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Theorem uc1pmon1p 22677
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 )
21ply1ring 18415 . . . 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 2457 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2457 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
71, 4, 5, 6ply1sclf 18452 . . . . 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 2457 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
11 uc1pmon1p.c . . . . . 6  |-  C  =  (Unic1p `  R )
129, 10, 11uc1pldg 22674 . . . . 5  |-  ( X  e.  C  ->  (
(coe1 `  X ) `  ( D `  X ) )  e.  (Unit `  R ) )
13 uc1pmon1p.i . . . . . 6  |-  I  =  ( invr `  R
)
1410, 13, 5ringinvcl 17451 . . . . 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 6033 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  e.  ( Base `  P
) )
171, 6, 11uc1pcl 22669 . . . 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, 19ringcl 17338 . . 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 1228 . 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 2457 . . . . . . . 8  |-  (RLReg `  R )  =  (RLReg `  R )
2423, 10unitrrg 18068 . . . . . . 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 17449 . . . . . . 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 3500 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  (
I `  ( (coe1 `  X ) `  ( D `  X )
) )  e.  (RLReg `  R ) )
299, 1, 23, 6, 19, 4deg1mul3 22641 . . . . 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 1228 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( D `  ( ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  =  ( D `  X
) )
319, 11uc1pdeg 22673 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( D `  X )  e.  NN0 )
3230, 31eqeltrd 2545 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  ( D `  ( ( A `  ( I `  ( (coe1 `  X ) `  ( D `  X ) ) ) )  .x.  X ) )  e. 
NN0 )
33 eqid 2457 . . . . 5  |-  ( 0g
`  P )  =  ( 0g `  P
)
349, 1, 33, 6deg1nn0clb 22615 . . . 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 5876 . . 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 2457 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
391, 6, 5, 4, 19, 38coe1sclmul 18449 . . . . 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 1228 . . . 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 5874 . . 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 10822 . . . . . . 7  |-  NN0  e.  _V
4342a1i 11 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  C )  ->  NN0  e.  _V )
44 fvex 5882 . . . . . . 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 2457 . . . . . . . 8  |-  (coe1 `  X
)  =  (coe1 `  X
)
4746, 6, 1, 5coe1f 18376 . . . . . . 7  |-  ( X  e.  ( Base `  P
)  ->  (coe1 `  X
) : NN0 --> ( Base `  R ) )
48 ffn 5737 . . . . . . 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 2458 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  X  e.  C )  /\  ( D `  X )  e.  NN0 )  ->  ( (coe1 `  X
) `  ( D `  X ) )  =  ( (coe1 `  X ) `  ( D `  X ) ) )
5143, 45, 49, 50ofc1 6562 . . . . 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 2457 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
5410, 13, 38, 53unitlinv 17452 . . . . 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 2498 . . 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 2502 . 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 22668 . 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 1180 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 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    C_ wss 3471   {csn 4032    X. cxp 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   NN0cn0 10816   Basecbs 14643   .rcmulr 14712   0gc0g 14856   1rcur 17279   Ringcrg 17324  Unitcui 17414   invrcinvr 17446  RLRegcrlreg 18053  algSccascl 18086  Poly1cpl1 18342  coe1cco1 18343   deg1 cdg1 22577  Monic1pcmn1 22651  Unic1pcuc1p 22652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-0g 14858  df-gsum 14859  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-mulg 16186  df-subg 16324  df-ghm 16391  df-cntz 16481  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-cring 17327  df-oppr 17398  df-dvdsr 17416  df-unit 17417  df-invr 17447  df-subrg 17553  df-lmod 17640  df-lss 17705  df-rlreg 18057  df-ascl 18089  df-psr 18131  df-mvr 18132  df-mpl 18133  df-opsr 18135  df-psr1 18345  df-vr1 18346  df-ply1 18347  df-coe1 18348  df-cnfld 18547  df-mdeg 22578  df-deg1 22579  df-mon1 22656  df-uc1p 22657
This theorem is referenced by:  ig1peu  22697
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