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Theorem 2idlcpbl 16002
Description: The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
2idlcpbl.x  |-  X  =  ( Base `  R
)
2idlcpbl.r  |-  E  =  ( R ~QG  S )
2idlcpbl.i  |-  I  =  (2Ideal `  R )
2idlcpbl.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
2idlcpbl  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  (
( A E C  /\  B E D )  ->  ( A  .x.  B ) E ( C  .x.  D ) ) )

Proof of Theorem 2idlcpbl
StepHypRef Expression
1 simpll 730 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Ring )
2 eqid 2296 . . . . . . . . . . . . 13  |-  (LIdeal `  R )  =  (LIdeal `  R )
3 eqid 2296 . . . . . . . . . . . . 13  |-  (oppr `  R
)  =  (oppr `  R
)
4 eqid 2296 . . . . . . . . . . . . 13  |-  (LIdeal `  (oppr `  R ) )  =  (LIdeal `  (oppr
`  R ) )
5 2idlcpbl.i . . . . . . . . . . . . 13  |-  I  =  (2Ideal `  R )
62, 3, 4, 52idlval 16001 . . . . . . . . . . . 12  |-  I  =  ( (LIdeal `  R
)  i^i  (LIdeal `  (oppr `  R
) ) )
76elin2 3372 . . . . . . . . . . 11  |-  ( S  e.  I  <->  ( S  e.  (LIdeal `  R )  /\  S  e.  (LIdeal `  (oppr
`  R ) ) ) )
87simplbi 446 . . . . . . . . . 10  |-  ( S  e.  I  ->  S  e.  (LIdeal `  R )
)
98ad2antlr 707 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  e.  (LIdeal `  R
) )
102lidlsubg 15983 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  S  e.  (LIdeal `  R )
)  ->  S  e.  (SubGrp `  R ) )
111, 9, 10syl2anc 642 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  e.  (SubGrp `  R
) )
12 2idlcpbl.x . . . . . . . . 9  |-  X  =  ( Base `  R
)
13 2idlcpbl.r . . . . . . . . 9  |-  E  =  ( R ~QG  S )
1412, 13eqger 14683 . . . . . . . 8  |-  ( S  e.  (SubGrp `  R
)  ->  E  Er  X )
1511, 14syl 15 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  E  Er  X )
16 simprl 732 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  A E C )
1715, 16ersym 6688 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  C E A )
18 rngabl 15386 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Abel )
1918ad2antrr 706 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Abel )
2012, 2lidlss 15977 . . . . . . . 8  |-  ( S  e.  (LIdeal `  R
)  ->  S  C_  X
)
219, 20syl 15 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  C_  X )
22 eqid 2296 . . . . . . . 8  |-  ( -g `  R )  =  (
-g `  R )
2312, 22, 13eqgabl 15147 . . . . . . 7  |-  ( ( R  e.  Abel  /\  S  C_  X )  ->  ( C E A  <->  ( C  e.  X  /\  A  e.  X  /\  ( A ( -g `  R
) C )  e.  S ) ) )
2419, 21, 23syl2anc 642 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C E A  <-> 
( C  e.  X  /\  A  e.  X  /\  ( A ( -g `  R ) C )  e.  S ) ) )
2517, 24mpbid 201 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  e.  X  /\  A  e.  X  /\  ( A ( -g `  R ) C )  e.  S ) )
2625simp2d 968 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  A  e.  X )
27 simprr 733 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  B E D )
2812, 22, 13eqgabl 15147 . . . . . . 7  |-  ( ( R  e.  Abel  /\  S  C_  X )  ->  ( B E D  <->  ( B  e.  X  /\  D  e.  X  /\  ( D ( -g `  R
) B )  e.  S ) ) )
2919, 21, 28syl2anc 642 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( B E D  <-> 
( B  e.  X  /\  D  e.  X  /\  ( D ( -g `  R ) B )  e.  S ) ) )
3027, 29mpbid 201 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( B  e.  X  /\  D  e.  X  /\  ( D ( -g `  R ) B )  e.  S ) )
3130simp1d 967 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  B  e.  X )
32 2idlcpbl.t . . . . 5  |-  .x.  =  ( .r `  R )
3312, 32rngcl 15370 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .x.  B )  e.  X )
341, 26, 31, 33syl3anc 1182 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A  .x.  B
)  e.  X )
3525simp1d 967 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  C  e.  X )
3630simp2d 968 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  D  e.  X )
3712, 32rngcl 15370 . . . 4  |-  ( ( R  e.  Ring  /\  C  e.  X  /\  D  e.  X )  ->  ( C  .x.  D )  e.  X )
381, 35, 36, 37syl3anc 1182 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  D
)  e.  X )
39 rnggrp 15362 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4039ad2antrr 706 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Grp )
4112, 32rngcl 15370 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  X  /\  B  e.  X )  ->  ( C  .x.  B )  e.  X )
421, 35, 31, 41syl3anc 1182 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  B
)  e.  X )
4312, 22grpnnncan2 14577 . . . . 5  |-  ( ( R  e.  Grp  /\  ( ( C  .x.  D )  e.  X  /\  ( A  .x.  B
)  e.  X  /\  ( C  .x.  B )  e.  X ) )  ->  ( ( ( C  .x.  D ) ( -g `  R
) ( C  .x.  B ) ) (
-g `  R )
( ( A  .x.  B ) ( -g `  R ) ( C 
.x.  B ) ) )  =  ( ( C  .x.  D ) ( -g `  R
) ( A  .x.  B ) ) )
