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Theorem uc1pmon1p 21622
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 17702 . . . 4  |-  ( R  e.  Ring  ->  P  e. 
Ring )
32adantr 465 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  P  e.  Ring )
4 uc1pmon1p.a . . . . . 6  |-  A  =  (algSc `  P )
5 eqid 2442 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2442 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
71, 4, 5, 6ply1sclf 17737 . . . . 5  |-  ( R  e.  Ring  ->  A :
( Base `  R ) --> ( Base `  P )
)
87adantr 465 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  A : ( Base `  R
) --> ( Base `  P
) )
9 uc1pmon1p.d . . . . . 6  |-  D  =  ( deg1  `  R )
10 eqid 2442 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
11 uc1pmon1p.c . . . . . 6  |-  C  =  (Unic1p `  R )
129, 10, 11uc1pldg 21619 . . . . 5  |-  ( X  e.  C  ->  (
(coe1 `  X ) `  ( D `  X ) )  e.  (Unit `  R ) )
13 uc1pmon1p.i . . . . . 6  |-  I  =  ( invr `  R
)
1410, 13, 5rnginvcl 16767 . . . . 5  |-  ( ( R  e.  Ring  /\  (
(coe1 `  X ) `  ( D `  X ) )  e.  (Unit `  R ) )  -> 
( I `  (
(coe1 `  X ) `  ( D `  X ) ) )  e.  (
Base `  R )
)
1512, 14sylan2 474 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  (
Base `  R )
)
168, 15ffvelrnd 5843 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  e.  ( Base `  P
) )
171, 6, 11uc1pcl 21614 . . . 4  |-  ( X  e.  C  ->  X  e.  ( Base `  P
) )
1817adantl 466 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  X  e.  ( Base `  P
) )
19 uc1pmon1p.t . . . 4  |-  .x.  =  ( .r `  P )
206, 19rngcl 16657 . . 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 1218 . 2  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  e.  (
Base `  P )
)
22 simpl 457 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  R  e.  Ring )
23 eqid 2442 . . . . . . . 8  |-  (RLReg `  R )  =  (RLReg `  R )
2423, 10unitrrg 17364 . . . . . . 7  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
2524adantr 465 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (Unit `  R )  C_  (RLReg `  R ) )
2610, 13unitinvcl 16765 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
(coe1 `  X ) `  ( D `  X ) )  e.  (Unit `  R ) )  -> 
( I `  (
(coe1 `  X ) `  ( D `  X ) ) )  e.  (Unit `  R ) )
2712, 26sylan2 474 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  (Unit `  R ) )
2825, 27sseldd 3356 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  (RLReg `  R ) )
299, 1, 23, 6, 19, 4deg1mul3 21586 . . . . 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 1218 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( D `  ( ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  =  ( D `  X
) )
319, 11uc1pdeg 21618 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( D `  X )  e.  NN0 )
3230, 31eqeltrd 2516 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( D `  ( ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  e. 
