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Theorem divsrng2 16647
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 2433 . . . 4  |-  ( u  e.  V  |->  [ u ]  .~  )  =  ( u  e.  V  |->  [ u ]  .~  )
4 divsrng2.r . . . . 5  |-  ( ph  ->  .~  Er  V )
5 fvex 5689 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2521 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 7113 . . . . 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 14463 . . 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 14464 . . 3  |-  ( ph  ->  ( u  e.  V  |->  [ u ]  .~  ) : V -onto-> ( V /.  .~  ) )
159adantr 462 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  R  e.  Ring )
16 simprl 748 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  x  e.  V )
172adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  V  =  ( Base `  R ) )
1816, 17eleqtrd 2509 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  x  e.  ( Base `  R ) )
19 simprr 749 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
y  e.  V )
2019, 17eleqtrd 2509 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
y  e.  ( Base `  R ) )
21 eqid 2433 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
2221, 11rngacl 16608 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x  .+  y )  e.  (
Base `  R )
)
2315, 18, 20, 22syl3anc 1211 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .+  y
)  e.  ( Base `  R ) )
2423, 17eleqtrrd 2510 . . . 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 14470 . . 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 16594 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x  .x.  y )  e.  (
Base `  R )
)
2815, 18, 20, 27syl3anc 1211 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .x.  y
)  e.  ( Base `  R ) )
2928, 17eleqtrrd 2510 . . . 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 14470 . . 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 16646 . 2  |-  ( ph  ->  ( U  e.  Ring  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  ( 1r `  U ) ) )
334, 6, 3divsfval 14468 . . . . 5  |-  ( ph  ->  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  [  .1.  ]  .~  )
3433eqcomd 2438 . . . 4  |-  ( ph  ->  [  .1.  ]  .~  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  ) )
3534eqeq1d 2441 . . 3  |-  ( ph  ->  ( [  .1.  ]  .~  =  ( 1r `  U )  <->  ( (
u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  ( 1r `  U ) ) )
3635anbi2d 696 . 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 1362    e. wcel 1755   _Vcvv 2962   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080    Er wer 7086   [cec 7087   /.cqs 7088   Basecbs 14157   +g cplusg 14221   .rcmulr 14222    /.s cqus 14426   Ringcrg 16577   1rcur 16579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-ec 7091  df-qs 7095  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-fz 11425  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-0g 14363  df-imas 14429  df-divs 14430  df-mnd 15398  df-grp 15525  df-minusg 15526  df-mgp 16566  df-rng 16580  df-ur 16582
This theorem is referenced by:  divs1  17239
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