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Theorem divscrng 15992
Description: The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
Hypotheses
Ref Expression
divscrng.u  |-  U  =  ( R  /.s  ( R ~QG  S
) )
divscrng.i  |-  I  =  (LIdeal `  R )
Assertion
Ref Expression
divscrng  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  U  e.  CRing )

Proof of Theorem divscrng
Dummy variables  a 
b  c  d  u  v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngrng 15351 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
21adantr 451 . . 3  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  R  e.  Ring )
3 simpr 447 . . . 4  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  S  e.  I )
4 divscrng.i . . . . . 6  |-  I  =  (LIdeal `  R )
54crng2idl 15991 . . . . 5  |-  ( R  e.  CRing  ->  I  =  (2Ideal `  R ) )
65adantr 451 . . . 4  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  I  =  (2Ideal `  R )
)
73, 6eleqtrd 2359 . . 3  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  S  e.  (2Ideal `  R )
)
8 divscrng.u . . . 4  |-  U  =  ( R  /.s  ( R ~QG  S
) )
9 eqid 2283 . . . 4  |-  (2Ideal `  R )  =  (2Ideal `  R )
108, 9divsrng 15988 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  (2Ideal `  R )
)  ->  U  e.  Ring )
112, 7, 10syl2anc 642 . 2  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  U  e.  Ring )
128a1i 10 . . . . . . 7  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  U  =  ( R  /.s  ( R ~QG  S ) ) )
13 eqidd 2284 . . . . . . 7  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  ( Base `  R )  =  ( Base `  R
) )
14 ovex 5883 . . . . . . . 8  |-  ( R ~QG  S )  e.  _V
1514a1i 10 . . . . . . 7  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  ( R ~QG  S )  e.  _V )
1612, 13, 15, 2divsbas 13447 . . . . . 6  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
( Base `  R ) /. ( R ~QG  S ) )  =  ( Base `  U
) )
1716eleq2d 2350 . . . . 5  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
x  e.  ( (
Base `  R ) /. ( R ~QG  S ) )  <->  x  e.  ( Base `  U )
) )
1816eleq2d 2350 . . . . 5  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
y  e.  ( (
Base `  R ) /. ( R ~QG  S ) )  <->  y  e.  ( Base `  U )
) )
1917, 18anbi12d 691 . . . 4  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
( x  e.  ( ( Base `  R
) /. ( R ~QG  S ) )  /\  y  e.  ( ( Base `  R
) /. ( R ~QG  S ) ) )  <->  ( x  e.  ( Base `  U
)  /\  y  e.  ( Base `  U )
) ) )
20 eqid 2283 . . . . . 6  |-  ( (
Base `  R ) /. ( R ~QG  S ) )  =  ( ( Base `  R
) /. ( R ~QG  S ) )
21 oveq2 5866 . . . . . . 7  |-  ( [ u ] ( R ~QG  S )  =  y  -> 
( x ( .r
`  U ) [ u ] ( R ~QG  S ) )  =  ( x ( .r `  U ) y ) )
22 oveq1 5865 . . . . . . 7  |-  ( [ u ] ( R ~QG  S )  =  y  -> 
( [ u ]
( R ~QG  S ) ( .r
`  U ) x )  =  ( y ( .r `  U
) x ) )
2321, 22eqeq12d 2297 . . . . . 6  |-  ( [ u ] ( R ~QG  S )  =  y  -> 
( ( x ( .r `  U ) [ u ] ( R ~QG  S ) )  =  ( [ u ]
( R ~QG  S ) ( .r
`  U ) x )  <->  ( x ( .r `  U ) y )  =  ( y ( .r `  U ) x ) ) )
24 oveq1 5865 . . . . . . . . 9  |-  ( [ v ] ( R ~QG  S )  =  x  -> 
( [ v ] ( R ~QG  S ) ( .r
`  U ) [ u ] ( R ~QG  S ) )  =  ( x ( .r `  U ) [ u ] ( R ~QG  S ) ) )
25 oveq2 5866 . . . . . . . . 9  |-  ( [ v ] ( R ~QG  S )  =  x  -> 
( [ u ]
