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Theorem deg1mul3le 19906
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 16569 . . . . . . 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 16604 . . . . . . . 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 5810 . . . . . 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 15604 . . . . . 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 2387 . . . . . 6  |-  (coe1 `  (
( A `  F
)  .x.  G )
)  =  (coe1 `  (
( A `  F
)  .x.  G )
)
1615, 6, 1, 5coe1f 16536 . . . . 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 3412 . . . . . 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 2387 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
241, 6, 5, 4, 12, 23coe1sclmulfv 16602 . . . . . . 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 2387 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
2827, 6, 1, 5coe1f 16536 . . . . . . . 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 3310 . . . . . . . 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 5803 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) ) )  ->  ( (coe1 `  G
) `  a )  =  ( 0g `  R ) )
3332oveq2d 6036 . . . . 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 2387 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
355, 23, 34rngrz 15628 . . . . . . 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 2423 . . . 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 5802 . . 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 5164 . . . . 5  |-  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  dom  (coe1 `  G
)
41 fdm 5535 . . . . . 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 3339 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  NN0 )
44 nn0ssre 10157 . . . . 5  |-  NN0  C_  RR
45 ressxr 9062 . . . . 5  |-  RR  C_  RR*
4644, 45sstri 3300 . . . 4  |-  NN0  C_  RR*
4743, 46syl6ss 3303 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  RR* )
48 supxrss 10843 . . 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 19886 . . 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 19886 . . 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 4183 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 1649    e. wcel 1717   _Vcvv 2899    \ cdif 3260    C_ wss 3263   {csn 3757   class class class wbr 4153   `'ccnv 4817   dom cdm 4818   "cima 4821   -->wf 5390   ` cfv 5394  (class class class)co 6020   supcsup 7380   RRcr 8922   RR*cxr 9052    < clt 9053    <_ cle 9054   NN0cn0 10153   Basecbs 13396   .rcmulr 13457   0gc0g 13650   Ringcrg 15587  algSccascl 16298  Poly1cpl1 16498  coe1cco1 16501   deg1 cdg1 19844
This theorem is referenced by:  hbtlem2  26997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-ofr 6245  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-fzo 11066  df-seq 11251  df-hash 11546  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-0g 13654  df-gsum 13655  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-mhm 14665  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-ghm 14931  df-cntz 15043  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-cring 15591  df-ur 15592  df-subrg 15793  df-lmod 15879  df-lss 15936  df-ascl 16301  df-psr 16344  df-mvr 16345  df-mpl 16346  df-opsr 16352  df-psr1 16503  df-vr1 16504  df-ply1 16505  df-coe1 16508  df-cnfld 16627  df-mdeg 19845  df-deg1 19846
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