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Theorem dvdsq1p 21766
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 21749 . . . . 5  |-  ( G  e.  C  ->  G  e.  B )
543ad2ant3 1011 . . . 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 16863 . . . 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 2463 . . . . 5  |-  ( ( q  .x.  G )  =  F  <->  F  =  ( q  .x.  G
) )
11 simprr 756 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  F  =  ( q  .x.  G ) )
12 simprl 755 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  q  e.  B )
13 simpl1 991 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  R  e.  Ring )
141ply1rng 17827 . . . . . . . . . . . . . . . . 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 16774 . . . . . . . . . . . . . . . 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 992 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  F  e.  B )
19 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  q  e.  B )
205adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  G  e.  B )
212, 7rngcl 16782 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Ring  /\  q  e.  B  /\  G  e.  B )  ->  (
q  .x.  G )  e.  B )
2215, 19, 20, 21syl3anc 1219 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( q  .x.  G )  e.  B
)
23 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  P )  =  ( 0g `  P
)
24 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ( -g `  P )  =  (
-g `  P )
252, 23, 24grpsubeq0 15732 . . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 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 619 . . . . . . . . . . . 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 5804 . . . . . . . . . . 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 991 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  R  e.  Ring )
31 eqid 2454 . . . . . . . . . . . . 13  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
3231, 1, 23deg1z 21692 . . . . . . . . . . . 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 2495 . . . . . . . . . 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 21753 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  NN0 )
36353adant2 1007 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  NN0 )
3736nn0red 10749 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  (
( deg1  `
 R ) `  G )  e.  RR )
3837adantr 465 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  (
( deg1  `
 R ) `  G )  e.  RR )
39 mnflt 11216 . . . . . . . . . . 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 4421 . . . . . . . . 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 21760 . . . . . . . . . 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 465 . . . . . . . . 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 912 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  ( F Q G )  =  q )
4645oveq1d 6216 . . . . . . 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 2498 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  ( q  e.  B  /\  F  =  (
q  .x.  G )
) )  ->  F  =  ( ( F Q G )  .x.  G ) )
4847expr 615 . . . . 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 217 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  /\  q  e.  B
)  ->  ( (
q  .x.  G )  =  F  ->  F  =  ( ( F Q G )  .x.  G
) ) )
5049rexlimdva 2947 . . 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 215 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  ->  F  =  ( ( F Q G )  .x.  G ) ) )
5242, 1, 2, 3q1pcl 21761 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F Q G )  e.  B )
532, 6, 7dvdsrmul 16864 . . . 4  |-  ( ( G  e.  B  /\  ( F Q G )  e.  B )  ->  G  .||  ( ( F Q G )  .x.  G ) )
545, 52, 53syl2anc 661 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  G  .||  ( ( F Q G )  .x.  G
) )
55 breq2 4405 . . 3  |-  ( F  =  ( ( F Q G )  .x.  G )  ->  ( G  .||  F  <->  G  .||  ( ( F Q G ) 
.x.  G ) ) )
5654, 55syl5ibrcom 222 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( F  =  ( ( F Q G )  .x.  G )  ->  G  .|| 
F ) )
5751, 56impbid 191 1  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  C )  ->  ( G  .||  F  <->  F  =  ( ( F Q G )  .x.  G
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2800   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   RRcr 9393   -oocmnf 9528    < clt 9530   NN0cn0 10691   Basecbs 14293   .rcmulr 14359   0gc0g 14498   Grpcgrp 15530   -gcsg 15533   Ringcrg 16769   ||rcdsr 16854  Poly1cpl1 17758   deg1 cdg1 21657  Unic1pcuc1p 21732  quot1pcq1p 21733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472  ax-addf 9473  ax-mulf 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-ofr 6432  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-tpos 6856  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-sup 7803  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-starv 14373  df-sca 14374  df-vsca 14375  df-tset 14377  df-ple 14378  df-ds 14380  df-unif 14381  df-0g 14500  df-gsum 14501  df-mre 14644  df-mrc 14645  df-acs 14647  df-mnd 15535  df-mhm 15584  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-mulg 15668  df-subg 15798  df-ghm 15865  df-cntz 15955  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-cring 16772  df-oppr 16839  df-dvdsr 16857  df-unit 16858  df-invr 16888  df-subrg 16987  df-lmod 17074  df-lss 17138  df-rlreg 17478  df-psr 17547  df-mvr 17548  df-mpl 17549  df-opsr 17551  df-psr1 17761  df-vr1 17762  df-ply1 17763  df-coe1 17764  df-cnfld 17945  df-mdeg 21658  df-deg1 21659  df-uc1p 21737  df-q1p 21738
This theorem is referenced by:  dvdsr1p  21767  fta1glem1  21771  fta1glem2  21772
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