . . 5  |-  NN0  C_  RR
49 ressxr 9427 . . . . 5  |-  RR  C_  RR*
5048, 49sstri 3365 . . . 4  |-  NN0  C_  RR*
5147, 50syl6ss 3368 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
(coe1 `  G ) supp  ( 0g `  R ) ) 
C_  RR* )
52 supxrss 11295 . . 3  |-  ( ( ( (coe1 `  ( ( A `
 F )  .x.  G ) ) supp  ( 0g `  R ) ) 
C_  ( (coe1 `  G
) supp  ( 0g `  R ) )  /\  ( (coe1 `  G ) supp  ( 0g `  R ) ) 
C_  RR* )  ->  sup ( ( (coe1 `  (
( A `  F
)  .x.  G )
) supp  ( 0g `  R ) ) , 
RR* ,  <  )  <_  sup ( ( (coe1 `  G
) supp  ( 0g `  R ) ) , 
RR* ,  <  ) )
5343, 51, 52syl2anc 661 . 2  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  sup ( ( (coe1 `  (
( A `  F
)  .x.  G )
) supp  ( 0g `  R ) ) , 
RR* ,  <  )  <_  sup ( ( (coe1 `  G
) supp  ( 0g `  R ) ) , 
RR* ,  <  ) )
54 deg1mul3le.d . . . 4  |-  D  =  ( deg1  `  R )
5554, 1, 6, 38, 15deg1val 21567 . . 3  |-  ( ( ( A `  F
)  .x.  G )  e.  B  ->  ( D `
 ( ( A `
 F )  .x.  G ) )  =  sup ( ( (coe1 `  ( ( A `  F )  .x.  G
) ) supp  ( 0g
`  R ) ) ,  RR* ,  <  )
)
5614, 55syl 16 . 2  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  ( ( A `  F )  .x.  G ) )  =  sup ( ( (coe1 `  ( ( A `  F )  .x.  G
) ) supp  ( 0g
`  R ) ) ,  RR* ,  <  )
)
5754, 1, 6, 38, 27deg1val 21567 . . 3  |-  ( G  e.  B  ->  ( D `  G )  =  sup ( ( (coe1 `  G ) supp  ( 0g
`  R ) ) ,  RR* ,  <  )
)
58573ad2ant3 1011 . 2  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  G )  =  sup ( ( (coe1 `  G ) supp  ( 0g
`  R ) ) ,  RR* ,  <  )
)
5953, 56, 583brtr4d 4322 1  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  ( ( A `  F )  .x.  G ) )  <_ 
( D `  G
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972    \ cdif 3325    C_ wss 3328   class class class wbr 4292   dom cdm 4840   -->wf 5414   ` cfv 5418  (class class class)co 6091   supp csupp 6690   supcsup 7690   RRcr 9281   RR*cxr 9417    < clt 9418    <_ cle 9419   NN0cn0 10579   Basecbs 14174   .rcmulr 14239   0gc0g 14378   Ringcrg 16645  algSccascl 17383  Poly1cpl1 17633  coe1cco1 17634   deg1 cdg1 21523
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-ofr 6321  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-0g 14380  df-gsum 14381  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mulg 15548  df-subg 15678  df-ghm 15745  df-cntz 15835  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-subrg 16863  df-lmod 16950  df-lss 17014  df-ascl 17386  df-psr 17423  df-mvr 17424  df-mpl 17425  df-opsr 17427  df-psr1 17636  df-vr1 17637  df-ply1 17638  df-coe1 17639  df-cnfld 17819  df-mdeg 21524  df-deg1 21525
This theorem is referenced by:  hbtlem2  29480
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