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Theorem 2idlcpbl 18393
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 758 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Ring )
2 eqid 2429 . . . . . . . . . . . . 13  |-  (LIdeal `  R )  =  (LIdeal `  R )
3 eqid 2429 . . . . . . . . . . . . 13  |-  (oppr `  R
)  =  (oppr `  R
)
4 eqid 2429 . . . . . . . . . . . . 13  |-  (LIdeal `  (oppr `  R ) )  =  (LIdeal `  (oppr
`  R ) )
5 2idlcpbl.i . . . . . . . . . . . . 13  |-  I  =  (2Ideal `  R )
62, 3, 4, 52idlval 18392 . . . . . . . . . . . 12  |-  I  =  ( (LIdeal `  R
)  i^i  (LIdeal `  (oppr `  R
) ) )
76elin2 3659 . . . . . . . . . . 11  |-  ( S  e.  I  <->  ( S  e.  (LIdeal `  R )  /\  S  e.  (LIdeal `  (oppr
`  R ) ) ) )
87simplbi 461 . . . . . . . . . 10  |-  ( S  e.  I  ->  S  e.  (LIdeal `  R )
)
98ad2antlr 731 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  e.  (LIdeal `  R
) )
102lidlsubg 18373 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  S  e.  (LIdeal `  R )
)  ->  S  e.  (SubGrp `  R ) )
111, 9, 10syl2anc 665 . . . . . . . 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 16818 . . . . . . . 8  |-  ( S  e.  (SubGrp `  R
)  ->  E  Er  X )
1511, 14syl 17 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  E  Er  X )
16 simprl 762 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  A E C )
1715, 16ersym 7383 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  C E A )
18 ringabl 17745 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Abel )
1918ad2antrr 730 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Abel )
2012, 2lidlss 18368 . . . . . . . 8  |-  ( S  e.  (LIdeal `  R
)  ->  S  C_  X
)
219, 20syl 17 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  C_  X )
22 eqid 2429 . . . . . . . 8  |-  ( -g `  R )  =  (
-g `  R )
2312, 22, 13eqgabl 17410 . . . . . . 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 665 . . . . . 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 213 . . . . 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 1018 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  A  e.  X )
27 simprr 764 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  B E D )
2812, 22, 13eqgabl 17410 . . . . . . 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 665 . . . . . 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 213 . . . . 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 1017 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  B  e.  X )
32 2idlcpbl.t . . . . 5  |-  .x.  =  ( .r `  R )
3312, 32ringcl 17729 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .x.  B )  e.  X )
341, 26, 31, 33syl3anc 1264 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A  .x.  B
)  e.  X )
3525simp1d 1017 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  C  e.  X )
3630simp2d 1018 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  D  e.  X )
3712, 32ringcl 17729 . . . 4  |-  ( ( R  e.  Ring  /\  C  e.  X  /\  D  e.  X )  ->  ( C  .x.  D )  e.  X )
381, 35, 36, 37syl3anc 1264 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  D
)  e.  X )
39 ringgrp 17720 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4039ad2antrr 730 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Grp )
4112, 32ringcl 17729 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  X  /\  B  e.  X )  ->  ( C  .x.  B )  e.  X )
421, 35, 31, 41syl3anc 1264 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  B
)  e.  X )
4312, 22grpnnncan2 16702 . . . . 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 1266 . . . 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, 31ringsubdi 17762 . . . . . 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 1019 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( D ( -g `  R ) B )  e.  S )
472, 12, 32lidlmcl 18376 . . . . . . 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 1265 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  ( D ( -g `  R
) B ) )  e.  S )
4945, 48eqeltrrd 2518 . . . . 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 2429 . . . . . . . 8  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5112, 32, 3, 50opprmul 17789 . . . . . . 7  |-  ( B ( .r `  (oppr `  R
) ) ( A ( -g `  R
) C ) )  =  ( ( A ( -g `  R
) C )  .x.  B )
5212, 32, 22, 1, 26, 35, 31rngsubdir 17763 . . . . . . 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 2482 . . . . . 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 ) ) )
543opprring 17794 . . . . . . . 8  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
5554ad2antrr 730 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
(oppr `  R )  e.  Ring )
567simprbi 465 . . . . . . . 8  |-  ( S  e.  I  ->  S  e.  (LIdeal `  (oppr
`  R ) ) )
5756ad2antlr 731 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  e.  (LIdeal `  (oppr `  R
) ) )
5825simp3d 1019 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A ( -g `  R ) C )  e.  S )
593, 12opprbas 17792 . . . . . . . 8  |-  X  =  ( Base `  (oppr `  R
) )
604, 59, 50lidlmcl 18376 . . . . . . 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 1265 . . . . . 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 2518 . . . . 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 18374 . . . . 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 1265 . . . 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 2518 . . 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 17410 . . . 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 665 . . 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 1188 . 2  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A  .x.  B
) E ( C 
.x.  D ) )
6968ex 435 1  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  (
( A E C  /\  B E D )  ->  ( A  .x.  B ) E ( C  .x.  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    C_ wss 3442   class class class wbr 4426   ` cfv 5601  (class class class)co 6305    Er wer 7368   Basecbs 15084   .rcmulr 15153   Grpcgrp 16620   -gcsg 16622  SubGrpcsubg 16762   ~QG cqg 16764   Abelcabl 17366   Ringcrg 17715  opprcoppr 17785  LIdealclidl 18328  2Idealc2idl 18390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-eqg 16767  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-subrg 17941  df-lmod 18028  df-lss 18091  df-sra 18330  df-rgmod 18331  df-lidl 18332  df-2idl 18391
This theorem is referenced by:  qus1  18394  qusrhm  18396  quscrng  18399
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