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Theorem deg1mul3le 20031
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 16634 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
323ad2ant1 978 . . . . . 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 16669 . . . . . . . 8  |-  ( R  e.  Ring  ->  A : K
--> B )
873ad2ant1 978 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  A : K --> B )
9 simp2 958 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  F  e.  K )
108, 9ffvelrnd 5863 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( A `  F )  e.  B )
11 simp3 959 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  G  e.  B )
12 deg1mul3le.t . . . . . . 7  |-  .x.  =  ( .r `  P )
136, 12rngcl 15669 . . . . . 6  |-  ( ( P  e.  Ring  /\  ( A `  F )  e.  B  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
143, 10, 11, 13syl3anc 1184 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
15 eqid 2435 . . . . . 6  |-  (coe1 `  (
( A `  F
)  .x.  G )
)  =  (coe1 `  (
( A `  F
)  .x.  G )
)
1615, 6, 1, 5coe1f 16601 . . . . 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 3461 . . . . . 6  |-  ( a  e.  ( NN0  \  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) )  ->  a  e.  NN0 )
19 simpl1 960 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  R  e.  Ring )
20 simpl2 961 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  F  e.  K
)
21 simpl3 962 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  G  e.  B
)
22 simpr 448 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  a  e.  NN0 )
23 eqid 2435 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
241, 6, 5, 4, 12, 23coe1sclmulfv 16667 . . . . . . 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 1189 . . . . . 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 461 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) ) )  ->  ( (coe1 `  (
( A `  F
)  .x.  G )
) `  a )  =  ( F ( .r `  R ) ( (coe1 `  G ) `  a ) ) )
27 eqid 2435 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
2827, 6, 1, 5coe1f 16601 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> K )
29283ad2ant3 980 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (coe1 `  G ) : NN0 --> K )
30 ssid 3359 . . . . . . . 8  |-  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) )
3130a1i 11 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) )
3229, 31suppssr 5856 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) ) )  ->  ( (coe1 `  G
) `  a )  =  ( 0g `  R ) )
3332oveq2d 6089 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) ) )  ->  ( F ( .r `  R ) ( (coe1 `  G ) `  a ) )  =  ( F ( .r
`  R ) ( 0g `  R ) ) )
34 eqid 2435 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
355, 23, 34rngrz 15693 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
36353adant3 977 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
3736adantr 452 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) ) )  ->  ( F ( .r `  R ) ( 0g `  R
) )  =  ( 0g `  R ) )
3826, 33, 373eqtrd 2471 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) ) )  ->  ( (coe1 `  (
( A `  F
)  .x.  G )
) `  a )  =  ( 0g `  R ) )
3917, 38suppss 5855 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  ( ( A `
 F )  .x.  G ) ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) )
40 cnvimass 5216 . . . . 5  |-  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  dom  (coe1 `  G
)
41 fdm 5587 . . . . . 6  |-  ( (coe1 `  G ) : NN0 --> K  ->  dom  (coe1 `  G
)  =  NN0 )
4229, 41syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  dom  (coe1 `  G )  =  NN0 )
4340, 42syl5sseq 3388 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  NN0 )
44 nn0ssre 10217 . . . . 5  |-  NN0  C_  RR
45 ressxr 9121 . . . . 5  |-  RR  C_  RR*
4644, 45sstri 3349 . . . 4  |-  NN0  C_  RR*
4743, 46syl6ss 3352 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  RR* )
48 supxrss 10903 . . 3  |-  ( ( ( `' (coe1 `  (
( A `  F
)  .x.  G )
) " ( _V 
\  { ( 0g
`  R ) } ) )  C_  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) )  /\  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) )  C_  RR* )  ->  sup ( ( `' (coe1 `  ( ( A `
 F )  .x.  G ) ) "
( _V  \  {
( 0g `  R
) } ) ) ,  RR* ,  <  )  <_  sup ( ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) ,  RR* ,  <  )
)
4939, 47, 48syl2anc 643 . 2  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  sup ( ( `' (coe1 `  ( ( A `  F )  .x.  G
) ) " ( _V  \  { ( 0g
`  R ) } ) ) ,  RR* ,  <  )  <_  sup ( ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) ,  RR* ,  <  ) )
50 deg1mul3le.d . . . 4  |-  D  =  ( deg1  `  R )
5150, 1, 6, 34, 15deg1val 20011 . . 3  |-  ( ( ( A `  F
)  .x.  G )  e.  B  ->  ( D `
 ( ( A `
 F )  .x.  G ) )  =  sup ( ( `' (coe1 `  ( ( A `
 F )  .x.  G ) ) "
( _V  \  {
( 0g `  R
) } ) ) ,  RR* ,  <  )
)
5214, 51syl 16 . 2  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  ( ( A `  F )  .x.  G ) )  =  sup ( ( `' (coe1 `  ( ( A `
 F )  .x.  G ) ) "
( _V  \  {
( 0g `  R
) } ) ) ,  RR* ,  <  )
)
5350, 1, 6, 34, 27deg1val 20011 . . 3  |-  ( G  e.  B  ->  ( D `  G )  =  sup ( ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) ,  RR* ,  <  )
)
54533ad2ant3 980 . 2  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  G )  =  sup ( ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) ,  RR* ,  <  )
)
5549, 52, 543brtr4d 4234 1  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  ( ( A `  F )  .x.  G ) )  <_ 
( D `  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    C_ wss 3312   {csn 3806   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   RRcr 8981   RR*cxr 9111    < clt 9112    <_ cle 9113   NN0cn0 10213   Basecbs 13461   .rcmulr 13522   0gc0g 13715   Ringcrg 15652  algSccascl 16363  Poly1cpl1 16563  coe1cco1 16566   deg1 cdg1 19969
This theorem is referenced by:  hbtlem2  27296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-subrg 15858  df-lmod 15944  df-lss 16001  df-ascl 16366  df-psr 16409  df-mvr 16410  df-mpl 16411  df-opsr 16417  df-psr1 16568  df-vr1 16569  df-ply1 16570  df-coe1 16573  df-cnfld 16696  df-mdeg 19970  df-deg1 19971
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