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Theorem dvdsq1p 20036
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 20019 . . . . 5  |-  ( G  e.  C  ->  G  e.  B )
543ad2ant3 980 . . . 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 15707 . . . 4  |-  ( G  e.  B  ->  ( G  .||  F  <->  E. q  e.  B  ( q  .x.  G )  =  F ) )
95, 8syl 16 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  <->  E. q  e.  B  ( q  .x.  G )  =  F ) )
10 eqcom 2406 . . . . 5  |-  ( ( q  .x.  G )  =  F  <->  F  =  ( q  .x.  G
) )
11 simprr 734 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  F  =  ( q  .x.  G ) )
12 simprl 733 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  q  e.  B )
13 simpl1 960 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  R  e.  Ring )
141ply1rng 16597 . . . . . . . . . . . . . . . . 17  |-  ( R  e.  Ring  ->  P  e. 
Ring )
1513, 14syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  P  e.  Ring )
16 rnggrp 15624 . . . . . . . . . . . . . . . 16  |-  ( P  e.  Ring  ->  P  e. 
Grp )
1715, 16syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  P  e.  Grp )
18 simpl2 961 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  F  e.  B )
19 simpr 448 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  q  e.  B )
205adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  G  e.  B )
212, 7rngcl 15632 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Ring  /\  q  e.  B  /\  G  e.  B )  ->  (
q  .x.  G )  e.  B )
2215, 19, 20, 21syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( q  .x.  G )  e.  B
)
23 eqid 2404 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  P )  =  ( 0g `  P
)
24 eqid 2404 . . . . . . . . . . . . . . . 16  |-  ( -g `  P )  =  (
-g `  P )
252, 23, 24grpsubeq0 14830 . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . 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 603 . . . . . . . . . . . 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 5691 . . . . . . . . . . 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 960 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  R  e.  Ring )
31 eqid 2404 . . . . . . . . . . . . 13  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
3231, 1, 23deg1z 19963 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( ( deg1  `  R ) `  ( 0g `  P ) )  =  -oo )
3330, 32syl 16 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( deg1  `
 R ) `  ( 0g `  P ) )  =  -oo )
3429, 33eqtrd 2436 . . . . . . . . . 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 20023 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  NN0 )
36353adant2 976 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  NN0 )
3736nn0red 10231 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  RR )
3837adantr 452 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( deg1  `
 R ) `  G )  e.  RR )
39 mnflt 10678 . . . . . . . . . . 11  |-  ( ( ( deg1  `  R ) `  G )  e.  RR  ->  -oo  <  ( ( deg1  `  R ) `  G
) )
4038, 39syl 16 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  -oo  <  ( ( deg1  `  R ) `  G ) )
4134, 40eqbrtrd 4192 . . . . . . . . 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 20030 . . . . . . . . . 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 452 . . . . . . . . 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 888 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  ( F Q G )  =  q )
4645oveq1d 6055 . . . . . . 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 2439 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  F  =  ( ( F Q G )  .x.  G ) )
4847expr 599 . . . . 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 209 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( (
q  .x.  G )  =  F  ->  F  =  ( ( F Q G )  .x.  G
) ) )
5049rexlimdva 2790 . . 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 207 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  ->  F  =  ( ( F Q G )  .x.  G ) ) )
5242, 1, 2, 3q1pcl 20031 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F Q G )  e.  B )
532, 6, 7dvdsrmul 15708 . . . 4  |-  ( ( G  e.  B  /\  ( F Q G )  e.  B )  ->  G  .||  ( ( F Q G )  .x.  G ) )
545, 52, 53syl2anc 643 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  G  .||  ( ( F Q G )  .x.  G
) )
55 breq2 4176 . . 3  |-  ( F  =  ( ( F Q G )  .x.  G )  ->  ( G  .||  F  <->  G  .||  ( ( F Q G ) 
.x.  G ) ) )
5654, 55syl5ibrcom 214 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F  =  ( ( F Q G )  .x.  G )  ->  G  .|| 
F ) )
5751, 56impbid 184 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   RRcr 8945    -oocmnf 9074    < clt 9076   NN0cn0 10177   Basecbs 13424   .rcmulr 13485   0gc0g 13678   Grpcgrp 14640   -gcsg 14643   Ringcrg 15615   ||rcdsr 15698  Poly1cpl1 16526   deg1 cdg1 19930  Unic1pcuc1p 20002  quot1pcq1p 20003
This theorem is referenced by:  dvdsr1p  20037  fta1glem1  20041  fta1glem2  20042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-subrg 15821  df-lmod 15907  df-lss 15964  df-rlreg 16298  df-psr 16372  df-mvr 16373  df-mpl 16374  df-opsr 16380  df-psr1 16531  df-vr1 16532  df-ply1 16533  df-coe1 16536  df-cnfld 16659  df-mdeg 19931  df-deg1 19932  df-uc1p 20007  df-q1p 20008
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