MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvdsq1p Unicode version

Theorem dvdsq1p 19562
Description: Divisibility in a polynomial ring is witnessed by the quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
dvdsq1p.p  |-  P  =  (Poly1 `  R )
dvdsq1p.d  |-  .||  =  (
||r `  P )
dvdsq1p.b  |-  B  =  ( Base `  P
)
dvdsq1p.c  |-  C  =  (Unic1p `  R )
dvdsq1p.t  |-  .x.  =  ( .r `  P )
dvdsq1p.q  |-  Q  =  (quot1p `  R )
Assertion
Ref Expression
dvdsq1p  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  <->  F  =  ( ( F Q G )  .x.  G
) ) )

Proof of Theorem dvdsq1p
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 dvdsq1p.p . . . . . 6  |-  P  =  (Poly1 `  R )
2 dvdsq1p.b . . . . . 6  |-  B  =  ( Base `  P
)
3 dvdsq1p.c . . . . . 6  |-  C  =  (Unic1p `  R )
41, 2, 3uc1pcl 19545 . . . . 5  |-  ( G  e.  C  ->  G  e.  B )
543ad2ant3 978 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  G  e.  B )
6 dvdsq1p.d . . . . 5  |-  .||  =  (
||r `  P )
7 dvdsq1p.t . . . . 5  |-  .x.  =  ( .r `  P )
82, 6, 7dvdsr2 15445 . . . 4  |-  ( G  e.  B  ->  ( G  .||  F  <->  E. q  e.  B  ( q  .x.  G )  =  F ) )
95, 8syl 15 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  <->  E. q  e.  B  ( q  .x.  G )  =  F ) )
10 eqcom 2298 . . . . 5  |-  ( ( q  .x.  G )  =  F  <->  F  =  ( q  .x.  G
) )
11 simprr 733 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  F  =  ( q  .x.  G ) )
12 simprl 732 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  q  e.  B )
13 simpl1 958 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  R  e.  Ring )
141ply1rng 16342 . . . . . . . . . . . . . . . . 17  |-  ( R  e.  Ring  ->  P  e. 
Ring )
1513, 14syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  P  e.  Ring )
16 rnggrp 15362 . . . . . . . . . . . . . . . 16  |-  ( P  e.  Ring  ->  P  e. 
Grp )
1715, 16syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  P  e.  Grp )
18 simpl2 959 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  F  e.  B )
19 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  q  e.  B )
205adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  G  e.  B )
212, 7rngcl 15370 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Ring  /\  q  e.  B  /\  G  e.  B )  ->  (
q  .x.  G )  e.  B )
2215, 19, 20, 21syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( q  .x.  G )  e.  B
)
23 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  P )  =  ( 0g `  P
)
24 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( -g `  P )  =  (
-g `  P )
252, 23, 24grpsubeq0 14568 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Grp  /\  F  e.  B  /\  ( q  .x.  G
)  e.  B )  ->  ( ( F ( -g `  P
) ( q  .x.  G ) )  =  ( 0g `  P
)  <->  F  =  (
q  .x.  G )
) )
2617, 18, 22, 25syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( ( F ( -g `  P
) ( q  .x.  G ) )  =  ( 0g `  P
)  <->  F  =  (
q  .x.  G )
) )
2726biimprd 214 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( F  =  ( q  .x.  G )  ->  ( F ( -g `  P
) ( q  .x.  G ) )  =  ( 0g `  P
) ) )
2827impr 602 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  ( F ( -g `  P
) ( q  .x.  G ) )  =  ( 0g `  P
) )
2928fveq2d 5545 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( deg1  `
 R ) `  ( F ( -g `  P
) ( q  .x.  G ) ) )  =  ( ( deg1  `  R
) `  ( 0g `  P ) ) )
30 simpl1 958 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  R  e.  Ring )
31 eqid 2296 . . . . . . . . . . . . 13  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
3231, 1, 23deg1z 19489 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( ( deg1  `  R ) `  ( 0g `  P ) )  =  -oo )
