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Theorem divsrng2 16702
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 5696 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2526 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 7117 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 60 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 divsrng2.x . . . 4  |-  ( ph  ->  R  e.  Ring )
101, 2, 3, 8, 9divsval 14472 . . 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 14473 . . 3  |-  ( ph  ->  ( u  e.  V  |->  [ u ]  .~  ) : V -onto-> ( V /.  .~  ) )
159adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  R  e.  Ring )
16 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  x  e.  V )
172adantr 465 . . . . . . 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 756 . . . . . . 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 16662 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x  .+  y )  e.  (
Base `  R )
)
2315, 18, 20, 22syl3anc 1218 . . . . 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 14479 . . 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 16648 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x  .x.  y )  e.  (
Base `  R )
)
2815, 18, 20, 27syl3anc 1218 . . . . 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 14479 . . 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 16701 . 2  |-  ( ph  ->  ( U  e.  Ring  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  ( 1r `  U ) ) )
334, 6, 3divsfval 14477 . . . . 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 703 . 2  |-  ( ph  ->  ( ( U  e. 
Ring  /\  [  .1.  ]  .~  =  ( 1r `  U ) )  <->  ( U  e.  Ring  /\  ( (
u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  ( 1r `  U ) ) ) )
3732, 36mpbird 232 1  |-  ( ph  ->  ( U  e.  Ring  /\ 
[  .1.  ]  .~  =  ( 1r `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   class class class wbr 4287    e. cmpt 4345   ` cfv 5413  (class class class)co 6086    Er wer 7090   [cec 7091   /.cqs 7092   Basecbs 14166   +g cplusg 14230   .rcmulr 14231    /.s cqus 14435   1rcur 16593   Ringcrg 16635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-ec 7095  df-qs 7099  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-0g 14372  df-imas 14438  df-divs 14439  df-mnd 15407  df-grp 15536  df-minusg 15537  df-mgp 16582  df-ur 16594  df-rng 16637
This theorem is referenced by:  divs1  17297
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