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Theorem dvdsq1p 20084
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 20067 . . . . 5  |-  ( G  e.  C  ->  G  e.  B )
543ad2ant3 981 . . . 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 15753 . . . 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 2439 . . . . 5  |-  ( ( q  .x.  G )  =  F  <->  F  =  ( q  .x.  G
) )
11 simprr 735 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  F  =  ( q  .x.  G ) )
12 simprl 734 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  q  e.  B )
13 simpl1 961 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  R  e.  Ring )
141ply1rng 16643 . . . . . . . . . . . . . . . . 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 15670 . . . . . . . . . . . . . . . 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 962 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  F  e.  B )
19 simpr 449 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  q  e.  B )
205adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  G  e.  B )
212, 7rngcl 15678 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Ring  /\  q  e.  B  /\  G  e.  B )  ->  (
q  .x.  G )  e.  B )
2215, 19, 20, 21syl3anc 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( q  .x.  G )  e.  B
)
23 eqid 2437 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  P )  =  ( 0g `  P
)
24 eqid 2437 . . . . . . . . . . . . . . . 16  |-  ( -g `  P )  =  (
-g `  P )
252, 23, 24grpsubeq0 14876 . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . 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 604 . . . . . . . . . . . 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 5733 . . . . . . . . . . 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 961 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  R  e.  Ring )
31 eqid 2437 . . . . . . . . . . . . 13  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
3231, 1, 23deg1z 20011 . . . . . . . . . . . 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 2469 . . . . . . . . . 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 20071 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  NN0 )
36353adant2 977 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  NN0 )
3736nn0red 10276 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  RR )
3837adantr 453 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( deg1  `
 R ) `  G )  e.  RR )
39 mnflt 10723 . . . . . . . . . . 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 4233 . . . . . . . . 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 20078 . . . . . . . . . 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 453 . . . . . . . . 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 889 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  ( F Q G )  =  q )
4645oveq1d 6097 . . . . . . 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 2472 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  F  =  ( ( F Q G )  .x.  G ) )
4847expr 600 . . . . 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 210 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( (
q  .x.  G )  =  F  ->  F  =  ( ( F Q G )  .x.  G
) ) )
5049rexlimdva 2831 . . 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 208 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  ->  F  =  ( ( F Q G )  .x.  G ) ) )
5242, 1, 2, 3q1pcl 20079 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F Q G )  e.  B )
532, 6, 7dvdsrmul 15754 . . . 4  |-  ( ( G  e.  B  /\  ( F Q G )  e.  B )  ->  G  .||  ( ( F Q G )  .x.  G ) )
545, 52, 53syl2anc 644 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  G  .||  ( ( F Q G )  .x.  G
) )
55 breq2 4217 . . 3  |-  ( F  =  ( ( F Q G )  .x.  G )  ->  ( G  .||  F  <->  G  .||  ( ( F Q G ) 
.x.  G ) ) )
5654, 55syl5ibrcom 215 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F  =  ( ( F Q G )  .x.  G )  ->  G  .|| 
F ) )
5751, 56impbid 185 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2707   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   RRcr 8990    -oocmnf 9119    < clt 9121   NN0cn0 10222   Basecbs 13470   .rcmulr 13531   0gc0g 13724   Grpcgrp 14686   -gcsg 14689   Ringcrg 15661   ||rcdsr 15744  Poly1cpl1 16572   deg1 cdg1 19978  Unic1pcuc1p 20050  quot1pcq1p 20051
This theorem is referenced by:  dvdsr1p  20085  fta1glem1  20089  fta1glem2  20090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-ofr 6307  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-fz 11045  df-fzo 11137  df-seq 11325  df-hash 11620  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-0g 13728  df-gsum 13729  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-mhm 14739  df-submnd 14740  df-grp 14813  df-minusg 14814  df-sbg 14815  df-mulg 14816  df-subg 14942  df-ghm 15005  df-cntz 15117  df-cmn 15415  df-abl 15416  df-mgp 15650  df-rng 15664  df-cring 15665  df-ur 15666  df-oppr 15729  df-dvdsr 15747  df-unit 15748  df-invr 15778  df-subrg 15867  df-lmod 15953  df-lss 16010  df-rlreg 16344  df-psr 16418  df-mvr 16419  df-mpl 16420  df-opsr 16426  df-psr1 16577  df-vr1 16578  df-ply1 16579  df-coe1 16582  df-cnfld 16705  df-mdeg 19979  df-deg1 19980  df-uc1p 20055  df-q1p 20056
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