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Theorem uc1pmon1p 20074
Description: Make a unitic polynomial monic by multiplying a factor to normalize the leading coefficient. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
uc1pmon1p.c  |-  C  =  (Unic1p `  R )
uc1pmon1p.m  |-  M  =  (Monic1p `  R )
uc1pmon1p.p  |-  P  =  (Poly1 `  R )
uc1pmon1p.t  |-  .x.  =  ( .r `  P )
uc1pmon1p.a  |-  A  =  (algSc `  P )
uc1pmon1p.d  |-  D  =  ( deg1  `  R )
uc1pmon1p.i  |-  I  =  ( invr `  R
)
Assertion
Ref Expression
uc1pmon1p  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  e.  M
)

Proof of Theorem uc1pmon1p
StepHypRef Expression
1 uc1pmon1p.p . . . . 5  |-  P  =  (Poly1 `  R )
21ply1rng 16642 . . . 4  |-  ( R  e.  Ring  ->  P  e. 
Ring )
32adantr 452 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  P  e.  Ring )
4 uc1pmon1p.a . . . . . 6  |-  A  =  (algSc `  P )
5 eqid 2436 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2436 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
71, 4, 5, 6ply1sclf 16677 . . . . 5  |-  ( R  e.  Ring  ->  A :
( Base `  R ) --> ( Base `  P )
)
87adantr 452 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  A : ( Base `  R
) --> ( Base `  P
) )
9 uc1pmon1p.d . . . . . 6  |-  D  =  ( deg1  `  R )
10 eqid 2436 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
11 uc1pmon1p.c . . . . . 6  |-  C  =  (Unic1p `  R )
129, 10, 11uc1pldg 20071 . . . . 5  |-  ( X  e.  C  ->  (
(coe1 `  X ) `  ( D `  X ) )  e.  (Unit `  R ) )
13 uc1pmon1p.i . . . . . 6  |-  I  =  ( invr `  R
)
1410, 13, 5rnginvcl 15781 . . . . 5  |-  ( ( R  e.  Ring  /\  (
(coe1 `  X ) `  ( D `  X ) )  e.  (Unit `  R ) )  -> 
( I `  (
(coe1 `  X ) `  ( D `  X ) ) )  e.  (
Base `  R )
)
1512, 14sylan2 461 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  (
Base `  R )
)
168, 15ffvelrnd 5871 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  e.  ( Base `  P
) )
171, 6, 11uc1pcl 20066 . . . 4  |-  ( X  e.  C  ->  X  e.  ( Base `  P
) )
1817adantl 453 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  X  e.  ( Base `  P
) )
19 uc1pmon1p.t . . . 4  |-  .x.  =  ( .r `  P )
206, 19rngcl 15677 . . 3  |-  ( ( P  e.  Ring  /\  ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  e.  ( Base `  P
)  /\  X  e.  ( Base `  P )
)  ->  ( ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X )  e.  (
Base `  P )
)
213, 16, 18, 20syl3anc 1184 . 2  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  e.  (
Base `  P )
)
22 simpl 444 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  R  e.  Ring )
23 eqid 2436 . . . . . . . 8  |-  (RLReg `  R )  =  (RLReg `  R )
2423, 10unitrrg 16353 . . . . . . 7  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
2524adantr 452 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (Unit `  R )  C_  (RLReg `  R ) )
2610, 13unitinvcl 15779 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
(coe1 `  X ) `  ( D `  X ) )  e.  (Unit `  R ) )  -> 
( I `  (
(coe1 `  X ) `  ( D `  X ) ) )  e.  (Unit `  R ) )
2712, 26sylan2 461 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  (Unit `  R ) )
2825, 27sseldd 3349 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  (RLReg `  R ) )
299, 1, 23, 6, 19, 4deg1mul3 20038 . . . . 5  |-  ( ( R  e.  Ring  /\  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  (RLReg `  R )  /\  X  e.  ( Base `  P
) )  ->  ( D `  ( ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  =  ( D `  X
) )
3022, 28, 18, 29syl3anc 1184 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( D `  ( ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  =  ( D `  X
) )
319, 11uc1pdeg 20070 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( D `  X )  e.  NN0 )
3230, 31eqeltrd 2510 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( D `  ( ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  e. 
NN0 )
33 eqid 2436 . . . . 5  |-  ( 0g
`  P )  =  ( 0g `  P
)
349, 1, 33, 6deg1nn0clb 20013 . . . 4  |-  ( ( R  e.  Ring  /\  (
( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  e.  (
Base `  P )
)  ->  ( (
( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  =/=  ( 0g `  P )  <->  ( D `  ( ( A `  ( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  e. 
NN0 ) )
3521, 34syldan 457 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( ( A `  ( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X )  =/=  ( 0g `  P )  <->  ( D `  ( ( A `  ( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  e. 