( R ~QG  S ) ( .r
`  U ) [ v ] ( R ~QG  S ) )  =  ( [ u ] ( R ~QG  S ) ( .r
`  U ) x ) )
2624, 25eqeq12d 2297 . . . . . . . 8  |-  ( [ v ] ( R ~QG  S )  =  x  -> 
( ( [ v ] ( R ~QG  S ) ( .r `  U
) [ u ]
( R ~QG  S ) )  =  ( [ u ]
( R ~QG  S ) ( .r
`  U ) [ v ] ( R ~QG  S ) )  <->  ( x
( .r `  U
) [ u ]
( R ~QG  S ) )  =  ( [ u ]
( R ~QG  S ) ( .r
`  U ) x ) ) )
27 eqid 2283 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
28 eqid 2283 . . . . . . . . . . . . 13  |-  ( .r
`  R )  =  ( .r `  R
)
2927, 28crngcom 15355 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  u  e.  ( Base `  R
)  /\  v  e.  ( Base `  R )
)  ->  ( u
( .r `  R
) v )  =  ( v ( .r
`  R ) u ) )
30293adant1r 1175 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  S  e.  I )  /\  u  e.  (
Base `  R )  /\  v  e.  ( Base `  R ) )  ->  ( u ( .r `  R ) v )  =  ( v ( .r `  R ) u ) )
31303expa 1151 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  S  e.  I
)  /\  u  e.  ( Base `  R )
)  /\  v  e.  ( Base `  R )
)  ->  ( u
( .r `  R
) v )  =  ( v ( .r
`  R ) u ) )
32 eceq1 6696 . . . . . . . . . 10  |-  ( ( u ( .r `  R ) v )  =  ( v ( .r `  R ) u )  ->  [ ( u ( .r `  R ) v ) ] ( R ~QG  S )  =  [ ( v ( .r `  R
) u ) ] ( R ~QG  S ) )
3331, 32syl 15 . . . . . . . . 9  |-  ( ( ( ( R  e. 
CRing  /\  S  e.  I
)  /\  u  e.  ( Base `  R )
)  /\  v  e.  ( Base `  R )
)  ->  [ (
u ( .r `  R ) v ) ] ( R ~QG  S )  =  [ ( v ( .r `  R
) u ) ] ( R ~QG  S ) )
344lidlsubg 15967 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  S  e.  (SubGrp `  R )
)
351, 34sylan 457 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  S  e.  (SubGrp `  R )
)
36 eqid 2283 . . . . . . . . . . . . 13  |-  ( R ~QG  S )  =  ( R ~QG  S )
3727, 36eqger 14667 . . . . . . . . . . . 12  |-  ( S  e.  (SubGrp `  R
)  ->  ( R ~QG  S
)  Er  ( Base `  R ) )
3835, 37syl 15 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  ( R ~QG  S )  Er  ( Base `  R ) )
3927, 36, 9, 282idlcpbl 15986 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  S  e.  (2Ideal `  R )
)  ->  ( (
a ( R ~QG  S ) c  /\  b ( R ~QG  S ) d )  ->  ( a ( .r `  R ) b ) ( R ~QG  S ) ( c ( .r `  R ) d ) ) )
402, 7, 39syl2anc 642 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
( a ( R ~QG  S ) c  /\  b
( R ~QG  S ) d )  ->  ( a ( .r `  R ) b ) ( R ~QG  S ) ( c ( .r `  R ) d ) ) )
4127, 28rngcl 15354 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  c  e.  ( Base `  R
)  /\  d  e.  ( Base `  R )
)  ->  ( c
( .r `  R
) d )  e.  ( Base `  R
) )
42413expb 1152 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  (
c  e.  ( Base `  R )  /\  d  e.  ( Base `  R
) ) )  -> 
( c ( .r
`  R ) d )  e.  ( Base `  R ) )
432, 42sylan 457 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  S  e.  I )  /\  ( c  e.  ( Base `  R
)  /\  d  e.  ( Base `  R )
) )  ->  (
c ( .r `  R ) d )  e.  ( Base `  R
) )
44 eqid 2283 . . . . . . . . . . 11  |-  ( .r
`  U )  =  ( .r `  U
)
4512, 13, 38, 2, 40, 43, 28, 44divsmulval 13457 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  S  e.  I )  /\  u  e.  (
Base `  R )  /\  v  e.  ( Base `  R ) )  ->  ( [ u ] ( R ~QG  S ) ( .r `  U
) [ v ] ( R ~QG  S ) )  =  [ ( u ( .r `  R ) v ) ] ( R ~QG  S ) )
46453expa 1151 . . . . . . . . 9  |-  ( ( ( ( R  e. 
