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Theorem divscrng 16311
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 15674 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
21adantr 452 . . 3  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  R  e.  Ring )
3 simpr 448 . . . 4  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  S  e.  I )
4 divscrng.i . . . . . 6  |-  I  =  (LIdeal `  R )
54crng2idl 16310 . . . . 5  |-  ( R  e.  CRing  ->  I  =  (2Ideal `  R ) )
65adantr 452 . . . 4  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  I  =  (2Ideal `  R )
)
73, 6eleqtrd 2512 . . 3  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  S  e.  (2Ideal `  R )
)
8 divscrng.u . . . 4  |-  U  =  ( R  /.s  ( R ~QG  S
) )
9 eqid 2436 . . . 4  |-  (2Ideal `  R )  =  (2Ideal `  R )
108, 9divsrng 16307 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  (2Ideal `  R )
)  ->  U  e.  Ring )
112, 7, 10syl2anc 643 . 2  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  U  e.  Ring )
128a1i 11 . . . . . . 7  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  U  =  ( R  /.s  ( R ~QG  S ) ) )
13 eqidd 2437 . . . . . . 7  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  ( Base `  R )  =  ( Base `  R
) )
14 ovex 6106 . . . . . . . 8  |-  ( R ~QG  S )  e.  _V
1514a1i 11 . . . . . . 7  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  ( R ~QG  S )  e.  _V )
1612, 13, 15, 2divsbas 13770 . . . . . 6  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
( Base `  R ) /. ( R ~QG  S ) )  =  ( Base `  U
) )
1716eleq2d 2503 . . . . 5  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
x  e.  ( (
Base `  R ) /. ( R ~QG  S ) )  <->  x  e.  ( Base `  U )
) )
1816eleq2d 2503 . . . . 5  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
y  e.  ( (
Base `  R ) /. ( R ~QG  S ) )  <->  y  e.  ( Base `  U )
) )
1917, 18anbi12d 692 . . . 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 2436 . . . . . 6  |-  ( (
Base `  R ) /. ( R ~QG  S ) )  =  ( ( Base `  R
) /. ( R ~QG  S ) )
21 oveq2 6089 . . . . . . 7  |-  ( [ u ] ( R ~QG  S )  =  y  -> 
( x ( .r
`  U ) [ u ] ( R ~QG  S ) )  =  ( x ( .r `  U ) y ) )
22 oveq1 6088 . . . . . . 7  |-  ( [ u ] ( R ~QG  S )  =  y  -> 
( [ u ]
( R ~QG  S ) ( .r
`  U ) x )  =  ( y ( .r `  U
) x ) )
2321, 22eqeq12d 2450 . . . . . 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 6088 . . . . . . . . 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 6089 . . . . . . . . 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 2450 . . . . . . . 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 2436 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
28 eqid 2436 . . . . . . . . . . . . 13  |-  ( .r
`  R )  =  ( .r `  R
)
2927, 28crngcom 15678 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  u  e.  ( Base `  R
)  /\  v  e.  ( Base `  R )
)  ->  ( u
( .r `  R
) v )  =  ( v ( .r
`  R ) u ) )
30293adant1r 1177 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  S  e.  I )  /\  u  e.  (
Base `  R )  /\  v  e.  ( Base `  R ) )  ->  ( u ( .r `  R ) v )  =  ( v ( .r `  R ) u ) )
31303expa 1153 . . . . . . . . . 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 6941 . . . . . . . . . 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 16 . . . . . . . . 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 16286 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  S  e.  (SubGrp `  R )
)
351, 34sylan 458 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  S  e.  (SubGrp `  R )
)
36 eqid 2436 . . . . . . . . . . . . 13  |-  ( R ~QG  S )  =  ( R ~QG  S )
3727, 36eqger 14990 . . . . . . . . . . . 12  |-  ( S  e.  (SubGrp `  R
)  ->  ( R ~QG  S
)  Er  ( Base `  R ) )
3835, 37syl 16 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  ( R ~QG  S )  Er  ( Base `  R ) )
3927, 36, 9, 282idlcpbl 16305 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 15677 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  c  e.  ( Base `  R
)  /\  d  e.  ( Base `  R )
)  ->  ( c
( .r `  R
) d )  e.  ( Base `  R
) )
42413expb 1154 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  (
c  e.  ( Base `  R )  /\  d  e.  ( Base `  R
) ) )  -> 
( c ( .r
`  R ) d )  e.  ( Base `  R ) )
432, 42sylan 458 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  S  e.  I )  /\  ( c  e.  ( Base `  R
)  /\  d  e.  ( Base `  R )
) )  ->  (
c ( .r `  R ) d )  e.  ( Base `  R
) )
44 eqid 2436 . . . . . . . . . . 11  |-  ( .r
`  U )  =  ( .r `  U
)
4512, 13, 38, 2, 40, 43, 28, 44divsmulval 13780 . . . . . . . . . 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 1153 . . . . . . . . 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 13780 . . . . . . . . . . 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 1153 . . . . . . . . . 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 780 . . . . . . . . 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 2479 . . . . . . . 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 6971 . . . . . . 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 780 . . . . . 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 6971 . . . . 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 602 . . . 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 227 . . 3  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  (
( x  e.  (
Base `  U )  /\  y  e.  ( Base `  U ) )  ->  ( x ( .r `  U ) y )  =  ( y ( .r `  U ) x ) ) )
5655ralrimivv 2797 . 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 2436 . . 3  |-  ( Base `  U )  =  (
Base `  U )
5857, 44iscrng2 15679 . 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 646 1  |-  ( ( R  e.  CRing  /\  S  e.  I )  ->  U  e.  CRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   class class class wbr 4212   ` cfv 5454  (class class class)co 6081    Er wer 6902   [cec 6903   /.cqs 6904   Basecbs 13469   .rcmulr 13530    /.s cqus 13731  SubGrpcsubg 14938   ~QG cqg 14940   Ringcrg 15660   CRingccrg 15661  LIdealclidl 16242  2Idealc2idl 16302
This theorem is referenced by:  zncrng2  16815
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-ec 6907  df-qs 6911  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-0g 13727  df-imas 13734  df-divs 13735  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-nsg 14942  df-eqg 14943  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-sra 16244  df-rgmod 16245  df-lidl 16246  df-rsp 16247  df-2idl 16303
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