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Theorem qusring2 17464
Description: The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
qusring2.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
qusring2.v  |-  ( ph  ->  V  =  ( Base `  R ) )
qusring2.p  |-  .+  =  ( +g  `  R )
qusring2.t  |-  .x.  =  ( .r `  R )
qusring2.o  |-  .1.  =  ( 1r `  R )
qusring2.r  |-  ( ph  ->  .~  Er  V )
qusring2.e1  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .+  b )  .~  (
p  .+  q )
) )
qusring2.e2  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .x.  b )  .~  (
p  .x.  q )
) )
qusring2.x  |-  ( ph  ->  R  e.  Ring )
Assertion
Ref Expression
qusring2  |-  ( 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 qusring2
Dummy variables  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qusring2.u . . . 4  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
2 qusring2.v . . . 4  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2454 . . . 4  |-  ( u  e.  V  |->  [ u ]  .~  )  =  ( u  e.  V  |->  [ u ]  .~  )
4 qusring2.r . . . . 5  |-  ( ph  ->  .~  Er  V )
5 fvex 5858 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2550 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 7327 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 60 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 qusring2.x . . . 4  |-  ( ph  ->  R  e.  Ring )
101, 2, 3, 8, 9qusval 15031 . . 3  |-  ( ph  ->  U  =  ( ( u  e.  V  |->  [ u ]  .~  )  "s  R ) )
11 qusring2.p . . 3  |-  .+  =  ( +g  `  R )
12 qusring2.t . . 3  |-  .x.  =  ( .r `  R )
13 qusring2.o . . 3  |-  .1.  =  ( 1r `  R )
141, 2, 3, 8, 9quslem 15032 . . 3  |-  ( ph  ->  ( u  e.  V  |->  [ u ]  .~  ) : V -onto-> ( V /.  .~  ) )
159adantr 463 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  R  e.  Ring )
16 simprl 754 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  x  e.  V )
172adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  V  =  ( Base `  R ) )
1816, 17eleqtrd 2544 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  ->  x  e.  ( Base `  R ) )
19 simprr 755 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
y  e.  V )
2019, 17eleqtrd 2544 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
y  e.  ( Base `  R ) )
21 eqid 2454 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
2221, 11ringacl 17421 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x  .+  y )  e.  (
Base `  R )
)
2315, 18, 20, 22syl3anc 1226 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .+  y
)  e.  ( Base `  R ) )
2423, 17eleqtrrd 2545 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .+  y
)  e.  V )
25 qusring2.e1 . . . 4  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .+  b )  .~  (
p  .+  q )
) )
264, 6, 3, 24, 25ercpbl 15038 . . 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, 12ringcl 17407 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x  .x.  y )  e.  (
Base `  R )
)
2815, 18, 20, 27syl3anc 1226 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .x.  y
)  e.  ( Base `  R ) )
2928, 17eleqtrrd 2545 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .x.  y
)  e.  V )
30 qusring2.e2 . . . 4  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .x.  b )  .~  (
p  .x.  q )
) )
314, 6, 3, 29, 30ercpbl 15038 . . 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, 9imasring 17463 . 2  |-  ( ph  ->  ( U  e.  Ring  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  ( 1r `  U ) ) )
334, 6, 3divsfval 15036 . . . . 5  |-  ( ph  ->  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  [  .1.  ]  .~  )
3433eqcomd 2462 . . . 4  |-  ( ph  ->  [  .1.  ]  .~  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .1.  ) )
3534eqeq1d 2456 . . 3  |-  ( ph  ->  ( [  .1.  ]  .~  =  ( 1r `  U )  <->  ( (
u  e.  V  |->  [ u ]  .~  ) `  .1.  )  =  ( 1r `  U ) ) )
3635anbi2d 701 . 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270    Er wer 7300   [cec 7301   /.cqs 7302   Basecbs 14716   +g cplusg 14784   .rcmulr 14785    /.s cqus 14994   1rcur 17348   Ringcrg 17393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-ec 7305  df-qs 7309  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-0g 14931  df-imas 14997  df-qus 14998  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-mgp 17337  df-ur 17349  df-ring 17395
This theorem is referenced by:  qus1  18078
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