NN0 )
33 eqid 2442 . . . . 5  |-  ( 0g
`  P )  =  ( 0g `  P
)
349, 1, 33, 6deg1nn0clb 21560 . . . 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 470 . . 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 232 . 2  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  =/=  ( 0g `  P ) )
3730fveq2d 5694 . . 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 2442 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
391, 6, 5, 4, 19, 38coe1sclmul 17734 . . . . 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 ) ) ) } )  oF ( .r
`  R ) (coe1 `  X ) ) )
4022, 15, 18, 39syl3anc 1218 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (coe1 `  ( ( A `  ( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  =  ( ( NN0  X.  { ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) } )  oF ( .r
`  R ) (coe1 `  X ) ) )
4140fveq1d 5692 . . 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 ) ) ) } )  oF ( .r
`  R ) (coe1 `  X ) ) `  ( D `  X ) ) )
42 nn0ex 10584 . . . . . . 7  |-  NN0  e.  _V
4342a1i 11 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  NN0  e.  _V )
44 fvex 5700 . . . . . . 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 2442 . . . . . . . 8  |-  (coe1 `  X
)  =  (coe1 `  X
)
4746, 6, 1, 5coe1f 17666 . . . . . . 7  |-  ( X  e.  ( Base `  P
)  ->  (coe1 `  X
) : NN0 --> ( Base `  R ) )
48 ffn 5558 . . . . . . 7  |-  ( (coe1 `  X ) : NN0 --> (
Base `  R )  ->  (coe1 `  X )  Fn 
NN0 )
4918, 47, 483syl 20 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (coe1 `  X )  Fn  NN0 )
50 eqidd 2443 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  X  e.  C )  /\  ( D `  X )  e.  NN0 )  ->  ( (coe1 `  X
) `  ( D `  X ) )  =  ( (coe1 `  X ) `  ( D `  X ) ) )
5143, 45, 49, 50ofc1 6342 . . . . 5  |-  ( ( ( R  e.  Ring  /\  X  e.  C )  /\  ( D `  X )  e.  NN0 )  ->  ( ( ( NN0  X.  { ( I `  ( (coe1 `  X ) `  ( D `  X )
) ) } )  oF ( .r
`  R ) (coe1 `  X ) ) `  ( D `  X ) )  =  ( ( I `  ( (coe1 `  X ) `  ( D `  X )
) ) ( .r
`  R ) ( (coe1 `  X ) `  ( D `  X ) ) ) )
5231, 51mpdan 668 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( ( NN0  X.  { ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) } )  oF ( .r
`  R ) (coe1 `  X ) ) `  ( D `  X ) )  =  ( ( I `  ( (coe1 `  X ) `  ( D `  X )
) ) ( .r
`  R ) ( (coe1 `  X ) `  ( D `  X ) ) ) )
53 eqid 2442 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
5410, 13, 38, 53unitlinv 16768 . . . . 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 474 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( I `  (
(coe1 `  X ) `  ( D `  X ) ) ) ( .r
`  R ) ( (coe1 `  X ) `  ( D `  X ) ) )  =  ( 1r `  R ) )
5652, 55eqtrd 2474 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( ( NN0  X.  { ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) } )  oF ( .r
`  R ) (coe1 `  X ) ) `  ( D `  X ) )  =  ( 1r
`  R ) )
5737, 41, 563eqtrd 2478 . 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 21613 . 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 1172 1  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
( A `  (
I `  ( (coe1 `  X ) `  ( D `  X )
) ) )  .x.  X )  e.  M
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   _Vcvv 2971    C_ wss 3327   {csn 3876    X. cxp 4837    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090    oFcof 6317   NN0cn0 10578   Basecbs 14173   .rcmulr 14238   0gc0g 14377   1rcur 16602   Ringcrg 16644  Unitcui 16730   invrcinvr 16762  RLRegcrlreg 17349  algSccascl 17382  Poly1cpl1 17632  coe1cco1 17633   deg1 cdg1 21522  Monic1pcmn1 21596  Unic1pcuc1p 21597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-ofr 6320  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-fz 11437  df-fzo 11548  df-seq 11806  df-hash 12103  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-0g 14379  df-gsum 14380  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-mhm 15463  df-submnd 15464  df-grp 15544  df-minusg 15545  df-sbg 15546  df-mulg 15547  df-subg 15677  df-ghm 15744  df-cntz 15834  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-rng 16646  df-cring 16647  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-subrg 16862  df-lmod 16949  df-lss 17013  df-rlreg 17353  df-ascl 17385  df-psr 17422  df-mvr 17423  df-mpl 17424  df-opsr 17426  df-psr1 17635  df-vr1 17636  df-ply1 17637  df-coe1 17638  df-cnfld 17818  df-mdeg 21523  df-deg1 21524  df-mon1 21601  df-uc1p 21602
This theorem is referenced by:  ig1peu  21642
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