4440, 38, 34, 42, 43syl13anc 1184 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( ( ( C 
.x.  D ) (
-g `  R )
( C  .x.  B
) ) ( -g `  R ) ( ( A  .x.  B ) ( -g `  R
) ( C  .x.  B ) ) )  =  ( ( C 
.x.  D ) (
-g `  R )
( A  .x.  B
) ) )
4512, 32, 22, 1, 35, 36, 31rngsubdi 15401 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  ( D ( -g `  R
) B ) )  =  ( ( C 
.x.  D ) (
-g `  R )
( C  .x.  B
) ) )
4630simp3d 969 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( D ( -g `  R ) B )  e.  S )
472, 12, 32lidlmcl 15985 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  (LIdeal `  R ) )  /\  ( C  e.  X  /\  ( D ( -g `  R ) B )  e.  S ) )  ->  ( C  .x.  ( D ( -g `  R
) B ) )  e.  S )
481, 9, 35, 46, 47syl22anc 1183 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  ( D ( -g `  R
) B ) )  e.  S )
4945, 48eqeltrrd 2371 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( ( C  .x.  D ) ( -g `  R ) ( C 
.x.  B ) )  e.  S )
50 eqid 2296 . . . . . . . 8  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5112, 32, 3, 50opprmul 15424 . . . . . . 7  |-  ( B ( .r `  (oppr `  R
) ) ( A ( -g `  R
) C ) )  =  ( ( A ( -g `  R
) C )  .x.  B )
5212, 32, 22, 1, 26, 35, 31rngsubdir 15402 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( ( A (
-g `  R ) C )  .x.  B
)  =  ( ( A  .x.  B ) ( -g `  R
) ( C  .x.  B ) ) )
5351, 52syl5eq 2340 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( B ( .r
`  (oppr
`  R ) ) ( A ( -g `  R ) C ) )  =  ( ( A  .x.  B ) ( -g `  R
) ( C  .x.  B ) ) )
543opprrng 15429 . . . . . . . 8  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
5554ad2antrr 706 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
(oppr `  R )  e.  Ring )
567simprbi 450 . . . . . . . 8  |-  ( S  e.  I  ->  S  e.  (LIdeal `  (oppr
`  R ) ) )
5756ad2antlr 707 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  e.  (LIdeal `  (oppr `  R
) ) )
5825simp3d 969 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A ( -g `  R ) C )  e.  S )
593, 12opprbas 15427 . . . . . . . 8  |-  X  =  ( Base `  (oppr `  R
) )
604, 59, 50lidlmcl 15985 . . . . . . 7  |-  ( ( ( (oppr
`  R )  e. 
Ring  /\  S  e.  (LIdeal `  (oppr
`  R ) ) )  /\  ( B  e.  X  /\  ( A ( -g `  R
) C )  e.  S ) )  -> 
( B ( .r
`  (oppr
`  R ) ) ( A ( -g `  R ) C ) )  e.  S )
6155, 57, 31, 58, 60syl22anc 1183 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( B ( .r
`  (oppr
`  R ) ) ( A ( -g `  R ) C ) )  e.  S )
6253, 61eqeltrrd 2371 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( ( A  .x.  B ) ( -g `  R ) ( C 
.x.  B ) )  e.  S )
632, 22lidlsubcl 15984 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  (LIdeal `  R ) )  /\  ( ( ( C 
.x.  D ) (
-g `  R )
( C  .x.  B
) )  e.  S  /\  ( ( A  .x.  B ) ( -g `  R ) ( C 
.x.  B ) )  e.  S ) )  ->  ( ( ( C  .x.  D ) ( -g `  R
) ( C  .x.  B ) ) (
-g `  R )
( ( A  .x.  B ) ( -g `  R ) ( C 
.x.  B ) ) )  e.  S )
641, 9, 49, 62, 63syl22anc 1183 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( ( ( C 
.x.  D ) (
-g `  R )
( C  .x.  B
) ) ( -g `  R ) ( ( A  .x.  B ) ( -g `  R
) ( C  .x.  B ) ) )  e.  S )
6544, 64eqeltrrd 2371 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( ( C  .x.  D ) ( -g `  R ) ( A 
.x.  B ) )  e.  S )
6612, 22, 13eqgabl 15147 . . . 4  |-  ( ( R  e.  Abel  /\  S  C_  X )  ->  (
( A  .x.  B
) E ( C 
.x.  D )  <->  ( ( A  .x.  B )  e.  X  /\  ( C 
.x.  D )  e.  X  /\  ( ( C  .x.  D ) ( -g `  R
) ( A  .x.  B ) )  e.  S ) ) )
6719, 21, 66syl2anc 642 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( ( A  .x.  B ) E ( C  .x.  D )  <-> 
( ( A  .x.  B )  e.  X  /\  ( C  .x.  D
)  e.  X  /\  ( ( C  .x.  D ) ( -g `  R ) ( A 
.x.  B ) )  e.  S ) ) )
6834, 38, 65, 67mpbir3and 1135 . 2  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A  .x.  B
) E ( C 
.x.  D ) )
6968ex 423 1  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  (
( A E C  /\  B E D )  ->  ( A  .x.  B ) E ( C  .x.  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    Er wer 6673   Basecbs 13164   .rcmulr 13225   Grpcgrp 14378   -gcsg 14381  SubGrpcsubg 14631   ~QG cqg 14633   Abelcabel 15106   Ringcrg 15353  opprcoppr 15420  LIdealclidl 15939  2Idealc2idl 15999
This theorem is referenced by:  divs1  16003  divsrhm  16005  divscrng  16008
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-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
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-iun 3923  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-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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-eqg 14636  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-subrg 15559  df-lmod 15645  df-lss 15706  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-2idl 16000
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