MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  deg1mul3le Unicode version

Theorem deg1mul3le 19502
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 16326 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
323ad2ant1 976 . . . . . 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 16361 . . . . . . . 8  |-  ( R  e.  Ring  ->  A : K
--> B )
873ad2ant1 976 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  A : K --> B )
9 simp2 956 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  F  e.  K )
10 ffvelrn 5663 . . . . . . 7  |-  ( ( A : K --> B  /\  F  e.  K )  ->  ( A `  F
)  e.  B )
118, 9, 10syl2anc 642 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( A `  F )  e.  B )
12 simp3 957 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  G  e.  B )
13 deg1mul3le.t . . . . . . 7  |-  .x.  =  ( .r `  P )
146, 13rngcl 15354 . . . . . 6  |-  ( ( P  e.  Ring  /\  ( A `  F )  e.  B  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
153, 11, 12, 14syl3anc 1182 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
16 eqid 2283 . . . . . 6  |-  (coe1 `  (
( A `  F
)  .x.  G )
)  =  (coe1 `  (
( A `  F
)  .x.  G )
)
1716, 6, 1, 5coe1f 16292 . . . . 5  |-  ( ( ( A `  F
)  .x.  G )  e.  B  ->  (coe1 `  (
( A `  F
)  .x.  G )
) : NN0 --> K )
1815, 17syl 15 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (coe1 `  ( ( A `  F )  .x.  G
) ) : NN0 --> K )
19 eldifi 3298 . . . . . 6  |-  ( a  e.  ( NN0  \  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) )  ->  a  e.  NN0 )
20 simpl1 958 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  R  e.  Ring )
21 simpl2 959 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  F  e.  K
)
22 simpl3 960 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  G  e.  B
)
23 simpr 447 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  a  e.  NN0 )
24 eqid 2283 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
251, 6, 5, 4, 13, 24coe1sclmulfv 16359 . . . . . . 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 ) ) )
2620, 21, 22, 23, 25syl121anc 1187 . . . . . 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 ) ) )
2719, 26sylan2 460 . . . . 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 ) ) )
28 eqid 2283 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
2928, 6, 1, 5coe1f 16292 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> K )
30293ad2ant3 978 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (coe1 `  G ) : NN0 --> K )
31 ssid 3197 . . . . . . . 8  |-  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) )
3231a1i 10 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) )
3330, 32suppssr 5659 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) ) )  ->  ( (coe1 `  G
) `  a )  =  ( 0g `  R ) )
3433oveq2d 5874 . . . . 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 ) ) )
35 eqid 2283 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
365, 24, 35rngrz 15378 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
37363adant3 975 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
3837adantr 451 . . . . 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 ) )
3927, 34, 383eqtrd 2319 . . . 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 ) )
4018, 39suppss 5658 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  ( ( A `
 F )  .x.  G ) ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  ( `' (coe1 `  G ) " ( _V  \  { ( 0g
`  R ) } ) ) )
41 cnvimass 5033 . . . . 5  |-  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  dom  (coe1 `  G
)
42 fdm 5393 . . . . . 6  |-  ( (coe1 `  G ) : NN0 --> K  ->  dom  (coe1 `  G
)  =  NN0 )
4330, 42syl 15 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  dom  (coe1 `  G )  =  NN0 )
4441, 43syl5sseq 3226 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  NN0 )
45 nn0ssre 9969 . . . . 5  |-  NN0  C_  RR
46 ressxr 8876 . . . . 5  |-  RR  C_  RR*
4745, 46sstri 3188 . . . 4  |-  NN0  C_  RR*
4844, 47syl6ss 3191 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) 
C_  RR* )
49 supxrss 10651 . . 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* ,  <  )
)
5040, 48, 49syl2anc 642 . 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* ,  <  ) )
51 deg1mul3le.d . . . 4  |-  D  =  ( deg1  `  R )
5251, 1, 6, 35, 16deg1val 19482 . . 3  |-  ( ( ( A `  F
)  .x.  G )  e.  B  ->  ( D `
 ( ( A `
 F )  .x.  G ) )  =  sup ( ( `' (coe1 `  ( ( A `
 F )  .x.  G ) ) "
( _V  \  {
( 0g `  R
) } ) ) ,  RR* ,  <  )
)
5315, 52syl 15 . 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* ,  <  )
)
5451, 1, 6, 35, 28deg1val 19482 . . 3  |-  ( G  e.  B  ->  ( D `  G )  =  sup ( ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) ,  RR* ,  <  )
)
55543ad2ant3 978 . 2  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  G )  =  sup ( ( `' (coe1 `  G ) "
( _V  \  {
( 0g `  R
) } ) ) ,  RR* ,  <  )
)
5650, 53, 553brtr4d 4053 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   RR*cxr 8866    < clt 8867    <_ cle 8868   NN0cn0 9965   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Ringcrg 15337  algSccascl 16052  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440
This theorem is referenced by:  hbtlem2  27328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442
  Copyright terms: Public domain W3C validator