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Theorem dvdsq1p 19546
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 19529 . . . . 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 15429 . . . 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 2285 . . . . 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 16326 . . . . . . . . . . . . . . . . 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 15346 . . . . . . . . . . . . . . . 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 15354 . . . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  P )  =  ( 0g `  P
)
24 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( -g `  P )  =  (
-g `  P )
252, 23, 24grpsubeq0 14552 . . . . . . . . . . . . . . 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 5529 . . . . . . . . . . 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 2283 . . . . . . . . . . . . 13  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
3231, 1, 23deg1z 19473 . . . . . . . . . . . 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 2315 . . . . . . . . . 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 19533 . . . . . . . . . . . . . 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 10019 . . . . . . . . . . . 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 10464 . . . . . . . . . . 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 4043 . . . . . . . . 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 19540 . . . . . . . . . 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 5873 . . . . . . 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 2318 . . . . . 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 2667 . . 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 19541 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F Q G )  e.  B )
532, 6, 7dvdsrmul 15430 . . . 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 4027 . . 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 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736    -oocmnf 8865    < clt 8867   NN0cn0 9965   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Grpcgrp 14362   -gcsg 14365   Ringcrg 15337   ||rcdsr 15420  Poly1cpl1 16252   deg1 cdg1 19440  Unic1pcuc1p 19512  quot1pcq1p 19513
This theorem is referenced by:  dvdsr1p  19547  fta1glem1  19551  fta1glem2  19552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-subrg 15543  df-lmod 15629  df-lss 15690  df-rlreg 16024  df-psr 16098  df-mvr 16099  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442  df-uc1p 19517  df-q1p 19518
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