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

Theorem divsrhm 16300
Description: If  S is a two-sided ideal in  R, then the "natural map" from elements to their cosets is a ring homomorphism from  R to  R  /  S. (Contributed by Mario Carneiro, 15-Jun-2015.)
Hypotheses
Ref Expression
divsrng.u  |-  U  =  ( R  /.s  ( R ~QG  S
) )
divsrng.i  |-  I  =  (2Ideal `  R )
divsrhm.x  |-  X  =  ( Base `  R
)
divsrhm.f  |-  F  =  ( x  e.  X  |->  [ x ] ( R ~QG  S ) )
Assertion
Ref Expression
divsrhm  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  F  e.  ( R RingHom  U )
)
Distinct variable groups:    x, I    x, R    x, S    x, U    x, X
Allowed substitution hint:    F( x)

Proof of Theorem divsrhm
Dummy variables  y 
z  a  b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 divsrhm.x . 2  |-  X  =  ( Base `  R
)
2 eqid 2435 . 2  |-  ( 1r
`  R )  =  ( 1r `  R
)
3 eqid 2435 . 2  |-  ( 1r
`  U )  =  ( 1r `  U
)
4 eqid 2435 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2435 . 2  |-  ( .r
`  U )  =  ( .r `  U
)
6 simpl 444 . 2  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  R  e.  Ring )
7 divsrng.u . . 3  |-  U  =  ( R  /.s  ( R ~QG  S
) )
8 divsrng.i . . 3  |-  I  =  (2Ideal `  R )
97, 8divsrng 16299 . 2  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  U  e.  Ring )
10 eqid 2435 . . . . . . . . 9  |-  (LIdeal `  R )  =  (LIdeal `  R )
11 eqid 2435 . . . . . . . . 9  |-  (oppr `  R
)  =  (oppr `  R
)
12 eqid 2435 . . . . . . . . 9  |-  (LIdeal `  (oppr `  R ) )  =  (LIdeal `  (oppr
`  R ) )
1310, 11, 12, 82idlval 16296 . . . . . . . 8  |-  I  =  ( (LIdeal `  R
)  i^i  (LIdeal `  (oppr `  R
) ) )
1413elin2 3523 . . . . . . 7  |-  ( S  e.  I  <->  ( S  e.  (LIdeal `  R )  /\  S  e.  (LIdeal `  (oppr
`  R ) ) ) )
1514simplbi 447 . . . . . 6  |-  ( S  e.  I  ->  S  e.  (LIdeal `  R )
)
1610lidlsubg 16278 . . . . . 6  |-  ( ( R  e.  Ring  /\  S  e.  (LIdeal `  R )
)  ->  S  e.  (SubGrp `  R ) )
1715, 16sylan2 461 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  S  e.  (SubGrp `  R )
)
18 eqid 2435 . . . . . 6  |-  ( R ~QG  S )  =  ( R ~QG  S )
191, 18eqger 14982 . . . . 5  |-  ( S  e.  (SubGrp `  R
)  ->  ( R ~QG  S
)  Er  X )
2017, 19syl 16 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( R ~QG  S )  Er  X
)
21 fvex 5734 . . . . . 6  |-  ( Base `  R )  e.  _V
221, 21eqeltri 2505 . . . . 5  |-  X  e. 
