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Theorem 2idlcpbl 15986
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 2283 . . . . . . . . . . . . 13  |-  (LIdeal `  R )  =  (LIdeal `  R )
3 eqid 2283 . . . . . . . . . . . . 13  |-  (oppr `  R
)  =  (oppr `  R
)
4 eqid 2283 . . . . . . . . . . . . 13  |-  (LIdeal `  (oppr `  R ) )  =  (LIdeal `  (oppr
`  R ) )
5 2idlcpbl.i . . . . . . . . . . . . 13  |-  I  =  (2Ideal `  R )
62, 3, 4, 52idlval 15985 . . . . . . . . . . . 12  |-  I  =  ( (LIdeal `  R
)  i^i  (LIdeal `  (oppr `  R
) ) )
76elin2 3359 . . . . . . . . . . 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 15967 . . . . . . . . 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 14667 . . . . . . . 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 6672 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  C E A )
18 rngabl 15370 . . . . . . . 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 15961 . . . . . . . 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 2283 . . . . . . . 8  |-  ( -g `  R )  =  (
-g `  R )
2312, 22, 13eqgabl 15131 . . . . . . 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 15131 . . . . . . 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 15354 . . . 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 15354 . . . 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 15346 . . . . . 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 15354 . . . . . 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 14561 . . . . 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 15385 . . . . . 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 15969 . . . . . . 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 2358 . . . . 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 2283 . . . . . . . 8  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5112, 32, 3, 50opprmul 15408 . . . . . . 7  |-  ( B ( .r `  (oppr `  R
) ) ( A ( -g `  R
) C ) )  =  ( ( A ( -g `  R
) C )  .x.  B )
5212, 32, 22, 1, 26, 35, 31rngsubdir 15386 . . . . . . 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 2327 . . . . . 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 15413 . . . . . . . 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 15411 . . . . . . . 8  |-  X  =  ( Base `  (oppr `  R
) )
604, 59, 50lidlmcl 15969 . . . . . . 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 2358 . . . . 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 15968 . . . . 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 2358 . . 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 15131 . . . 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 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    Er wer 6657   Basecbs 13148   .rcmulr 13209   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615   ~QG cqg 14617   Abelcabel 15090   Ringcrg 15337  opprcoppr 15404  LIdealclidl 15923  2Idealc2idl 15983
This theorem is referenced by:  divs1  15987  divsrhm  15989  divscrng  15992
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-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
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-iun 3907  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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-eqg 14620  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-subrg 15543  df-lmod 15629  df-lss 15690  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-2idl 15984
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