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Theorem deg1mul3le 22245
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 18053 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
323ad2ant1 1012 . . . . . 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 18090 . . . . . . . 8  |-  ( R  e.  Ring  ->  A : K
--> B )
873ad2ant1 1012 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  A : K --> B )
9 simp2 992 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  F  e.  K )
108, 9ffvelrnd 6013 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( A `  F )  e.  B )
11 simp3 993 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  G  e.  B )
12 deg1mul3le.t . . . . . . 7  |-  .x.  =  ( .r `  P )
136, 12rngcl 16992 . . . . . 6  |-  ( ( P  e.  Ring  /\  ( A `  F )  e.  B  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
143, 10, 11, 13syl3anc 1223 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
15 eqid 2460 . . . . . 6  |-  (coe1 `  (
( A `  F
)  .x.  G )
)  =  (coe1 `  (
( A `  F
)  .x.  G )
)
1615, 6, 1, 5coe1f 18014 . . . . 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 3619 . . . . . 6  |-  ( a  e.  ( NN0  \  (
(coe1 `  G ) supp  ( 0g `  R ) ) )  ->  a  e.  NN0 )
19 simpl1 994 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  R  e.  Ring )
20 simpl2 995 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  F  e.  K
)
21 simpl3 996 . . . . . . 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 2460 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
241, 6, 5, 4, 12, 23coe1sclmulfv 18088 . . . . . . 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 1228 . . . . . 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 2460 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
2827, 6, 1, 5coe1f 18014 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> K )
29283ad2ant3 1014 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (coe1 `  G ) : NN0 --> K )
30 ssid 3516 . . . . . . . 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 10790 . . . . . . . 8  |-  NN0  e.  _V
3332a1i 11 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  NN0  e.  _V )
34 fvex 5867 . . . . . . . 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 6921 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( (coe1 `  G
) supp  ( 0g `  R ) ) ) )  ->  ( (coe1 `  G ) `  a
)  =  ( 0g
`  R ) )
3736oveq2d 6291 . . . . 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 2460 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
395, 23, 38rngrz 17016 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
40393adant3 1011 . . . . . 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 2505 . . . 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 6920 . . 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 6904 . . . . 5  |-  ( (coe1 `  G ) supp  ( 0g
`  R ) ) 
C_  dom  (coe1 `  G
)
45 fdm 5726 . . . . . 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 3545 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
(coe1 `  G ) supp  ( 0g `  R ) ) 
C_  NN0 )
48 nn0ssre 10788 . . . . 5  |-  NN0  C_  RR
49 ressxr 9626 . . . . 5  |-  RR  C_  RR*
5048, 49sstri 3506 . . . 4  |-  NN0  C_  RR*
5147, 50syl6ss 3509 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
(coe1 `  G ) supp  ( 0g `  R ) ) 
C_  RR* )
52 supxrss 11513 . . 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 22224 . . 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 22224 . . 3  |-  ( G  e.  B  ->  ( D `  G )  =  sup ( ( (coe1 `  G ) supp  ( 0g
`  R ) ) ,  RR* ,  <  )
)
58573ad2ant3 1014 . 2  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  G )  =  sup ( ( (coe1 `  G ) supp  ( 0g
`  R ) ) ,  RR* ,  <  )
)
5953, 56, 583brtr4d 4470 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 968    = wceq 1374    e. wcel 1762   _Vcvv 3106    \ cdif 3466    C_ wss 3469   class class class wbr 4440   dom cdm 4992   -->wf 5575   ` cfv 5579  (class class class)co 6275   supp csupp 6891   supcsup 7889   RRcr 9480   RR*cxr 9616    < clt 9617    <_ cle 9618   NN0cn0 10784   Basecbs 14479   .rcmulr 14545   0gc0g 14684   Ringcrg 16979  algSccascl 17724  Poly1cpl1 17980  coe1cco1 17981   deg1 cdg1 22180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-ofr 6516  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-0g 14686  df-gsum 14687  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-mhm 15770  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mulg 15854  df-subg 15986  df-ghm 16053  df-cntz 16143  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-subrg 17203  df-lmod 17290  df-lss 17355  df-ascl 17727  df-psr 17769  df-mvr 17770  df-mpl 17771  df-opsr 17773  df-psr1 17983  df-vr1 17984  df-ply1 17985  df-coe1 17986  df-cnfld 18185  df-mdeg 22181  df-deg1 22182
This theorem is referenced by:  hbtlem2  30666
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