MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divsgrp2 Structured version   Unicode version

Theorem divsgrp2 15666
Description: Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
divsgrp2.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
divsgrp2.v  |-  ( ph  ->  V  =  ( Base `  R ) )
divsgrp2.p  |-  ( ph  ->  .+  =  ( +g  `  R ) )
divsgrp2.r  |-  ( ph  ->  .~  Er  V )
divsgrp2.x  |-  ( ph  ->  R  e.  X )
divsgrp2.e  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .+  b )  .~  (
p  .+  q )
) )
divsgrp2.1  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
divsgrp2.2  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  .~  ( x  .+  ( y  .+  z
) ) )
divsgrp2.3  |-  ( ph  ->  .0.  e.  V )
divsgrp2.4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  .~  x )
divsgrp2.5  |-  ( (
ph  /\  x  e.  V )  ->  N  e.  V )
divsgrp2.6  |-  ( (
ph  /\  x  e.  V )  ->  ( N  .+  x )  .~  .0.  )
Assertion
Ref Expression
divsgrp2  |-  ( ph  ->  ( U  e.  Grp  /\ 
[  .0.  ]  .~  =  ( 0g `  U ) ) )
Distinct variable groups:    a, b, p, q, x, y, z, 
.~    .0. , a, b, p, q, x    N, p    R, p, q    .+ , a,
b, p, q, x, y    ph, a, b, p, q, x, y, z    V, a, b, p, q, x, y, z    U, a, b, p, q, x, y, z
Allowed substitution hints:    .+ ( z)    R( x, y, z, a, b)    N( x, y, z, q, a, b)    X( x, y, z, q, p, a, b)    .0. ( y,
z)

Proof of Theorem divsgrp2
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 divsgrp2.u . . . 4  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
2 divsgrp2.v . . . 4  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2441 . . . 4  |-  ( u  e.  V  |->  [ u ]  .~  )  =  ( u  e.  V  |->  [ u ]  .~  )
4 divsgrp2.r . . . . 5  |-  ( ph  ->  .~  Er  V )
5 fvex 5698 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2529 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 7121 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 60 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 divsgrp2.x . . . 4  |-  ( ph  ->  R  e.  X )
101, 2, 3, 8, 9divsval 14476 . . 3  |-  ( ph  ->  U  =  ( ( u  e.  V  |->  [ u ]  .~  )  "s  R ) )
11 divsgrp2.p . . 3  |-  ( ph  ->  .+  =  ( +g  `  R ) )
121, 2, 3, 8, 9divslem 14477 . . 3  |-  ( ph  ->  ( u  e.  V  |->  [ u ]  .~  ) : V -onto-> ( V /.  .~  ) )
13 divsgrp2.1 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
14133expb 1183 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .+  y
)  e.  V )
15 divsgrp2.e . . . 4  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .+  b )  .~  (
p  .+  q )
) )
164, 6, 3, 14, 15ercpbl 14483 . . 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 ) ) ) )
174adantr 462 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  .~  Er  V )
18 divsgrp2.2 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  .~  ( x  .+  ( y  .+  z
) ) )
1917, 18erthi 7143 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  [ ( ( x 
.+  y )  .+  z ) ]  .~  =  [ ( x  .+  ( y  .+  z
) ) ]  .~  )
206adantr 462 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  e.  _V )
2117, 20, 3divsfval 14481 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( ( x  .+  y ) 
.+  z ) )  =  [ ( ( x  .+  y ) 
.+  z ) ]  .~  )
2217, 20, 3divsfval 14481 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( x 
.+  ( y  .+  z ) ) )  =  [ ( x 
.+  ( y  .+  z ) ) ]  .~  )
2319, 21, 223eqtr4d 2483 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( ( x  .+  y ) 
.+  z ) )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  (
x  .+  ( y  .+  z ) ) ) )
24 divsgrp2.3 . . 3  |-  ( ph  ->  .0.  e.  V )
254adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .~  Er  V )
26 divsgrp2.4 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  .~  x )
2725, 26erthi 7143 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ (  .0.  .+  x ) ]  .~  =  [ x ]  .~  )
286adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  V  e.  _V )
2925, 28, 3divsfval 14481 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  (  .0.  .+  x ) )  =  [ (  .0.  .+  x ) ]  .~  )
3025, 28, 3divsfval 14481 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  x )  =  [ x ]  .~  )
3127, 29, 303eqtr4d 2483 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  (  .0.  .+  x ) )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  x ) )
32 divsgrp2.5 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  N  e.  V )
33 divsgrp2.6 . . . . . 6  |-  ( (
ph  /\  x  e.  V )  ->  ( N  .+  x )  .~  .0.  )
3425, 33ersym 7109 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .0.  .~  ( N  .+  x
) )
3525, 34erthi 7143 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [  .0.  ]  .~  =  [ ( N  .+  x ) ]  .~  )
3625, 28, 3divsfval 14481 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
3725, 28, 3divsfval 14481 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  ( N  .+  x ) )  =  [ ( N  .+  x ) ]  .~  )
3835, 36, 373eqtr4rd 2484 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  ( N  .+  x ) )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )
)
3910, 2, 11, 12, 16, 9, 13, 23, 24, 31, 32, 38imasgrp2 15663 . 2  |-  ( ph  ->  ( U  e.  Grp  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) )
404, 6, 3divsfval 14481 . . . . 5  |-  ( ph  ->  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
4140eqcomd 2446 . . . 4  |-  ( ph  ->  [  .0.  ]  .~  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  ) )
4241eqeq1d 2449 . . 3  |-  ( ph  ->  ( [  .0.  ]  .~  =  ( 0g `  U )  <->  ( (
u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) )
4342anbi2d 698 . 2  |-  ( ph  ->  ( ( U  e. 
Grp  /\  [  .0.  ]  .~  =  ( 0g
`  U ) )  <-> 
( U  e.  Grp  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) ) )
4439, 43mpbird 232 1  |-  ( ph  ->  ( U  e.  Grp  /\ 
[  .0.  ]  .~  =  ( 0g `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090    Er wer 7094   [cec 7095   /.cqs 7096   Basecbs 14170   +g cplusg 14234   0gc0g 14374    /.s cqus 14439   Grpcgrp 15406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-0g 14376  df-imas 14442  df-divs 14443  df-mnd 15411  df-grp 15538
This theorem is referenced by:  divsgrp  15729  frgp0  16250  pi1grplem  20580
  Copyright terms: Public domain W3C validator