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Theorem divsrhm 16005
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 2296 . 2  |-  ( 1r
`  R )  =  ( 1r `  R
)
3 eqid 2296 . 2  |-  ( 1r
`  U )  =  ( 1r `  U
)
4 eqid 2296 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2296 . 2  |-  ( .r
`  U )  =  ( .r `  U
)
6 simpl 443 . 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 16004 . 2  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  U  e.  Ring )
10 eqid 2296 . . . . . . . . 9  |-  (LIdeal `  R )  =  (LIdeal `  R )
11 eqid 2296 . . . . . . . . 9  |-  (oppr `  R
)  =  (oppr `  R
)
12 eqid 2296 . . . . . . . . 9  |-  (LIdeal `  (oppr `  R ) )  =  (LIdeal `  (oppr
`  R ) )
1310, 11, 12, 82idlval 16001 . . . . . . . 8  |-  I  =  ( (LIdeal `  R
)  i^i  (LIdeal `  (oppr `  R
) ) )
1413elin2 3372 . . . . . . 7  |-  ( S  e.  I  <->  ( S  e.  (LIdeal `  R )  /\  S  e.  (LIdeal `  (oppr
`  R ) ) ) )
1514simplbi 446 . . . . . 6  |-  ( S  e.  I  ->  S  e.  (LIdeal `  R )
)
1610lidlsubg 15983 . . . . . 6  |-  ( ( R  e.  Ring  /\  S  e.  (LIdeal `  R )
)  ->  S  e.  (SubGrp `  R ) )
1715, 16sylan2 460 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  S  e.  (SubGrp `  R )
)
18 eqid 2296 . . . . . 6  |-  ( R ~QG  S )  =  ( R ~QG  S )
191, 18eqger 14683 . . . . 5  |-  ( S  e.  (SubGrp `  R
)  ->  ( R ~QG  S
)  Er  X )
2017, 19syl 15 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( R ~QG  S )  Er  X
)
21 fvex 5555 . . . . . 6  |-  ( Base `  R )  e.  _V
221, 21eqeltri 2366 . . . . 5  |-  X  e. 
_V
2322a1i 10 . . . 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 13465 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( F `  ( 1r `  R ) )  =  [ ( 1r `  R ) ] ( R ~QG  S ) )
267, 8, 2divs1 16003 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( U  e.  Ring  /\  [
( 1r `  R
) ] ( R ~QG  S )  =  ( 1r
`  U ) ) )
2726simprd 449 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  [ ( 1r `  R ) ] ( R ~QG  S )  =  ( 1r `  U ) )
2825, 27eqtrd 2328 . 2  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( F `  ( 1r `  R ) )  =  ( 1r `  U
) )
297a1i 10 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  U  =  ( R  /.s  ( R ~QG  S ) ) )
301a1i 10 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  X  =  ( Base `  R
) )
311, 18, 8, 42idlcpbl 16002 . . . . 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 15370 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  y  e.  X  /\  z  e.  X )  ->  (
y ( .r `  R ) z )  e.  X )
33323expb 1152 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
( .r `  R
) z )  e.  X )
3433adantlr 695 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( y ( .r
`  R ) z )  e.  X )
3534caovclg 6028 . . . . 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 13473 . . . 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 1152 . . 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 451 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( R ~QG  S )  Er  X
)
3922a1i 10 . . . . 5  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  ->  X  e.  _V )
4038, 39, 24divsfval 13465 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  y
)  =  [ y ] ( R ~QG  S ) )
4138, 39, 24divsfval 13465 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  z
)  =  [ z ] ( R ~QG  S ) )
4240, 41oveq12d 5892 . . 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 13465 . . 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 2339 . 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 15386 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Abel )
4645adantr 451 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  R  e.  Abel )
47 ablnsg 15155 . . . . 5  |-  ( R  e.  Abel  ->  (NrmSGrp `  R
)  =  (SubGrp `  R ) )
4846, 47syl 15 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  (NrmSGrp `  R )  =  (SubGrp `  R ) )
4917, 48eleqtrrd 2373 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  S  e.  (NrmSGrp `  R )
)
501, 7, 24divsghm 14735 . . 3  |-  ( S  e.  (NrmSGrp `  R
)  ->  F  e.  ( R  GrpHom  U ) )
5149, 50syl 15 . 2  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  F  e.  ( R  GrpHom  U ) )
521, 2, 3, 4, 5, 6, 9, 28, 44, 51isrhm2d 15522 1  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  F  e.  ( R RingHom  U )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    Er wer 6673   [cec 6674   Basecbs 13164   .rcmulr 13225    /.s cqus 13424  SubGrpcsubg 14631  NrmSGrpcnsg 14632   ~QG cqg 14633    GrpHom cghm 14696   Abelcabel 15106   Ringcrg 15353   1rcur 15355  opprcoppr 15420   RingHom crh 15510  LIdealclidl 15939  2Idealc2idl 15999
This theorem is referenced by:  znzrh2  16515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-divs 13428  df-mnd 14383  df-mhm 14431  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-nsg 14635  df-eqg 14636  df-ghm 14697  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-rnghom 15512  df-subrg 15559  df-lmod 15645  df-lss 15706  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-2idl 16000
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