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Theorem deg1mul3le 22686
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 )
21ply1ring 18487 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
323ad2ant1 1015 . . . . . 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 18524 . . . . . . . 8  |-  ( R  e.  Ring  ->  A : K
--> B )
873ad2ant1 1015 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  A : K --> B )
9 simp2 995 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  F  e.  K )
108, 9ffvelrnd 6008 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( A `  F )  e.  B )
11 simp3 996 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  G  e.  B )
12 deg1mul3le.t . . . . . . 7  |-  .x.  =  ( .r `  P )
136, 12ringcl 17410 . . . . . 6  |-  ( ( P  e.  Ring  /\  ( A `  F )  e.  B  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
143, 10, 11, 13syl3anc 1226 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
( A `  F
)  .x.  G )  e.  B )
15 eqid 2454 . . . . . 6  |-  (coe1 `  (
( A `  F
)  .x.  G )
)  =  (coe1 `  (
( A `  F
)  .x.  G )
)
1615, 6, 1, 5coe1f 18448 . . . . 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 3612 . . . . . 6  |-  ( a  e.  ( NN0  \  (
(coe1 `  G ) supp  ( 0g `  R ) ) )  ->  a  e.  NN0 )
19 simpl1 997 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  R  e.  Ring )
20 simpl2 998 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  F  e.  K
)
21 simpl3 999 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  G  e.  B
)
22 simpr 459 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  NN0 )  ->  a  e.  NN0 )
23 eqid 2454 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
241, 6, 5, 4, 12, 23coe1sclmulfv 18522 . . . . . . 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 1231 . . . . . 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 472 . . . . 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 2454 . . . . . . . . 9  |-  (coe1 `  G
)  =  (coe1 `  G
)
2827, 6, 1, 5coe1f 18448 . . . . . . . 8  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> K )
29283ad2ant3 1017 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (coe1 `  G ) : NN0 --> K )
30 ssid 3508 . . . . . . . 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 10797 . . . . . . . 8  |-  NN0  e.  _V
3332a1i 11 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  NN0  e.  _V )
34 fvex 5858 . . . . . . . 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 6923 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  /\  a  e.  ( NN0  \  ( (coe1 `  G
) supp  ( 0g `  R ) ) ) )  ->  ( (coe1 `  G ) `  a
)  =  ( 0g
`  R ) )
3736oveq2d 6286 . . . . 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 2454 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
395, 23, 38ringrz 17434 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
40393adant3 1014 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
4140adantr 463 . . . . 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 2499 . . . 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 6922 . . 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 5717 . . . . . 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 3537 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
(coe1 `  G ) supp  ( 0g `  R ) ) 
C_  NN0 )
48 nn0ssre 10795 . . . . 5  |-  NN0  C_  RR
49 ressxr 9626 . . . . 5  |-  RR  C_  RR*
5048, 49sstri 3498 . . . 4  |-  NN0  C_  RR*
5147, 50syl6ss 3501 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  (
(coe1 `  G ) supp  ( 0g `  R ) ) 
C_  RR* )
52 supxrss 11527 . . 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 659 . 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 22665 . . 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 22665 . . 3  |-  ( G  e.  B  ->  ( D `  G )  =  sup ( ( (coe1 `  G ) supp  ( 0g
`  R ) ) ,  RR* ,  <  )
)
58573ad2ant3 1017 . 2  |-  ( ( R  e.  Ring  /\  F  e.  K  /\  G  e.  B )  ->  ( D `  G )  =  sup ( ( (coe1 `  G ) supp  ( 0g
`  R ) ) ,  RR* ,  <  )
)
5953, 56, 583brtr4d 4469 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458    C_ wss 3461   class class class wbr 4439   dom cdm 4988   -->wf 5566   ` cfv 5570  (class class class)co 6270   supp csupp 6891   supcsup 7892   RRcr 9480   RR*cxr 9616    < clt 9617    <_ cle 9618   NN0cn0 10791   Basecbs 14719   .rcmulr 14788   0gc0g 14932   Ringcrg 17396  algSccascl 18158  Poly1cpl1 18414  coe1cco1 18415   deg1 cdg1 22621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-ofr 6514  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-0g 14934  df-gsum 14935  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-submnd 16169  df-grp 16259  df-minusg 16260  df-sbg 16261  df-mulg 16262  df-subg 16400  df-ghm 16467  df-cntz 16557  df-cmn 17002  df-abl 17003  df-mgp 17340  df-ur 17352  df-ring 17398  df-cring 17399  df-subrg 17625  df-lmod 17712  df-lss 17777  df-ascl 18161  df-psr 18203  df-mvr 18204  df-mpl 18205  df-opsr 18207  df-psr1 18417  df-vr1 18418  df-ply1 18419  df-coe1 18420  df-cnfld 18619  df-mdeg 22622  df-deg1 22623
This theorem is referenced by:  hbtlem2  31317
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