3330, 32syl 15 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( deg1  `
 R ) `  ( 0g `  P ) )  =  -oo )
3429, 33eqtrd 2328 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( deg1  `
 R ) `  ( F ( -g `  P
) ( q  .x.  G ) ) )  =  -oo )
3531, 3uc1pdeg 19549 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  NN0 )
36353adant2 974 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  NN0 )
3736nn0red 10035 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  RR )
3837adantr 451 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( deg1  `
 R ) `  G )  e.  RR )
39 mnflt 10480 . . . . . . . . . . 11  |-  ( ( ( deg1  `  R ) `  G )  e.  RR  ->  -oo  <  ( ( deg1  `  R ) `  G
) )
4038, 39syl 15 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  -oo  <  ( ( deg1  `  R ) `  G ) )
4134, 40eqbrtrd 4059 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( deg1  `
 R ) `  ( F ( -g `  P
) ( q  .x.  G ) ) )  <  ( ( deg1  `  R
) `  G )
)
42 dvdsq1p.q . . . . . . . . . . 11  |-  Q  =  (quot1p `  R )
4342, 1, 2, 31, 24, 7, 3q1peqb 19556 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  (
( q  e.  B  /\  ( ( deg1  `  R ) `  ( F ( -g `  P ) ( q 
.x.  G ) ) )  <  ( ( deg1  `  R ) `  G
) )  <->  ( F Q G )  =  q ) )
4443adantr 451 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( q  e.  B  /\  ( ( deg1  `  R ) `  ( F ( -g `  P ) ( q 
.x.  G ) ) )  <  ( ( deg1  `  R ) `  G
) )  <->  ( F Q G )  =  q ) )
4512, 41, 44mpbi2and 887 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  ( F Q G )  =  q )
4645oveq1d 5889 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( F Q G )  .x.  G )  =  ( q  .x.  G ) )
4711, 46eqtr4d 2331 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  F  =  ( ( F Q G )  .x.  G ) )
4847expr 598 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( F  =  ( q  .x.  G )  ->  F  =  ( ( F Q G )  .x.  G ) ) )
4910, 48syl5bi 208 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( (
q  .x.  G )  =  F  ->  F  =  ( ( F Q G )  .x.  G
) ) )
5049rexlimdva 2680 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( E. q  e.  B  ( q  .x.  G
)  =  F  ->  F  =  ( ( F Q G )  .x.  G ) ) )
519, 50sylbid 206 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  ->  F  =  ( ( F Q G )  .x.  G ) ) )
5242, 1, 2, 3q1pcl 19557 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F Q G )  e.  B )
532, 6, 7dvdsrmul 15446 . . . 4  |-  ( ( G  e.  B  /\  ( F Q G )  e.  B )  ->  G  .||  ( ( F Q G )  .x.  G ) )
545, 52, 53syl2anc 642 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  G  .||  ( ( F Q G )  .x.  G
) )
55 breq2 4043 . . 3  |-  ( F  =  ( ( F Q G )  .x.  G )  ->  ( G  .||  F  <->  G  .||  ( ( F Q G ) 
.x.  G ) ) )
5654, 55syl5ibrcom 213 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F  =  ( ( F Q G )  .x.  G )  ->  G  .|| 
F ) )
5751, 56impbid 183 1  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  <->  F  =  ( ( F Q G )  .x.  G
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752    -oocmnf 8881    < clt 8883   NN0cn0 9981   Basecbs 13164   .rcmulr 13225   0gc0g 13416   Grpcgrp 14378   -gcsg 14381   Ringcrg 15353   ||rcdsr 15436  Poly1cpl1 16268   deg1 cdg1 19456  Unic1pcuc1p 19528  quot1pcq1p 19529
This theorem is referenced by:  dvdsr1p  19563  fta1glem1  19567  fta1glem2  19568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-subrg 15559  df-lmod 15645  df-lss 15706  df-rlreg 16040  df-psr 16114  df-mvr 16115  df-mpl 16116  df-opsr 16122  df-psr1 16273  df-vr1 16274  df-ply1 16275  df-coe1 16278  df-cnfld 16394  df-mdeg 19457  df-deg1 19458  df-uc1p 19533  df-q1p 19534
  Copyright terms: Public domain W3C validator