MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2idlcpbl Structured version   Unicode version

Theorem 2idlcpbl 16307
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 732 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Ring )
2 eqid 2438 . . . . . . . . . . . . 13  |-  (LIdeal `  R )  =  (LIdeal `  R )
3 eqid 2438 . . . . . . . . . . . . 13  |-  (oppr `  R
)  =  (oppr `  R
)
4 eqid 2438 . . . . . . . . . . . . 13  |-  (LIdeal `  (oppr `  R ) )  =  (LIdeal `  (oppr
`  R ) )
5 2idlcpbl.i . . . . . . . . . . . . 13  |-  I  =  (2Ideal `  R )
62, 3, 4, 52idlval 16306 . . . . . . . . . . . 12  |-  I  =  ( (LIdeal `  R
)  i^i  (LIdeal `  (oppr `  R
) ) )
76elin2 3533 . . . . . . . . . . 11  |-  ( S  e.  I  <->  ( S  e.  (LIdeal `  R )  /\  S  e.  (LIdeal `  (oppr
`  R ) ) ) )
87simplbi 448 . . . . . . . . . 10  |-  ( S  e.  I  ->  S  e.  (LIdeal `  R )
)
98ad2antlr 709 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  e.  (LIdeal `  R
) )
102lidlsubg 16288 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  S  e.  (LIdeal `  R )
)  ->  S  e.  (SubGrp `  R ) )
111, 9, 10syl2anc 644 . . . . . . . 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 14992 . . . . . . . 8  |-  ( S  e.  (SubGrp `  R
)  ->  E  Er  X )
1511, 14syl 16 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  E  Er  X )
16 simprl 734 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  A E C )
1715, 16ersym 6919 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  C E A )
18 rngabl 15695 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Abel )
1918ad2antrr 708 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Abel )
2012, 2lidlss 16282 . . . . . . . 8  |-  ( S  e.  (LIdeal `  R
)  ->  S  C_  X
)
219, 20syl 16 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  C_  X )
22 eqid 2438 . . . . . . . 8  |-  ( -g `  R )  =  (
-g `  R )
2312, 22, 13eqgabl 15456 . . . . . . 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 644 . . . . . 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 203 . . . . 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 971 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  A  e.  X )
27 simprr 735 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  B E D )
2812, 22, 13eqgabl 15456 . . . . . . 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 644 . . . . . 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 203 . . . . 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 970 . . . 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 15679 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .x.  B )  e.  X )
341, 26, 31, 33syl3anc 1185 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A  .x.  B
)  e.  X )
3525simp1d 970 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  C  e.  X )
3630simp2d 971 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  D  e.  X )
3712, 32rngcl 15679 . . . 4  |-  ( ( R  e.  Ring  /\  C  e.  X  /\  D  e.  X )  ->  ( C  .x.  D )  e.  X )
381, 35, 36, 37syl3anc 1185 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  D
)  e.  X )
39 rnggrp 15671 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4039ad2antrr 708 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Grp )
4112, 32rngcl 15679 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  X  /\  B  e.  X )  ->  ( C  .x.  B )  e.  X )
421, 35, 31, 41syl3anc 1185 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  B
)  e.  X )
4312, 22grpnnncan2 14886 . . . . 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 1187 . . . 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 15710 . . . . . 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 972 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( D ( -g `  R ) B )  e.  S )
472, 12, 32lidlmcl 16290 . . . . . . 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 1186 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  ( D ( -g `  R
) B ) )  e.  S )
4945, 48eqeltrrd 2513 . . . . 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 2438 . . . . . . . 8  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5112, 32, 3, 50opprmul 15733 . . . . . . 7  |-  ( B ( .r `  (oppr `  R
) ) ( A ( -g `  R
) C ) )  =  ( ( A ( -g `  R
) C )  .x.  B )
5212, 32, 22, 1, 26, 35, 31rngsubdir 15711 . . . . . . 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 ) ) )
543opprrng 15738 . . . . . . . 8  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
5554ad2antrr 708 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
(oppr `  R )  e.  Ring )
567simprbi 452 . . . . . . . 8  |-  ( S  e.  I  ->  S  e.  (LIdeal `  (oppr
`  R ) ) )
5756ad2antlr 709 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  e.  (LIdeal `  (oppr `  R
) ) )
5825simp3d 972 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A ( -g `  R ) C )  e.  S )
593, 12opprbas 15736 . . . . . . . 8  |-  X  =  ( Base `  (oppr `  R
) )
604, 59, 50lidlmcl 16290 . . . . . . 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 1186 . . . . . 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 2513 . . . . 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 16289 . . . . 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 1186 . . . 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 2513 . . 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 15456 . . . 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 644 . . 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 1138 . 2  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A  .x.  B
) E ( C 
.x.  D ) )
6968ex 425 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   class class class wbr 4214   ` cfv 5456  (class class class)co 6083    Er wer 6904   Basecbs 13471   .rcmulr 13532   Grpcgrp 14687   -gcsg 14690  SubGrpcsubg 14940   ~QG cqg 14942   Abelcabel 15415   Ringcrg 15662  opprcoppr 15729  LIdealclidl 16244  2Idealc2idl 16304
This theorem is referenced by:  divs1  16308  divsrhm  16310  divscrng  16313
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 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-eqg 14945  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-subrg 15868  df-lmod 15954  df-lss 16011  df-sra 16246  df-rgmod 16247  df-lidl 16248  df-2idl 16305
  Copyright terms: Public domain W3C validator