CRing  /\  S  e.  I
)  /\  u  e.  ( Base `  R )
)  /\  v  e.  ( Base `  R )
)  ->  ( [
u ] ( R ~QG  S ) ( .r `  U ) [ v ] ( R ~QG  S ) )  =  [ ( u ( .r `  R ) v ) ] ( R ~QG  S ) )
4712, 13, 38, 2, 40, 43, 28, 44divsmulval 13457 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  S  e.  I )  /\  v  e.  (
Base `  R )  /\  u  e.  ( Base `  R ) )  ->  ( [ v ] ( R ~QG  S ) ( .r `  U
) [ u ]
( R ~QG  S ) )  =  [ ( v ( .r `  R ) u ) ] ( R ~QG  S ) )
48473expa 1151 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  S  e.  I
)  /\  v  e.  ( Base `  R )
)  /\  u  e.  ( Base `  R )
)  ->  ( [
v ] ( R ~QG  S ) ( .r `  U ) [ u ] ( R ~QG  S ) )  =  [ ( v ( .r `  R ) u ) ] ( R ~QG  S ) )
4948an32s 779 . . . . . . . . 9  |-  ( ( ( ( R  e. 
CRing  /\  S  e.  I
)  /\  u  e.  ( Base `  R )
)  /\  v  e.  ( Base `  R )
)  ->  ( [
v ] ( R ~QG  S ) ( .r `  U ) [ u ] ( R ~QG  S ) )  =  [ ( v ( .r `  R ) u ) ] ( R ~QG  S ) )
5033, 46, 493eqtr4rd 2326 . . . . . . . 8  |-  ( ( ( ( R  e. 
CRing  /\  S  e.  I
)  /\  u  e.  ( Base `  R )
)  /\  v  e.  ( Base `  R )
)  ->  ( [
v ] ( R ~QG  S ) ( .r `  U ) [ u ] ( R ~QG  S ) )  =  ( [ u ] ( R ~QG  S ) ( .r `  U ) [ v ] ( R ~QG  S ) ) )
5120, 26, 50ectocld 6726 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  S  e.  I
)  /\  u  e.  ( Base `  R )
)  /\  x  e.  ( ( Base `  R
) /. ( R ~QG  S ) ) )  -> 
( x ( .r
`  U ) [ u ] ( R ~QG  S ) )  =  ( [ u ] ( R ~QG  S ) ( .r
`  U ) x ) )
5251an32s 779 . . . . . 6  |-  ( ( ( ( R  e. 
CRing  /\  S  e.  I
)  /\  x  e.  ( ( Base `  R
) /. ( R ~QG  S ) ) )  /\  u  e.  ( Base `  R ) )  -> 
( x ( .r
`  U ) [ u ] ( R ~QG  S ) )  =  ( [ u ] ( R ~QG  S ) ( .r
`  U ) x ) )
5320, 23, 52ectocld 6726 . . . . 5  |-  ( ( ( ( R  e. 
CRing  /\  S  e.  I
)  /\  x  e.  ( ( Base `  R
) /. ( R ~QG  S ) ) )  /\  y  e.  ( ( Base `  R ) /. ( R ~QG  S ) ) )  ->  ( x ( .r `  U ) y )  =  ( y ( .r `  U ) x ) )
5453expl 601 . . . 4  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
( x  e.  ( ( Base `  R
) /. ( R ~QG  S ) )  /\  y  e.  ( ( Base `  R
) /. ( R ~QG  S ) ) )  -> 
( x ( .r
`  U ) y )  =  ( y ( .r `  U
) x ) ) )
5519, 54sylbird 226 . . 3  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
( x  e.  (
Base `  U )  /\  y  e.  ( Base `  U ) )  ->  ( x ( .r `  U ) y )  =  ( y ( .r `  U ) x ) ) )
5655ralrimivv 2634 . 2  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  A. x  e.  ( Base `  U
) A. y  e.  ( Base `  U
) ( x ( .r `  U ) y )  =  ( y ( .r `  U ) x ) )
57 eqid 2283 . . 3  |-  ( Base `  U )  =  (
Base `  U )
5857, 44iscrng2 15356 . 2  |-  ( U  e.  CRing 
<->  ( U  e.  Ring  /\ 
A. x  e.  (
Base `  U ) A. y  e.  ( Base `  U ) ( x ( .r `  U ) y )  =  ( y ( .r `  U ) x ) ) )
5911, 56, 58sylanbrc 645 1  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  U  e.  CRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    Er wer 6657   [cec 6658   /.cqs 6659   Basecbs 13148   .rcmulr 13209    /.s cqus 13408  SubGrpcsubg 14615   ~QG cqg 14617   Ringcrg 15337   CRingccrg 15338  LIdealclidl 15923  2Idealc2idl 15983
This theorem is referenced by:  zncrng2  16488
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-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-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619  df-eqg 14620  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984
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