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Theorem 2idlcpbl 17321
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 753 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Ring )
2 eqid 2443 . . . . . . . . . . . . 13  |-  (LIdeal `  R )  =  (LIdeal `  R )
3 eqid 2443 . . . . . . . . . . . . 13  |-  (oppr `  R
)  =  (oppr `  R
)
4 eqid 2443 . . . . . . . . . . . . 13  |-  (LIdeal `  (oppr `  R ) )  =  (LIdeal `  (oppr
`  R ) )
5 2idlcpbl.i . . . . . . . . . . . . 13  |-  I  =  (2Ideal `  R )
62, 3, 4, 52idlval 17320 . . . . . . . . . . . 12  |-  I  =  ( (LIdeal `  R
)  i^i  (LIdeal `  (oppr `  R
) ) )
76elin2 3546 . . . . . . . . . . 11  |-  ( S  e.  I  <->  ( S  e.  (LIdeal `  R )  /\  S  e.  (LIdeal `  (oppr
`  R ) ) ) )
87simplbi 460 . . . . . . . . . 10  |-  ( S  e.  I  ->  S  e.  (LIdeal `  R )
)
98ad2antlr 726 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  e.  (LIdeal `  R
) )
102lidlsubg 17302 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  S  e.  (LIdeal `  R )
)  ->  S  e.  (SubGrp `  R ) )
111, 9, 10syl2anc 661 . . . . . . . 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 15736 . . . . . . . 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 755 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  A E C )
1715, 16ersym 7118 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  C E A )
18 rngabl 16679 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Abel )
1918ad2antrr 725 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Abel )
2012, 2lidlss 17296 . . . . . . . 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 2443 . . . . . . . 8  |-  ( -g `  R )  =  (
-g `  R )
2312, 22, 13eqgabl 16324 . . . . . . 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 661 . . . . . 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 210 . . . . 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 1001 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  A  e.  X )
27 simprr 756 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  B E D )
2812, 22, 13eqgabl 16324 . . . . . . 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 661 . . . . . 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 210 . . . . 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 1000 . . . 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 16663 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .x.  B )  e.  X )
341, 26, 31, 33syl3anc 1218 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A  .x.  B
)  e.  X )
3525simp1d 1000 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  C  e.  X )
3630simp2d 1001 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  D  e.  X )
3712, 32rngcl 16663 . . . 4  |-  ( ( R  e.  Ring  /\  C  e.  X  /\  D  e.  X )  ->  ( C  .x.  D )  e.  X )
381, 35, 36, 37syl3anc 1218 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  D
)  e.  X )
39 rnggrp 16655 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
4039ad2antrr 725 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  R  e.  Grp )
4112, 32rngcl 16663 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  X  /\  B  e.  X )  ->  ( C  .x.  B )  e.  X )
421, 35, 31, 41syl3anc 1218 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( C  .x.  B
)  e.  X )
4312, 22grpnnncan2 15626 . . . . 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 1220 . . . 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 16695 . . . . . 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 1002 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( D ( -g `  R ) B )  e.  S )
472, 12, 32lidlmcl 17304 . . . . . . 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 1219 . . . . . 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 2443 . . . . . . . 8  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5112, 32, 3, 50opprmul 16723 . . . . . . 7  |-  ( B ( .r `  (oppr `  R
) ) ( A ( -g `  R
) C ) )  =  ( ( A ( -g `  R
) C )  .x.  B )
5212, 32, 22, 1, 26, 35, 31rngsubdir 16696 . . . . . . 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 2487 . . . . . 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 16728 . . . . . . . 8  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
5554ad2antrr 725 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
(oppr `  R )  e.  Ring )
567simprbi 464 . . . . . . . 8  |-  ( S  e.  I  ->  S  e.  (LIdeal `  (oppr
`  R ) ) )
5756ad2antlr 726 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  ->  S  e.  (LIdeal `  (oppr `  R
) ) )
5825simp3d 1002 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A ( -g `  R ) C )  e.  S )
593, 12opprbas 16726 . . . . . . . 8  |-  X  =  ( Base `  (oppr `  R
) )
604, 59, 50lidlmcl 17304 . . . . . . 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 1219 . . . . . 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 17303 . . . . 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 1219 . . . 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 16324 . . . 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 661 . . 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 1171 . 2  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( A E C  /\  B E D ) )  -> 
( A  .x.  B
) E ( C 
.x.  D ) )
6968ex 434 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3333   class class class wbr 4297   ` cfv 5423  (class class class)co 6096    Er wer 7103   Basecbs 14179   .rcmulr 14244   Grpcgrp 15415   -gcsg 15418  SubGrpcsubg 15680   ~QG cqg 15682   Abelcabel 16283   Ringcrg 16650  opprcoppr 16719  LIdealclidl 17256  2Idealc2idl 17318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-subg 15683  df-eqg 15685  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-subrg 16868  df-lmod 16955  df-lss 17019  df-sra 17258  df-rgmod 17259  df-lidl 17260  df-2idl 17319
This theorem is referenced by:  divs1  17322  divsrhm  17324  divscrng  17327
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