NN0 ) )
3632, 35mpbird 224 . 2  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  =/=  ( 0g `  P ) )
3730fveq2d 5732 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
(coe1 `  ( ( A `
 ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) ) `  ( D `  ( ( A `  ( I `
 ( (coe1 `  X
) `  ( D `  X ) ) ) )  .x.  X ) ) )  =  ( (coe1 `  ( ( A `
 ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) ) `  ( D `  X ) ) )
38 eqid 2436 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
391, 6, 5, 4, 19, 38coe1sclmul 16674 . . . . 5  |-  ( ( R  e.  Ring  /\  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  (
Base `  R )  /\  X  e.  ( Base `  P ) )  ->  (coe1 `  ( ( A `
 ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  =  ( ( NN0  X.  { ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) } )  o F ( .r
`  R ) (coe1 `  X ) ) )
4022, 15, 18, 39syl3anc 1184 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (coe1 `  ( ( A `  ( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  =  ( ( NN0  X.  { ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) } )  o F ( .r
`  R ) (coe1 `  X ) ) )
4140fveq1d 5730 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
(coe1 `  ( ( A `
 ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) ) `  ( D `  X ) )  =  ( ( ( NN0  X.  {
( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) } )  o F ( .r
`  R ) (coe1 `  X ) ) `  ( D `  X ) ) )
42 nn0ex 10227 . . . . . . 7  |-  NN0  e.  _V
4342a1i 11 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  NN0  e.  _V )
44 fvex 5742 . . . . . . 7  |-  ( I `
 ( (coe1 `  X
) `  ( D `  X ) ) )  e.  _V
4544a1i 11 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  _V )
46 eqid 2436 . . . . . . . 8  |-  (coe1 `  X
)  =  (coe1 `  X
)
4746, 6, 1, 5coe1f 16609 . . . . . . 7  |-  ( X  e.  ( Base `  P
)  ->  (coe1 `  X
) : NN0 --> ( Base `  R ) )
48 ffn 5591 . . . . . . 7  |-  ( (coe1 `  X ) : NN0 --> (
Base `  R )  ->  (coe1 `  X )  Fn 
NN0 )
4918, 47, 483syl 19 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (coe1 `  X )  Fn  NN0 )
50 eqidd 2437 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  X  e.  C )  /\  ( D `  X )  e.  NN0 )  ->  ( (coe1 `  X
) `  ( D `  X ) )  =  ( (coe1 `  X ) `  ( D `  X ) ) )
5143, 45, 49, 50ofc1 6327 . . . . 5  |-  ( ( ( R  e.  Ring  /\  X  e.  C )  /\  ( D `  X )  e.  NN0 )  ->  ( ( ( NN0  X.  { ( I `  ( (coe1 `  X ) `  ( D `  X )
) ) } )  o F ( .r
`  R ) (coe1 `  X ) ) `  ( D `  X ) )  =  ( ( I `  ( (coe1 `  X ) `  ( D `  X )
) ) ( .r
`  R ) ( (coe1 `  X ) `  ( D `  X ) ) ) )
5231, 51mpdan 650 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( ( NN0  X.  { ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) } )  o F ( .r
`  R ) (coe1 `  X ) ) `  ( D `  X ) )  =  ( ( I `  ( (coe1 `  X ) `  ( D `  X )
) ) ( .r
`  R ) ( (coe1 `  X ) `  ( D `  X ) ) ) )
53 eqid 2436 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
5410, 13, 38, 53unitlinv 15782 . . . . 5  |-  ( ( R  e.  Ring  /\  (
(coe1 `  X ) `  ( D `  X ) )  e.  (Unit `  R ) )  -> 
( ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) ( .r
`  R ) ( (coe1 `  X ) `  ( D `  X ) ) )  =  ( 1r `  R ) )
5512, 54sylan2 461 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) ( .r
`  R ) ( (coe1 `  X ) `  ( D `  X ) ) )  =  ( 1r `  R ) )
5652, 55eqtrd 2468 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( ( NN0  X.  { ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) } )  o F ( .r
`  R ) (coe1 `  X ) ) `  ( D `  X ) )  =  ( 1r
`  R ) )
5737, 41, 563eqtrd 2472 . 2  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
(coe1 `  ( ( A `
 ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) ) `  ( D `  ( ( A `  ( I `
 ( (coe1 `  X
) `  ( D `  X ) ) ) )  .x.  X ) ) )  =  ( 1r `  R ) )
58 uc1pmon1p.m . . 3  |-  M  =  (Monic1p `  R )
591, 6, 33, 9, 58, 53ismon1p 20065 . 2  |-  ( ( ( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  e.  M  <->  ( ( ( A `  ( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X )  e.  (
Base `  P )  /\  ( ( A `  ( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X )  =/=  ( 0g `  P )  /\  ( (coe1 `  ( ( A `
 ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) ) `  ( D `  ( ( A `  ( I `
 ( (coe1 `  X
) `  ( D `  X ) ) ) )  .x.  X ) ) )  =  ( 1r `  R ) ) )
6021, 36, 57, 59syl3anbrc 1138 1  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  e.  M
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    C_ wss 3320   {csn 3814    X. cxp 4876    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   NN0cn0 10221   Basecbs 13469   .rcmulr 13530   0gc0g 13723   Ringcrg 15660   1rcur 15662  Unitcui 15744   invrcinvr 15776  RLRegcrlreg 16339  algSccascl 16371  Poly1cpl1 16571  coe1cco1 16574   deg1 cdg1 19977  Monic1pcmn1 20048  Unic1pcuc1p 20049
This theorem is referenced by:  ig1peu  20094
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-addf 9069  ax-mulf 9070
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-ofr 6306  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-subrg 15866  df-lmod 15952  df-lss 16009  df-rlreg 16343  df-ascl 16374  df-psr 16417  df-mvr 16418  df-mpl 16419  df-opsr 16425  df-psr1 16576  df-vr1 16577  df-ply1 16578  df-coe1 16581  df-cnfld 16704  df-mdeg 19978  df-deg1 19979  df-mon1 20053  df-uc1p 20054
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