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Theorem divsrng2 15728
Description: The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
divsrng2.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
divsrng2.v  |-  ( ph  ->  V  =  ( Base `  R ) )
divsrng2.p  |-  .+  =  ( +g  `  R )
divsrng2.t  |-  .x.  =  ( .r `  R )
divsrng2.o  |-  .1.  =  ( 1r `  R )
divsrng2.r  |-  ( ph  ->  .~  Er  V )
divsrng2.e1  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .+  b )  .~  (
p  .+  q )
) )
divsrng2.e2  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .x.  b )  .~  (
p  .x.  q )
) )
divsrng2.x  |-  ( ph  ->  R  e.  Ring )
Assertion
Ref Expression
divsrng2  |-  ( ph  ->  ( U  e.  Ring  /\ 
[  .1.  ]  .~  =  ( 1r `  U ) ) )
Distinct variable groups:    q, p,  .+    .1. , p, q    a, b, p, q, U    V, a, b, p, q    .~ , a, b, p, q    ph, a,
b, p, q    .x. , p, q    R, p, q
Allowed substitution hints:    .+ ( a, b)    R( a, b)    .x. ( a, b)    .1. ( a, b)

Proof of Theorem divsrng2
Dummy variables  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 divsrng2.u . . . 4  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
2 divsrng2.v . . . 4  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2438 . . . 4  |-  ( u  e.  V  |->  [ u ]  .~  )  =  ( u  e.  V  |->  [ u ]  .~  )
4 divsrng2.r . . . . 5  |-  ( ph  ->  .~  Er  V )
5 fvex 5744 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2526 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 6931 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 59 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 divsrng2.x . . . 4  |-  ( ph  ->  R  e.  Ring )
101, 2, 3, 8, 9divsval 13769 . . 3  |-  ( ph  ->  U  =  ( ( u  e.  V  |->  [ u ]  .~  )  "s  R ) )
11 divsrng2.p . . 3  |-  .+  =  ( +g  `  R )
12 divsrng2.t . . 3  |-  .x.  =  ( .r `  R )
13 divsrng2.o . . 3  |-  .1.  =  ( 1r `  R )
141, 2, 3, 8, 9divslem 13770 . . 3  |-  ( ph  ->  ( u  e.  V  |->  [ u ]  .~  ) : V -onto-> ( V /.  .~  ) )
159adantr 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  R  e.  Ring )
16 simprl 734 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  x  e.  V )
172adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  V  =  ( Base `  R ) )
1816, 17eleqtrd 2514 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  x  e.  ( Base `  R ) )
19 simprr 735 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
y  e.  V )
2019, 17eleqtrd 2514 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
y  e.  ( Base `  R ) )
21 eqid 2438 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
2221, 11rngacl 15693 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x  .+  y )  e.  (
Base `  R )
)
2315, 18, 20, 22syl3anc 1185 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .+  y
)  e.  ( Base `  R ) )
2423, 17eleqtrrd 2515 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .+  y
)  e.  V )
25 divsrng2.e1 . . . 4  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .+  b )  .~  (
p  .+  q )
) )
264, 6, 3, 24, 25ercpbl 13776 . . 3  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( ( u  e.  V  |->  [ u ]  .~  ) `  a )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  p
)  /\  ( (
u  e.  V  |->  [ u ]  .~  ) `  b )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  q )
)  ->  ( (
u  e.  V  |->  [ u ]  .~  ) `  ( a  .+  b
) )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  ( p  .+  q ) ) ) )
2721, 12rngcl 15679 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x  .x.  y )  e.  (
Base `  R )
)
2815, 18, 20, 27syl3anc 1185 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .x.  y
)  e.  ( Base `  R ) )
2928, 17eleqtrrd 2515 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .x.  y
)  e.  V )
30 divsrng2.e2 . . . 4  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .x.  b )  .~  (
p  .x.  q )
) )
314, 6, 3, 29, 30ercpbl 13776 . . 3  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( ( u  e.  V  |->  [ u ]  .~  ) `  a )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  p
)  /\  ( (
u  e.  V  |->  [ u ]  .~  ) `  b )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  q )
)  ->  ( (
u  e.  V  |->  [ u ]  .~  ) `  ( a  .x.  b
) )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  ( p  .x.  q ) ) ) )
3210, 2, 11, 12, 13, 14, 26, 31, 9imasrng 15727 . 2  |-  ( ph  ->  ( U  e.  Ring  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  ( 1r `  U ) ) )
334, 6, 3divsfval 13774 . . . . 5  |-  ( ph  ->  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  [  .1.  ]  .~  )
3433eqcomd 2443 . . . 4  |-  ( ph  ->  [  .1.  ]  .~  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  ) )
3534eqeq1d 2446 . . 3  |-  ( ph  ->  ( [  .1.  ]  .~  =  ( 1r `  U )  <->  ( (
u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  ( 1r `  U ) ) )
3635anbi2d 686 . 2  |-  ( ph  ->  ( ( U  e. 
Ring  /\  [  .1.  ]  .~  =  ( 1r `  U ) )  <->  ( U  e.  Ring  /\  ( (
u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  ( 1r `  U ) ) ) )
3732, 36mpbird 225 1  |-  ( ph  ->  ( U  e.  Ring  /\ 
[  .1.  ]  .~  =  ( 1r `  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   class class class wbr 4214    e. cmpt 4268   ` cfv 5456  (class class class)co 6083    Er wer 6904   [cec 6905   /.cqs 6906   Basecbs 13471   +g cplusg 13531   .rcmulr 13532    /.s cqus 13733   Ringcrg 15662   1rcur 15664
This theorem is referenced by:  divs1  16308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-ec 6909  df-qs 6913  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-0g 13729  df-imas 13736  df-divs 13737  df-mnd 14692  df-grp 14814  df-minusg 14815  df-mgp 15651  df-rng 15665  df-ur 15667
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