_V
2322a1i 11 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  X  e.  _V )
24 divsrhm.f . . . 4  |-  F  =  ( x  e.  X  |->  [ x ] ( R ~QG  S ) )
2520, 23, 24divsfval 13764 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( F `  ( 1r `  R ) )  =  [ ( 1r `  R ) ] ( R ~QG  S ) )
267, 8, 2divs1 16298 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( U  e.  Ring  /\  [
( 1r `  R
) ] ( R ~QG  S )  =  ( 1r
`  U ) ) )
2726simprd 450 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  [ ( 1r `  R ) ] ( R ~QG  S )  =  ( 1r `  U ) )
2825, 27eqtrd 2467 . 2  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( F `  ( 1r `  R ) )  =  ( 1r `  U
) )
297a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  U  =  ( R  /.s  ( R ~QG  S ) ) )
301a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  X  =  ( Base `  R
) )
311, 18, 8, 42idlcpbl 16297 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  (
( a ( R ~QG  S ) c  /\  b
( R ~QG  S ) d )  ->  ( a ( .r `  R ) b ) ( R ~QG  S ) ( c ( .r `  R ) d ) ) )
321, 4rngcl 15669 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  y  e.  X  /\  z  e.  X )  ->  (
y ( .r `  R ) z )  e.  X )
33323expb 1154 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
( .r `  R
) z )  e.  X )
3433adantlr 696 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( y ( .r
`  R ) z )  e.  X )
3534caovclg 6231 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( c  e.  X  /\  d  e.  X ) )  -> 
( c ( .r
`  R ) d )  e.  X )
3629, 30, 20, 6, 31, 35, 4, 5divsmulval 13772 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  y  e.  X  /\  z  e.  X
)  ->  ( [
y ] ( R ~QG  S ) ( .r `  U ) [ z ] ( R ~QG  S ) )  =  [ ( y ( .r `  R ) z ) ] ( R ~QG  S ) )
37363expb 1154 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( [ y ] ( R ~QG  S ) ( .r
`  U ) [ z ] ( R ~QG  S ) )  =  [
( y ( .r
`  R ) z ) ] ( R ~QG  S ) )
3820adantr 452 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( R ~QG  S )  Er  X
)
3922a1i 11 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  ->  X  e.  _V )
4038, 39, 24divsfval 13764 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  y
)  =  [ y ] ( R ~QG  S ) )
4138, 39, 24divsfval 13764 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  z
)  =  [ z ] ( R ~QG  S ) )
4240, 41oveq12d 6091 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( ( F `  y ) ( .r
`  U ) ( F `  z ) )  =  ( [ y ] ( R ~QG  S ) ( .r `  U ) [ z ] ( R ~QG  S ) ) )
4338, 39, 24divsfval 13764 . . 3  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  (
y ( .r `  R ) z ) )  =  [ ( y ( .r `  R ) z ) ] ( R ~QG  S ) )
4437, 42, 433eqtr4rd 2478 . 2  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  (
y ( .r `  R ) z ) )  =  ( ( F `  y ) ( .r `  U
) ( F `  z ) ) )
45 rngabl 15685 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Abel )
4645adantr 452 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  R  e.  Abel )
47 ablnsg 15454 . . . . 5  |-  ( R  e.  Abel  ->  (NrmSGrp `  R
)  =  (SubGrp `  R ) )
4846, 47syl 16 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  (NrmSGrp `  R )  =  (SubGrp `  R ) )
4917, 48eleqtrrd 2512 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  S  e.  (NrmSGrp `  R )
)
501, 7, 24divsghm 15034 . . 3  |-  ( S  e.  (NrmSGrp `  R
)  ->  F  e.  ( R  GrpHom  U ) )
5149, 50syl 16 . 2  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  F  e.  ( R  GrpHom  U ) )
521, 2, 3, 4, 5, 6, 9, 28, 44, 51isrhm2d 15821 1  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  F  e.  ( R RingHom  U )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    e. cmpt 4258   ` cfv 5446  (class class class)co 6073    Er wer 6894   [cec 6895   Basecbs 13461   .rcmulr 13522    /.s cqus 13723  SubGrpcsubg 14930  NrmSGrpcnsg 14931   ~QG cqg 14932    GrpHom cghm 14995   Abelcabel 15405   Ringcrg 15652   1rcur 15654  opprcoppr 15719   RingHom crh 15809  LIdealclidl 16234  2Idealc2idl 16294
This theorem is referenced by:  znzrh2  16818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-0g 13719  df-imas 13726  df-divs 13727  df-mnd 14682  df-mhm 14730  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-nsg 14934  df-eqg 14935  df-ghm 14996  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-rnghom 15811  df-subrg 15858  df-lmod 15944  df-lss 16001  df-sra 16236  df-rgmod 16237  df-lidl 16238  df-2idl 16295
  Copyright terms: Public domain W3C validator