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Theorem divsrhm 15989
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 2283 . 2  |-  ( 1r
`  R )  =  ( 1r `  R
)
3 eqid 2283 . 2  |-  ( 1r
`  U )  =  ( 1r `  U
)
4 eqid 2283 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2283 . 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 15988 . 2  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  U  e.  Ring )
10 eqid 2283 . . . . . . . . 9  |-  (LIdeal `  R )  =  (LIdeal `  R )
11 eqid 2283 . . . . . . . . 9  |-  (oppr `  R
)  =  (oppr `  R
)
12 eqid 2283 . . . . . . . . 9  |-  (LIdeal `  (oppr `  R ) )  =  (LIdeal `  (oppr
`  R ) )
1310, 11, 12, 82idlval 15985 . . . . . . . 8  |-  I  =  ( (LIdeal `  R
)  i^i  (LIdeal `  (oppr `  R
) ) )
1413elin2 3359 . . . . . . 7  |-  ( S  e.  I  <->  ( S  e.  (LIdeal `  R )  /\  S  e.  (LIdeal `  (oppr
`  R ) ) ) )
1514simplbi 446 . . . . . 6  |-  ( S  e.  I  ->  S  e.  (LIdeal `  R )
)
1610lidlsubg 15967 . . . . . 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 2283 . . . . . 6  |-  ( R ~QG  S )  =  ( R ~QG  S )
191, 18eqger 14667 . . . . 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 5539 . . . . . 6  |-  ( Base `  R )  e.  _V
221, 21eqeltri 2353 . . . . 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 13449 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( F `  ( 1r `  R ) )  =  [ ( 1r `  R ) ] ( R ~QG  S ) )
267, 8, 2divs1 15987 . . . 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 2315 . 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 15986 . . . . 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 15354 . . . . . . . 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 6012 . . . . 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 13457 . . . 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 13449 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  y
)  =  [ y ] ( R ~QG  S ) )
4138, 39, 24divsfval 13449 . . . 4  |-  ( ( ( R  e.  Ring  /\  S  e.  I )  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  z
)  =  [ z ] ( R ~QG  S ) )
4240, 41oveq12d 5876 . . 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 13449 . . 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 2326 . 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 15370 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Abel )
4645adantr 451 . . . . 5  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  R  e.  Abel )
47 ablnsg 15139 . . . . 5  |-  ( R  e.  Abel  ->  (NrmSGrp `  R
)  =  (SubGrp `  R ) )
4846, 47syl 15 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  (NrmSGrp `  R )  =  (SubGrp `  R ) )
4917, 48eleqtrrd 2360 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  S  e.  (NrmSGrp `  R )
)
501, 7, 24divsghm 14719 . . 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 15506 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 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    Er wer 6657   [cec 6658   Basecbs 13148   .rcmulr 13209    /.s cqus 13408  SubGrpcsubg 14615  NrmSGrpcnsg 14616   ~QG cqg 14617    GrpHom cghm 14680   Abelcabel 15090   Ringcrg 15337   1rcur 15339  opprcoppr 15404   RingHom crh 15494  LIdealclidl 15923  2Idealc2idl 15983
This theorem is referenced by:  znzrh2  16499
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-int 3863  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-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  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-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  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-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-rnghom 15496  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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